All language subtitles for College-Algebra-Full-Course-with-Python-Code_en

This college algebra course is different than other college

ones, you will be learning all the concepts from an experienced

But in this course, the instructor Ed Protowski will also show you

concepts in Python. This course is for everybody, but especially

science. Hello and welcome to the Algebra with Python course. My

guide on this adventure. So we're going to learn about algebra and

all of your algebra. Because there's so many different things that

You learn these formulas. Hey, anytime you have a formula, you can

So that's what we're going to do. We're going to write some code

That's Python, a Jupyter notebook, essentially in your Google

we'll do is look at how to set that up. We're going to go through

look at the math. We're going to look in the actual lessons. I'll

and we'll look at the concepts, how to do the math on the board.

then how to write the code to do this. And you'll be able to

In fact, I hope you do create your own notebook or two along the

resources that you can use. You can also, you know, once we set it

download the CoLab, the Google CoLab app, and you have access to

your phone. So, you know, you can make your own like, you know,

access to on your phone, you know, solve all kinds of things, you

You know what, you know, building these resources, but also

you go further in math and in writing code and eventually leading

data science, that you really understand these concepts and you

that you're creating. So that's really, that's the idea. That's

going to be a lot like this. I'm going to talk about the math, you

on the board and then we'll flip it and we'll look at how to do

all the resources for you. And then in addition to the core

then what we'll do is I'll have another video or two of extra

So, you know, you know how to write the code, you're building your

and then we'll just work through a bunch of extra problems, you

And it'll be really great. You know, you'll learn algebra, you'll

and you'll have all these skills under your belt that you can use

we're going to build upon these in future courses. So I hope you

to set up Google CoLab in your Google Drive. So in your Google

in folders and, you know, you have a, you'll set up a folder for

you can have all the documents and notebooks and everything there,

and you will have access to all these documents you see, you know,

if we take a look at the little yellow infinity symbol, that's the

And, you know, if you want to create a new Google CoLab, you're in

be, click on new, and it won't be here, and you go to more, and

So Google CoLab. Now, if you don't have it already, if you have

then you go down to connect more apps. And, you know, it's begins

you know, here. But if not, then you can search, you know, and

you just do Co, it comes up. But if you type in CoLab, then it'll

mine's already installed. But if it wasn't, you'd see it there,

button, instead of saying uninstall, it would say install. And

And there we go. Then when you go to new, and then you can go to

and it loads. There you go. So as you see, I like the dark theme

that's just what I picked. You know, you can change that. So this

where you're going to write the code, you know, it's set up to

addition to that, you have the ability to add more code or more

another block of code, you can add more text. And if you click up

And, you know, if you type text, now, if I hit enter, that just

I have to click somewhere else to get out of that. So there you

doing that. Also, if you might have noticed this appear, if you

you can add code or text. And that works anywhere. You see, so you

in between these two. And there we go. And that text actually,

actually use some a few HTML tags to, you know, make new line

And then if you get really fancy, you can use the latex math

formulas. But we'll get to that soon enough. Alright, so you have

you can do, you know, your code, your text. And one of the other

like any document, give it a name. And you'll notice the familiar

Google Docs, you can, there we go. Let's just call this week one

you know, give it a name, you know, you don't want to leave all

might be, you might forget, and it might be hard to find later.

whatever you'd like to put in there, print. Okay. And, you know,

code until you run it. And you can click this run button, or if

notebooks, you can hit shift enter, that'll run it also. And there

here. Notice you can also do a lot of math right there in the

put in math formulas, you see hit enter, it just gives you a new

of math formulas. And remember, the add, subtract, multiply,

this asterisk, two of them. So there you go. So that would be

And you see it'll, it'll do the math right there. If you wanted to

you have to import something. And we'll get to imports as we go

But notice you can, you know, you can have your text, you can

for a lot of simple things. You can just do that right in the

One of the other interesting things you can do is, and if I make

I'll just make that B for a certain reason, then you have a table

Now this has nothing because I don't have any headings or

table of contents looks for. If you click on a new section, then

it as new section and call it good right there. You see it shows

And if you double click, now this B is underneath that section. So

it becomes minimized within that section. And then here, A, if I

then take a look at the section heading. If I double click on it,

would be normally used for a Python comment, but we're, but we're

It would be normally used for a Python comment, but since we're in

just use it to indicate a new section. So I can go up to here,

section. You probably don't need the space, but I like it. And

out of that. So now I have A and I have my new section here. So I

I can put some, I can put some code here and I can, in this code,

arrow over there? I can move it up. And then maybe I'll make B,

make B its own section. I'll click somewhere else. And so I can

and now B is its own section. That kind of didn't matter because I

of it. There we go. Put text in here, click somewhere else, and

So you see, I can have these different sections collapsed, and

of contents, A, new section, B. So if you give them names more

them more than A and B. It becomes useful because you can find

And then you just click to expand each section in whatever else.

section, you can have code, text, you know, and as many of those

goes until you create a new section. So some good useful things

Google CoLab notebooks. You might decide that you might like that

you know, keep them in sections. So that helps. Also, the Google

the Google Drive, it does save automatically, but it doesn't do it

and you do have a save here, and also the normal shortcuts, Ctrl

work. And that way, sometimes, you know, if it goes a while in

it, it might actually prompt you to say that you might need to

to think about. You know, you actually do actually make an effort

CoLab. So we set up our code. We can do math right there in the

sections, table of contents, and all these sections, you know,

expand or collapse them all at once if you want. You know, that's

all the code we're going to work on, you know, you also have down

that looks like, you know, XML or HTML tags, and it's a bunch of

you know, copy and paste, try these right in your Google CoLab

things to explore. And there we have it. You're going to be

You might want to build one for algebra and, you know, just

in which case, definitely make sections because that's going to be

to find what you want. So you could do that. You could make a

for each unit or each week or however you want to organize it. But

these resources that you will have access to. And as well as

you can also download and install the app. So that'll be useful.

That when you first run it, you need to do a few things to go

authentication and such. But, you know, it doesn't take that

All right, so we have this, you're going to be building your stuff

in your Google CoLab and it's going to be exciting. We'll get to a

a lot of interesting things. So let's get into the math next.

So we can't talk about proportions without talking about ratios.

word for a fraction. So this two out of four, I can write that as,

but I could say two fourths. I can call it a fraction, but it's

And then a proportion is two equal ratios. So if I know that these

things I can do with this. And one of the things that works and

multiply. So if these two are equal ratios, then I can cross

I can do two times 10 equals four times five. And that's another

get it out of the proportion mode and I can use this to double

So two times 10, we see 20 equals 20 and then that's true. So, you

works. We can always do this. If these were other numbers, I can

cross multiplying, because if that doesn't work, then that

if one of these numbers is unknown. So if I have three out of six

other ways to solve this, but this is the go-to way because it

when I do my cross multiplying, then I can do this three times

go. Three times four is 12 and then six times X is six X. And then

going to do is going to be dividing by six. So then I get that X

we can jump ahead and say, okay, let's I know that when I multiply

step. It's going to be six X. So the next thing I'm doing is

just put this in one step. So I multiply the diagonal that I can

because I know that the next step and then my answer is X. So we

in writing code. I can see the proportion and I can then just put

for X. And again, that's it's the go-to way it works every time.

for these numbers, cross multiply and then solve for X. One of the

some of the other applications. If I have, let's say I'm

kilometers. So if I have one mile is about 1.6 kilometers, then I

ratio that I know. So as long as I have miles and then kilometers,

out the other. This unknown can be either place. So if I have two

is that? You see, we can make use of this. I know that I'm cross

by one. And then, so there we go. So X would be 3.2. And again,

two times 1.6 dividing by the other number, which in this case,

So then I can solve. So this is what we're going to use. And we're

how to put this into code. And so prompt for these four numbers or

and then figure out the fourth one. So let's take a look at how

at the code. So we're going to see here how much easier it is to

our problems. And just like proportions, we have that cross

time. So I'm going to put just a display here to remind us that

And here's what it looks like. N1, D1 equals N2 over D2. So N for

And if I have a proportion, I would know one of the ratios. So I

And then N2 or D2, one of those I would not know, but I would know

comment here in the code, put a zero in for the unknown value,

these values would be zero. That would just, you know, you

that, that zero is everything out, or if it's in the denominator,

never occur in something that we're really trying to solve. So I

just change these numbers, put in my numbers that I know, and I'll

know. And then I'll put two if statements. So I have if N2 equals

the setup of the if statement in Python. If N2 at the double

I have a colon. And then the indentation afterward is one, two,

some people use a tab, I would hesitate to not use a tab. They

of spaces. So indent four spaces. And then I have the answer is,

multiplying. So you see D2 times N1 divided by D1, that cross

and then we'll print out the answer. And we'll do the same thing

so my if statement, if D2 equals zero, colon, four spaces indent,

notice the cross multiplying. So D2 is what I don't know. So I'd

by N1. And then I'd print out that answer. So there we go. And

equals what over 16. And then if I would run this, and it will

we go. N2 equals eight. Or if I had this, I'll keep this and maybe

Rio, and if we run it, there we go. Really straightforward. We

that we know these we know these formulas, and we can just put

display something to set it up, and cross multiply, solve any

Now that we've worked through the core skills in this unit, let's

And I'm going to work through extra problems using the CoLab

you can apply these resources that you're building and use these

up in a textbook or in day-to-day life. So we're going to go

We're going to work through some extra problems here relating to

working with proportions and ratios. And one of the things that

numbers. And in math, I might write it like this, but then you

And Python, remember, you can even do math in the print statement.

two thirds becomes one plus two thirds, and you see two divided by

sometimes I use parentheses, extra parentheses, but we really

operations. So you could have one plus two thirds, plus three and

plus four divided by five. And, you know, you can change up minus

print statement there and it would output the answer. Now, later

going to also talk about converting that output. If it was a weird

number, but we'll get to that. So you can do this, but even also

if you wanted to, instead of print and you don't need the

there, you can store them as a variable. So if you had something

variable and then use that later on, or any one of these, you

as a variable, because some of these, like two thirds, that will

mixed number, or the one and two thirds that will come out as that

later on, you might want to keep as many of those decimal places

it as a variable will do that for you without you having to write

So really, that's the whole thing. We're trying to build, you

you might use a calculator. Hey, might as well use Google CoLab

formulas in there, store variables and everything. So for all the

work with mixed numbers. Yeah. Print it out, store it as a

work with the rest of the problem. So if this comes out as a

let's take a look at one other thing here. If this comes out, so

one and two thirds. So we know the one, but if I look at the two

you know, 0.6 and it keeps repeating. You know, it'll keep

how would you know that that would be two thirds? Well, one of the

throw in some other variables here. Let's call, let's call this X.

decimals. So if it repeats each, um, each place, then I just need

You see, and that keeps going because if this keeps going and then

if I subtract them 10 X minus one X, so that would be nine X

it, but I am subtracting this bottom row minus the top row because

line up and it would subtract. So multiplying it by 10 moves it

and when I subtract 10 X minus one X is nine X, then that all

how do I get to X divide both sides by nine? So we get six over

we can reduce divided by three is two divided by three is three.

any single digit that repeats is a factor of nine. So if I have,

and actually I might use the same, same formula, but if I have

um, point four and it's a four that keeps repeating. So then 10

and as many fours as you want. So then when we subtract 10 X minus

I subtract this, everything after the decimal subtracts. And so

by nine X equals four over nine. Now, so this works for any number

but this is kind of interesting then, um, how, what if you get a

Because would that be then nine over nine? Yes, maybe. So if I

and so then 10 times that would be nine point and as many nines as

we'll do the same thing. So nine X equals, so you see if this is

this still follows 10 times that. And then when we subtract

it all lines up, it becomes nine X equals nine over nine, which is

So you see some algebra acrobatics. How do I show that point nine,

And these are kind of the things that we want to bring in. Um,

can do, you know, this works out nicely just seeing it on the

different things to convert decimals to fractions, you can see

But yeah, some math acrobatics, this is kind of what we want to be

We want to also sometimes use the code to show different ways to

And when we see it get to a solution in different ways, it kind of

reproves, Hey, this works. This is the true solution.

So these are some of the things that we can do, um, with

you know, some of the other things we'll do, we, you know, we

convert converting, uh, money like.

Yeah. So if I have, you know, let's say, um, you know, one U S

I just looked this up. It's 1.29 Canadian dollars. So if I want to

ratio and proportion as long as you know, I know one of these, so

And then as long as I know one of these, I can figure out the

And in the code, you remember, we just add it, whichever one I

I put a zero there because either way zero over anything is zero.

You know, the zero in the numerator, the answer is zero and zero

undefined. You know, you could look up different, uh, images of

things like going into the abyss and all kinds of fun things like

But yeah, we don't want to divide by zero. So if I know one of

zero, that just really would, we would never arrive at that

that that's why we would put a zero in for the one we want to

have it set up, you know, then I can figure out, oh, okay. Well,

um, how many, I'll put the question mark here since we're not in

okay, if one U S is 1.29 Canadian, then what's one Canadian hour

multiply divide by 1.29 and you'll get that exchange rate. So

to this one, one U S to how many Canadian, we can change it to one

you know, cause very often you find exchange rates and they'll

And maybe you want to find it in the other direction. So these are

use this for. We can also do things like miles to kilometers or,

you know, other unit conversion you want, but you know, if you

is 1.6 kilometers, um, it's that exactly 1.6, you know, there's

another nine meters or something, but okay. And then there we go.

then as long as I have miles and I have kilometers, then it'll

put that in the code. So now that you have these tools, you can

have that ratio of one aspect of it, then you can, you know, plug

figure out the other. So, you know, here, you know, some, some of

of the things we want to do in the extra practice section. You'll

that, you know, we're going to get to the core, um, core skills in

your ongoing, uh, CoLab notebook. So you have some resources and

um, extra practice, extra problem section where we then use some

and let's apply this and let's solve a bunch of extra problems. So

and we will, uh, we'll go on to the next unit.

Here, we're going to look at four different ways to solve for the

And in algebra, we often call this X just because X works nicely,

if I have, I had to put four different situations here, add,

simple numbers because you might be able to do this in your head

explanation is you want to see what you're doing and do the

are equal, both sides of the equal sign, if you do the same thing

So that works. What am I doing? I want to do the opposite to

is do the same thing to both sides of the equal sign. I'm going to

algebra operations here. So if we have X plus three equals five,

the goal is that I get X equals something. So I would like that X

on the left side of the equal sign. So what am I doing? I'm adding

opposite, subtract three, and then I'm going to do the same thing

three is zero. I don't have to write plus zero. Five minus three

So it's like Sherlock Holmes eliminating possibilities, and then

So the next one, if I'm subtracting, I'm going to do the opposite.

something. So X minus two, if I add two, that cancels. That's why

same thing to the other side. So negative two plus two is zero.

Have to write plus zero. And then 10 plus two is 12. There we go,

very similar. Also notice in writing math, most textbooks, you

see a multiplication symbol at some point. Just the fact that

that I'm multiplying. Now we have to remember this in Python

multiplication symbol. But here three X means three times X. So to

so I have to want to divide. And notice I'm going to write that

and then divided by three here. Now here's something else about

by three is one, but I don't have to put one X. You could, but you

you won't see it. One X. Any other number there, I'd have to write

we often don't write that. So then 12 divided by three is four.

Somehow, whatever I teach this, everybody's like, get it, get it,

thing. I'm dividing. So I'm going to do the opposite, which is

I might write it, how can I write multiplied by four? I might even

just saying it's next to that, so that means I'm multiplying.

I'll put in parentheses. So divided by four times four cancels

X. And then here, two times four is eight. So with some simple

add, subtract, multiply, and divide. And the key is no matter what

what else you encounter, how complicated the numbers are, it's the

So we recognize this method, then whatever number comes up, then

this I can do in my head, but supposing I have instead of X plus

7.2 equals 11.1 or something like that. So supposing I have

how could I solve it? Oh, I have an algebra method. What am I

in this case, 7.2 from both sides. And so then I get X equals, and

complicated, I know what I'm doing on the calculator. I'm

and then I know what I'm doing on the calculator. There we go. So

anytime I have a two-step equation. And in the two-step equations,

order of operations in reverse, because the addition is the

like 4X plus 6 equals 22, so two steps, I'm still going to combine

we were talking about. I'm just going to do one than the other.

Like I said, like order of operations in reverse, the addition or

easiest to do first. If I then subtract 6 from both sides, so if I

I have 4X equals 16. And now I get to my second step, divide by 4,

divide by 4, then I get X equals 4. So we have our two-step

are, you're going to get quick enough at this that it's not even

or anything to solve this. But some of these, as they get more

there's other factoring and more elaborate problems, then it would

because I'll show you now how we can just put this in Python,

and it'll output the answer. So let's take a look at the code. So

solve for X in Python because we can just import the Sympy library

math library. You don't have to install anything. You can just

Google Colab works nicely behind the scenes like that. And then

symbols. And from Sympy.solvers, we're going to import solve. So

X as a math symbol and solve for that. And solve is going to be

then I'll define X equals symbols X. And that's defining that X is

I just put this comment here. But then the next thing we have, I

equals X minus two. And this is the function set equal to zero. So

is what we're solving for. And then that's the equation. So then I

statement, print X equals, and Python syntax, then print that

And here's where we see solve eq, the equation, and use X as the

So we're going to solve that X minus two equals zero. And you see

It'll print an array of whatever your answer is. This just had one

But if I had something like X, X squared, X squared minus two, now

answer. Kind of interesting how that'll show up. You see square

make that a decimal. But if we make this something that works out

that, see it'll give me an array of my two answers. X squared

equation. And negative two or positive two, either one works. And

something like two X, I can't just write two X. I have to put, I

that's Python syntax. Two X minus four. And then if we run that,

There we go. Much easier. And all this stays the same. You just

run it, and you can solve anything. Just make sure it's that

Now that we've worked through the core skills in this unit, let's

And I'm going to work through extra problems using the Colab

you can apply these resources that you're building, and use these

You can use these to solve problems that might come up in a

So we're going to go through some more extra problems here. All

bit deeper, looking at different ways we can solve for X here,

code that we were talking about before. So we import Sympy, so we

and then we're still from Sympy, import symbols, and some

to define that symbol X. And this is the simplest thing. We can

just put my equation right here. Here's my equation. And then I

and I'm solving for X. Notice I don't even really need a print

function is the last thing there. And then when we run it, we see

half. And notice the brackets around it, because it's a finite

And if I have something that solves for even, you know, something

plus 1. Now this is setting this equal to zero. So 2X squared plus

solution. We can actually take a look at that. And you see the

and that is I, the imaginary number. So it will give us this.

square root display in a better way, but we'll get to that later.

you know, it gives us two solutions as this finite set of these

you know, minus 1, or I can even make it minus, you know, 4, we'll

But there's still going to be square root solutions. There we go.

will output. But you get this. You can put the equation in there

pretty much anything you can type in there. All right. But let's

fancier. What if we prompt for input? So let's say I want this to

rather than somebody going in and finding that correct line and

we can just prompt for the input here. And notice the function is

display this. Enter equation. And I even have it as zero equals to

equal to zero. You could add another print statement saying make

you know, if that's what you'd like. And this input does come in

are using these symbols, it will be able to interpret that. And

and just putting that into a print statement. There we go. So we

the equation. And maybe I just make it three x minus six. How

good. One solution. Three x minus six, because if x is two, then

But again, if, you know, I have multiple solutions, or maybe if I

maybe I want it to be a little bit fancier. So, again, this solve

so we can do more. All right, we're going to import all the rest

knowing that it's a finite set, I'm going to store that answer as

And then this particular equation, I know this has one solution,

display solution index zero. And that's going to be the first

to be the only solution. So when we run it, there we go. And

it a little bit nicer output, just like having that nice user

having it display zero equals and then waiting for the prompt. So,

do this. And, you know, here's saving this. Okay, that's great.

And what if I don't know how many answers I'll have? So I'm going

so if I enter the solution, now, maybe I'll just put the first one

this one next soon enough. So I'll just take the same solution

And just like we were doing before, I'm going to solve it and

notice my loop here and Python has this great

method of iterating through everything in that list. So for S in

item in that solution set, whether it be one, whether it be

it'll iterate through that. So S is that solution. And then it'll

And each time, I'm just going to print x equals, and then I'm

we make use of the Python shortcuts. There we go. So 2x minus 4,

through and did this. So what if I have something like something

what if I have something like something that has multiple

want to prompt for this. All right. So what if I just plug in

I know is going to have three solutions, because the three

So let's see what this does. Again, loops through, and it will

So we see we're building some of these things behind the scenes

solutions and output it in a nice way. So we see, you know, if

you're going to use, this is the one to do it. Because, you know,

you take out this, this was my demonstration. But there we go. And

So this could be your code. And that will work for anything,

you or you or anybody else using this to enter the enter the

we're building some tools here that we can use to solve. All

Now, some interesting things. We're also this time, we're going to

and we're going to use x, y, but we'll come back to the y. And we

because you can use this for multiple equations, but I just call

two x plus 10. All right. Now, we know that this is set equal to

the go-to way of doing this. But then this simpy syntax, we're

define it. EQ first, comma zero, saying that this is set equal to

for x. That's great. And this solution I just called sol.

And since I have this one, I know that the solution is, you know,

just put salt. You know, that's my finite set. And I want the

actually, these are factored results, but this is actually going

because I can get that answer. X is negative five. All right. Now,

Now, I have to have y there. If I define y as a variable. Now,

this is equal to y. And you see that works. So spelling it out

But now I now I'll solve for x. So that's kind of interesting. And

Python could actually do some of this factoring for you. You know,

algebraically. So now it's solved. Why over two minus five. Pretty

like that. We can rearrange things. And then if I have this, you

around a little bit or whatever. But some different things you can

to zero. Make this equal to zero. But this gives you this other

to y. And then you can do some other things you can solve for it.

as you incorporate this into, you know, some of the notebooks that

around with this and, you know, try some different things. You

some different some different formulas or different equations in a

try this? Like, we could take a look at this and say, all right,

what about if I have two x, then, then if I want to solve it for

So there we go. And if that's included in y, then I still might

this is equal to zero, because that's kind of the default way for

back and solve for x, and it should end up being very similar.

do something very similar, because whether that was, you know,

it's equal to y or with this minus y saying it's equal to zero,

And, you know, we could even just to show you, we can even say

this. There we go, x is y over two, all kinds of different things

rearrange this, you can put that in this setup here. And then, you

set equal to now? And what am I solving for here? And in this

zero for that solution set each time. So these are some cool

the factoring for you. Well, here's some other factoring.

you know, we can divide out common factors. Now, you might see

ten times y, plus four. Now, supposing this is another thing in

You know, I'm not saying that's equal to zero or anything. I just

I don't want to factor it. Look at that. And SymPy also does the

So you see, there was a common factor of two. And that worked.

and let's say this is three, which then doesn't factor at all. So

you know, it will just output that because it's not factorable.

interesting is if I have something else, like x squared, and let's

you may or may not be familiar with that factoring, but I can

I look at that. It'll factor that out. That's the correct

So it will do a lot of really interesting factoring for you. So

we can, you know, just to show you that we can even take it to the

X to the third minus two x squared minus five x plus six. So we

there we go. Look at that. It breaks it down into these factors.

you know, knowing that you can set your factoring, you know, you

factoring, you can make your own factoring calculator here. All

at some of the functions here. So when we have, you know, we have

you've been doing some of this practice with these functions. So I

you know, looking at these and this different input, how we can,

you know, remember you were doing the practicing the one step

the one step addition, one step multiply. There we go. Now, that

is because you can't do string to a fraction. This print, this

we take this input, now other inputs, if it's actually a number or

and you can cast it as a float. So that's a number that could

it's going to be a fraction, this float won't work because that

parse nicely. So then this will give an error. Now, just because

just a display. If I just disobeyed that and put in a number, then

But if I, as soon as they use that divided by, I need that. So

is how do I convert the string input, including fractions to a

functions work. I'm going to use this DEF to define it. And then

I give it the name of the function. I need the open and closed

and this particular one will also have input. So I define this

That's going to be the input and then colon at the end. So once I

then everything under this is indented for spaces. Yep. And I

use indentations. So then I have an if statement and if, and then

is true, colon, and then that's indented for spaces. Okay. So I

and I'm testing to see if this fraction bar is in there. And

in string, cause that's what I called it. That's my input. If it's

Each string has this split function. So you can split it and say,

I'm going to split it. And then that creates two different arrays.

input. So it's going to be two string arrays or two or one array

going to be ND. I call that for numerator denominator. So then ND

everything before that. And ND one is going to be everything after

if somebody put a fraction in there, then ND zero, everything

So I'll cast it as a float, store it as N for the numerator. And

that. So I will cast it as a float, store it as D for denominator.

we had to detect that split that string. And then each of these

them as float numbers. And now these are numbers N and D. And now

N divided by D. So you see, because that slash doesn't work for

like work around it. And now I have ANS as this answer. And then I

this, this function doesn't have a print statement, just returns

comes up. But it just, you know, wherever this function is, it'll

So that's if it has that. And if it doesn't, because I want to be

either way. So anything else, colon, indent, then I won't have a

in string cast as a float. That works. So anything else you put

return ANS. Now, where this comes in, I'll come back to that one,

So here's this function, one step more. And I put this common in

else using this, that it uses this function. So we need to make

All right, so here's, here's the idea to generate, you know, some

random. And then here, I'm going to get random dot Rand int from

a random integer, and I'll store it as variable A. So that's going

including one, but not including 11. So one could show up, 10

So that's, that's the way the random integer works. It includes

And then B, I just wanted to make this two to 24. So given these

notice what we're going to print here. So AX equals B. So we see

times X. And I just want to display this. So I didn't need to

to show the multiplying. All right, so AX equals B. And then you

You want to divide both sides by A. So you know that, and we're

you know, X equals, and whatever your input is, we'll store it as

And then the actual answer at this point is going to be, we know

so the actual answer is going to be B divided by A. So then this

still a string. And that's where we're going to use that in our

string frack. So see, it's going to take that answer, run it

run it through that function, and return right here, that answer.

So that way we can easily compare it because this function will

the double equal sign to compare, and we're going to compare it to

So there we go. So that's the return statement, wherever I don't

print or output anything, it just returns that value right here

And if it's, if it is good, we're going to print correct. If not,

but then print out the answer. So there we go. And now it'll loop

it once, and then, you know, we can create a loop to have as many

practice one-step equations. So there we go, and then test them

And then test them each time. So, you know, we have the simplest

is another very similar one. There we go, import random, display

random integers, and yes, you can go through negative numbers if

All right. And what are we printing out here? X plus A equals B,

you know that to solve for X, subtract A. So we'll take your

But also we're going to calculate exactly B minus A. What is it

we don't need to do anything else. We just say, if A and S equals

And here, see, we can cast the float, cast the input as a float

expecting anything fancy. You see the other one, knowing that I'm

to address all the other things that could come up. And there we

If not, and then we print out the answer. Cool. So some different

one-step addition with negative numbers, just we just did in a

a bunch of things that we can do, just practicing your own ability

more into some other functions later, but, you know, good to see

yeah, here we go. Just some of the things you can do, factor or

that you want to be able to do and, you know, set up, you know,

your own CoLab notebook. So by now, you might have some things

You might have some things to, you know, solve for X and, you

those, you know, convert fraction input to something, you know,

copy these, tweak them, make use of them on your own. And, you

library as well as really understanding the code as you're

your understanding of the math, understanding of the code gets

here. All right, so I think we're ready to go on to the next unit.

decimals and how to convert from one to the other. Well,

could be one out of 10. If I say out of, that means divide. So any

denominator, you can do that on your calculator and it most likely

decimal. There you go. So fractions, decimals, you know, pretty

you know, pretty easy. Calculator, do the division and then, you

what about doing it in the reverse process? So when we look at

one out of 10, I could call it one tenth or 0.1 is what you would

I would still call that one tenth because that first decimal place

be one tenth. Or then if I had one out of a hundred, then there we

So tenths, the second one, the hundredths. So whether I write it

I would still call it one hundredth with the th. Then, you know, I

you know, third decimal is would be thousandths. So, and it would

we have that, you know, tenths, hundredths, thousandths, then that

if we want to convert it back the other way. So if I would, if I

you know, 0.234, so something that I wanted to convert, then I

hundredths, thousandths. That's where it ends. So then I could

And I could reduce that, but that still is the fraction that I'm

thousand. Or if I had something, you know, like 0.4 one place,

four out of ten. And again, I could reduce it or I could leave it

that that's converting my decimal right to a fraction. And when we

we're going to look at how to see here. I'm just looking at this

hundredths, thousandths, et cetera. It could keep going. You could

but I'm looking at these remembering the place values and then

have ten with one followed by three zeros, you know, one decimal

one followed by one zero. And we get into the code, we're going to

in a different way rather than just, well, there's a few ways you

count how many places, or we're going to, you know, one of the

logarithm way, probably the string and just counting it, that's

straightforward. We'll do that one. But there we go. Converting

and then, you know, how do I reduce it? Okay. So we think we have

if I have a repeating decimal? So supposing I have 0.3333 and it

And mathematically, if I was writing this, you put the line over

So I really could just keep it as one, but I want to show you

three. So that keeps repeating. Okay. So we were just talking

places and then put it over that. But if it keeps repeating, where

at a trick how to do this. We want to get to the end of the

repeats once, then if I multiply this, let's, let me call this X

this down a little bit further. This is, that's going to make

X is my 0.3333 repeating. All right. So if that's my value, I only

the decimal place once. And then that cancel out. So if I have 10

And these keep repeating. Now, the reason I do that is because of

everything that repeats now is going to line up. And so if I

what can I do on the left? 10 X minus X is nine X equal sign still

minus zero point, all this that cancels all this, however many

it all cancels and my answer's three. And then one step of

nine. So I get X equals three over nine because nine divided by

three divided by nine. And I could reduce that as one third if I

over nine, you know, that's nice, like bonus math insight. If I

it's that number over nine. That's the fraction. So if I had four

if I had four, if I had, you know, very similar type, type of

something like, uh, if I had zero point four repeating, so then 10

point or four, et cetera. They both keep repeating. Do the same

And when I subtract these, all the repeating decimals cancel. So

four is exactly four. And when I divide both sides by nine, X

So there we go. And, uh, so one more bonus here. If I had nine

we could do the same thing. And you would think 0.999. That's not

in calculus, we can look at limits and how 0.9 repeating. If it

purposes, that is one. But if we did this, it would be nine's

and you get X equals one. So it kind of works out nicely, but this

You know, notice we're not looking at code here. Now we're just

something that repeats, then that's what we need to do. We need to

had something that repeated, um, you know, three, two, two places

that one. And just to give you an idea of a different way, if I

then if I have, uh, so X equals, uh, 0.090909, something like

it's the zero nine that repeats. So I need to multiply it by a

X because that would move the decimal place twice and that makes

So when I do that now, same method, if I subtract these, that's 99

cancels. So then X equals 99, uh, nine over 99 and fine, perfectly

reduce this divided by nine. So that's one out of 11. We'll get

code for that too, but you see, this is, this is one way. If you

maybe you're right the way you write the code, it might seem to

you don't find some way around it, this is the way around it. So

decimal, but now we're going to, what we're going to do is we're

to take any decimal. Just here's the decimal input. How can I find

is and reframe that as a fraction so that given the decimal and

related to that is percents because remember the first is 10th

are the hundredths and that's really what percent means out of a

thing. And then just, I'm always looking at the first two decimal

So, you know, very similar, but just the first two decimal places.

and we'll see how to convert any decimal to a fraction. All right.

convert decimals to fractions with code. And first we'll get some

going to be useful. So if I have 10 two different exponents, let's

exponents. So 10 to the first power, we see this first one, 10 to

10 to the third. We'll see these come up. What happens when we

10 or anything else to the zero exponent, and then we'll take a

or negative two or negative three. So this will just print these

what this does. So we see 10 to the first, second, or third

to the zero is 1. And then the negative exponents are how many

So these are the things we're looking for and we're going to make

our decimals to fractions. So the next thing, we'll take a look at

And we have this formula here, user input, open parentheses, and

will display. And then you'll see then a box, you know, then

going to store that as text as that's the variable. For our

you that you can enter a number and we're going to print it out. I

to remind you it comes in as a string. We can't do any math with

do, any string, I can figure out the length of that string. So

And I just have, again, this comment here, if I tried to do math

error. So let's take a look. So you see, enter a number. And then

like 0.123, I hit enter. You see 0.123, the length of it is four,

these are four characters, the decimal point counts as one

with this. So I, you know, if I wanted to enter a number, now, if

now I can do math to it. And I can cast it as a float or an int

mind, you know, things I might want to do with this, a float can

usually my default cast. But, you know, if you definitely know

we would use that. But we're going to enter a number, cast it as a

the variable number num. Now I can do math to it. I'm going to

There we go. If I enter a number, there we go. And if I have

now I can add. All right, or whatever other math operation you

an int, then you can do math to it. And the float, because it can

operation that there might be a chance of a decimal use float.

together. So converting this input, which will come in as a

and here's what I'm going to print out. Enter a decimal number to

that that's input and that's going to be stored as digits. And we

And we know that that's going to be a string. So to get the number

So I'm going to use this, the length of that input digits, and I'm

first. So notice that happens first. Then I'm going to subtract

that decimal point is going to count when I do the length of that

because I don't want that decimal. So now whatever decimal enter

that's going to be an integer that's going to be the exponent. And

that is still a string. Now I'm going to cast that exactly as a

So now I know how many decimal places it took to make that and the

So let's talk about our fraction. My numerator is going to be n

10 to that exponent. So in my previous example, point one, two,

you know, that was four, the length of that was four, so minus

my exponent. And then when I multiply it by that exponent, it

know it's going to be a whole number, I will cast it as an

numerator. And then the denominator is going to be 10 times that

what we were doing before we were doing the code. You know, how

and then, you know, my numerator is that times 10 to that many

denominator is just 10 to that exponent. So three decimal places

and then the numerator denominator would be 10 to the third, and

Now, that's numerator and denominator for any fraction. Percent is

places. So whatever n is, I'm going to multiply it by 100, because

That decimal, move it over tw- move decimal place twice, which is

see the output, the decimal. There we go. We just repeat the

and I'm just going to print this out to show you it's numerator

percent, and then I'll print out the percent symbol. There we go.

something else, but here I'm putting it in quotes. It'll just be

to enter a decimal number to convert, and there we go. Point, uh,

The decimal, point, one, two, five, and then the fraction is 125

and the percent's 12.5. And we can run this again and do this for

Um, you know, point two, enter. So it's point two, two out of 10,

And there we go. The same things we were doing, and this is what

look at throughout this course. You know, the steps that we're

this down, and we're pretty much doing the same steps, but then

code, then, you know, you have a formula that you can reuse. Okay,

script here, and I'll be able to convert any decimal to a

Now that we've worked through the core skills in this unit, let's

and I'm going to work through extra problems using the Colab

apply these resources that you're building, and use these to solve

have, problems that might come up in a textbook or in day-to-day

some more extra problems here. So now I want to show you how to

You know, this is what we're talking about here. You're making

the things that I'm showing you, and you'll have access to all

but you're going to put together yours. So you're going to click

the Google Colab, so go over to Google Colaboratory, click on it,

so actually I will call it that. I'm going to put the underscore,

your calculator, and you just hit enter or click somewhere else,

you can do to put together your calculator based on things that

and then things that we do further on in the class, then you can

go to, let's say, the week one notebook where you solve for

and here's some things you can do. So if you click on this text

let's copy that first. So you click on that text and go all the

This looks like copy, but it actually doesn't do what you're

Click on the three dots, copy that cell, and when you go back to

I'm just going to click in the open space here and paste it, Ctrl

it's kind of cool. So, you know, Ctrl V for the Windows, Command V

either one, and then we can go back. That's your proportion set up

over all the way over the three dots, copy cell, and your

So you can have this, you know, we don't need this. If you have an

the trash can over there. So there we go. Now you have your text

and then it really is then self-explanatory. You know, you see the

N2, D2, and even the comment there, put the zero in for the

each of these. Then if we look at week two, now we were doing a

and maybe you don't need to do the simplest one. Maybe you like

that. Doing more with the solution, how we split it up. Now, I

might be the one that I copy as one of them. And what you can do

because notice this one, I personally like this one. It prompts

that it's something equal to zero. You know, there we go. We

this also prints out multiple solutions if you have them. So this

And when you go back into yours, now if I just paste this, that's

And notice it didn't paste where I wanted it to be. So watch what

in here and use these arrows. I'm going to move it down, and

Now, we wanted some text in there. So right above it, you hover

solve for X. Maybe I'll just add that text there. So there's more

That's nice. Now, this solve in other ways, that was pretty good.

factoring. So you could include something like this, solve in

you know, simpy.factor. Now, I might include this because you'll

the one of the other ones. So if we copy this, and this is just

do this. And this is just factor when there's, you know, nothing

we'll copy this cell, and paste this down here. Now, maybe I'm

factor. And down here, then I will paste it. Didn't show up where

arrow, move it down. So, you know, there we go. We have set

But we can even make this, you know, equation to factor. And see,

want. There we go. And then, you know, you can factor things. So

the next week. Now, notice to put it all together, because when we

of these leading up to converting a decimal to a fraction. So this

You're prompting the person to enter in this, and then we're going

decimal, the fraction, the percent. So that's nice. Maybe we copy

back to. So then we go down to here. Maybe I'll paste that here.

some text ahead of it. Decimal to fraction. So you see now you're

here. And we don't need that output right now. Now, we have one

for a variable. And notice it's when I paste this, I might put it

remember we did this, then you have the left side, which could be

of the equation, which could be anything. You know, this would be

you would need to change here. Change the variable you want to

your basic common variables that you would use, but you know, you

if you want, but you might not need to change that line. And then

even if it is something that you can just, you know, solve for a

It doesn't come down to an exact number here. So that might be a

example in here. You know, copy this. Now, we might put this in

factor solve for x, and factor solve for a variable. Maybe we put

And notice you can just click and add code. I'll click right in

And this will be pretty much right where I want it to be. Then I

add some text, solve for a variable.

Now, these are just some examples here of how you could put this

So that then you have your resource here, proportion. There we go.

solve for a variable, convert decimal to fraction to percent. And

you could add more into this here. Now, supposing you want you you

then you add all this, that's this is a lot of scrolling. So one

is now I'm going to start with proportion. I like so right around

there's multiple ways to do this. You could insert, you go up top,

a section header cell. So I'll do it one way this way. I'll do

And actually, I want that above. So I noticed this, when you

some of this. These text boxes accept HTML, look at that the br to

and they accept latex, which is how to format various things

all kinds of things. So in this new section, and you see, because

that's not a comment in Python, it's the new section. And I might

Okay, so now, you see, it takes that and all these cells become a

I don't want that. But what I want to do is, if this next one, I

this section, and I don't need another one, I think that'll be the

There we go. So I'm just going to put this here. Now that's a new

So we see proportions now just has the two cells, the text about

And solve for x has just one cell because that section said it

There we go factor. And maybe you want to say something more than

but you can make this a section heading. Solve for a variable.

this together. So you see solve for a variable.

Convert decimal to fraction. You know, that probably I didn't need

solve for a variable. All the rest was explained in the comments.

And convert decimal to fraction.

So now this just has the one cell. And the advantage of doing this

you see these lines. Now we have the table of contents here.

There we go. And I could change, you know, that's long enough. I

two lines, but that fits so that it's a longer title, but that's

table of contents that you can actually go through. And you see

x. You know, different things. And in the view, you could change

You don't have to, but you could. And then you see you have all

So these are some things you can do in putting together your

Putting together your Google CoLab notebook. You know, I want this

that you're building. So then you're learning how to do all this,

your resources here that you can then refer to. Oh, what do I need

Enter what I need to enter and solve it. You know, you're

So that's what I want you to do for this unit, is take a look at

and put it together in your own notebook. Some of the things we'll

and we'll, you know, make some of these into functions that you

But this would be a good place. Put together your notebook of

but you know, you can tweak it to the way you like it. All right.

So what does it mean to be a function? A function means it in math

some input and then doing something to it. And actually there are

don't have to take input, but in math we will. So like we were

being X, the output will be Y. So each function we want to take X

to say that the output is Y. And here we're dealing with all

and then outputting Y. So you see we have like solving for X we

have a second variable and we still want the idea that just like

in science and in research, we can call X the independent variable

first. And then Y is the dependent variable because it depends on

want to solve this or do anything to it, then I'm going to plug in

and I can pick any value. I might start with zero. So I'm going to

is zero. And then what happens when I plug that in? So that'd be Y

And you see I'll just put in zero in parentheses where I plug in

it highlights that that's what I just plugged in. And also the

I'm still multiplying. So when we do this two times zero zero plus

three and my output is Y. And if you remember then that's an XY

Y is three and I'd be able to plot that point and I can keep

what other numbers can I plug in? I'm even just going to keep

X equals one. Plug in one. Two times one is two plus three is

And then we'll plug that in and I can plug in as many or as few. I

integers. I tend to do that so you can see the numbers a little

But you know you don't have to worry about doing any other

But if I have this I can put in whatever numbers I want as many or

numbers in there and we can plot these on the graph which we'll

But this is the essence of a function that X comes before Y. I

output the Y value. So another function notation that a lot of

another variable Y and this connects a lot with the way the code's

F of X and F for function. So two X plus three. So if you see

Y equals two X plus three or F of X equals two X plus three. This

what we read F of X, F for function. We solve it the same and this

your computer science. So if I have the output called F of X, one

into writing the code is that in algebra we tend to just say F for

or other code you can actually give your function a name and it

function. It can be add, divide, calculate the area, something

long as there's no spaces. So if we have this and then sometimes

into two I could write in F of two. And so again the usefulness of

what what am I already defining as my input and then I know what

two times two plus three. Two times two is four plus three is

I know that answer and when I see that X is two, Y is seven. And

relates to the code and how we can write even a little bit more

and how we can define the input there. Then we'll look at graphing

look at the code. So with the code and functions we can do very

by hand writing out the math. Here I'm just going to define one X

the function that Y value. Y equals four times X plus three. And

make sure we put the times. So I'm just going to plug in that Y X

in for this X value. So it's four times five plus three and then

my solution X comma Y. So we see that and before I print that out.

but I can loop this. So I'm going to print out this line here

tab Y. So that's defining my columns here X and Y. And then what

all those values. So for X in range 11 so X is going to go 0

take that same function and each time then I'm going to plug in

I'm going to print out the X Y values. And in this case again I'm

So this is the essence of what we're doing with a function. Each

solve for Y and then I have an X Y value and this is going to

So when I run this we'll see it'll come out first of all we'll

you see five twenty three five comma twenty three that first

Y and all of our X Y values if X is zero Y is three if X is one Y

five twenty three that shows up on the list too. So that's

lot of other functions beyond math you can do things with and we

of Python function definition for our math. Here we can on this

space and Python I can name my function anything but I'm going to

F of X and then here I have a colon. So this is going to define

is indented four spaces and I have Y equals my four times X plus

and what am I going to do then I'll return the Y value. So

Y value. So you see this in use if I print here if I print five so

my X value comma and then I'm going to call this function right

take that and run it through there we go. Same type of thing same

define this and when we run this it's just going to print that 523

loop I could just take the same loop here and I'm going to copy

loop if I wanted to and in this case I'm going to print because I

defined I don't need this and I could really just put put that

I could just have right here since I defined that function X F of

and I don't have my headings here but I think we can use our

one value but then in the loop just went through and looped so

illustrate a function in code and you know what we can do with

almost limitless you know you can have a function do anything and

more complex than this you know give it a word but I wanted to

classic math notation and functions in Python whether I actually

that still works as a function or whether I define it and next

so now when we talk about graphing functions we'll use this

and I'll use the x and y notation so we have our input and output

values and what y values match up so if you ever come across

you can see that if I plug in both of these x and y that should

have a few of our points here and then now we're going to see how

plane our x y coordinate plane it's Cartesian because Rene

zero zero at the center and how much I count for x positive to the

it comes first like x comes first in the alphabet then my y value

and my coordinate pair would be my x y so this zero three would

zero so I'm not going left or right at all up to three plot the

write this but I'll show you so then that would be the point zero

so I would go over one up five plot the point and then that's one

now I just plotted three points but I'll tell you if you really

decimals or other numbers that are in between you could bet that

one point something two point something they would all end up

there's no x value that can't go into this well whatever weird

number anything so all the numbers would be on this line so with

going to call it linear and we'll get into other types of

points and then we want to look at these patterns what type of

so what we're going to do now is look at in the python code how to

to make a basic blank graph we're going to import this library

we're going to we can import this as plt so I can reference this

out and here's what we're going to do the first line is these are

fig for the figure axe for the axis so these two variables and

and then that sets this up and we'll do plt.show the base there

show and then that gives a basic graph these are all the defaults

and you know we have this set up so we're going to build upon that

here notice that was all just the first quadrant everything's

positive and even only one goes up to one so again import matplot

first lines but now let's take a look at these dimensions now I

and I'm going to set this axis all right and it gives you the

going to show a better way in a second but you can do this on one

when we show it I want it to go from negative 10 we see across the

up to 10 zero in the middle and on the y-axis negative 10 up to 10

great but I have no lines I have no axis and you know not the

here's a better way to set the dimensions is I'll define them

as variables x min x max y min y max and that way all right I want

and I can set them here easy to see it and change it if I want to

really what I'm doing when I set the axis size x min x max y min y

that it's taking and it becomes a lot clearer when we set it up

I can set this and then if I wanted to I say oh for this one this

so maybe I want that y maximum to be 20 instead of 10 and then

size of this still looks the same but it adjusted accordingly and

and on my x-axis negative 10 to 10

and so let's display axis lines here if I have my dimensions

well that's the one we already did here the window size but I'm

this line here I have x min x max as my array of x values see

array of x values and then my array of y values and the default is

just plot two x values and two y values it'll draw the line from x

in quotes makes it blue so we're going to do the same thing with

are going to be 0 at 0 y min 0 y max and then it'll get make that

just because I like blue and there we go so now we have this that

so from everything we do now we have an axis we have a point of

so everything that we plot is going to you know start with this

set all this up we might not even need to change any of this and

with an x y-axis all right so how do we plot one point so we see

and I made it blue b in quotes so how do I plot a point well I

I have an array of one value which is five for the x value one

and then the r makes it red and the zero makes it a circle

other options you can do you can do a dash you can do different

to make this a red circle and that's how we make make that circle

here if I have these two and so when I run this it'll still plot

do five four and then that's going to be a red dot all right so if

now connecting with what we were doing with the functions here I'm

and here in rain in my range I'm going to have the range from 10

from this value my y value y equals 0.5 x plus one and in this

going to do plot plt dot plot and I'm going to plot that x y value

I'm going to plot these x y values and again that's the essence of

these different values and when I plot all these points I'm going

that this all lines up and I'll tell you if I plotted points that

x values there would be they would also line up that's why it's a

of it's going to be a straight line and then every point on this

this function it's going to work so my loop to graph a function

to do the function and the table so like we were doing before I'm

graph that I have I'm going to print the x tab y for the heading

I'll use the same function here I'm going to plot the point and

and all these it it actually won't plot all the it'll kind of save

them all it's really just going to add them all to the table and

really doesn't show the graph until the last thing outside the

when you have graphs I think Python just likes to have that

first and then show the graph so then we have this and we have our

put one number in the range it'll do zero to that number but I can

or I could even just go back to those original values that I

x min x max you see and then but as you know you can put any two

do the range between there and you see it just abruptly starts

I could change that to make this x min and I can make this x max

last point so I'll do x max plus one and I can have that range and

of values and it'll graph the full dimensions so there we go we

I have whatever math function I have plug in different values and

numerical output see the graphing output and now we see the

now that we've worked through the core skills in this unit let's

problems and I'm going to work through extra problems using the

see how you can apply these resources that you're building and use

might come up in a textbook or in day-to-day life so we're going

extra problems here so let's talk about more ways to graph and

because we want to be able to display our data our functions so we

can loop through inputs you know I can define my graph and we

y min and max and we talked about how we can loop through I get x

going to plug this in and plot that value and that's the idea of

x value in this range plug it in for y plug it in here get y and I

as a red dot after doing all that then we show the graph but what

a loop for our input what if we could use an array and we'll

but we have this numpy array and it's not that long of a word but

convention that we import it as np and so you'll see a lot of

np dot this or that so that's what we're going to do and again

so given this numpy array here's what we're going to do then given

how many points I want in my array and in this case it's going to

min I feel like that's probably enough for our purposes but that

10 or whatever you want and then we see so then we want to make as

these values as we can that way if that changes if I change my

accordingly and I don't have to go through and find all these

range times two so that's two points through that you'll see that

just fine but you can always increase that so now they have any

of x values and since I really am only using this once I'm going

my array of x values and so numpy we have np dot linspace so we

going to set up our array you know space for linear graphing so it

I start where do I end and how many points do I have so we see we

and then I get my x value so this saves us the time of having to

for us here's all the graphs and we're going to talk about how to

in a minute so now I'll just I like to define my x value up there

in the range so I have my y value and for our purposes here we'll

it y equals but you can graph multiple things on the same axis and

whatever you'd like so I just define this relatively simple linear

when I plot I'll plot x and y and if you remember before we were

in this case with only one element but here you see I define my x

and therefore y becomes an array of y values so I can just

they each do represent an array in this case then I decided to

quotes so there we go and when we graph this then we can see I

about like that would be 2x plus 1 with the red line all right so

things we can customize here now yes r g b work well you know I

line r g for green and if I want it to be a black line then I will

those all work as you know simplified ways to define the color of

put anything python will actually assign a color to each line and

you plot a few lines they each get a different color but if you

want it specifically to be this line for this color you can define

those major colors you can actually just write in colors then you

abbreviation for pink but if I want my line to be pink I can write

every color that will be acceptable here but it's going to be a

a line of a particular color type it in try it and there we go so

to define the color of line because remember this plot when this

it's a line so if I want it to be a point then I would have to

array but as each an array of one value and then the the letter

circle or the carrot for a triangle or something like that so or a

if we just wanted to be a point we'll see that in a second here so

and our line but we're you know we're not done customizing it

a look at these tick marks you know zero and then every 2.5 maybe

we'll change some of these things and maybe you know this seems

the grid lines so here's some other things we can do all right I'm

certainly matplotlibrary but I'm also doing numpy because we use

graphing and same thing I'll define you know the window up here

window so for now I'll keep this you know the same axis I just

now notice when we do up here figure and axis so the labels would

axe dot set x label axe dot set y label axe dot set title there we

graph and I didn't mean that some graph like whatever I meant more

charlotte's web or some graph so anyway so we have these and I

later and I have a few other things here so I still have one y

now notice this I'm going to define this as a label you know it

rather than a color but notice no color I can also do a color but

what color python assigns to us and then I decided to plot a point

zero red or that's actually oh the lower case o so it's going to

this one's going to have a label too and I'm going to plot

then notice I can define the y value up here and then say y right

in here directly either way whichever you feel like doing I don't

options here and again we have this label also so let's just take

late set x label y label and the title and then we'll see these

so there we go and you know we get nice orange a different shade

and I define that as a red dot so there we go but what if all

what if I want the axis to show the grid it looks kind of plain

when I comment that out and this one notice true is a capital T

it will actually show the grid lines and you see it does line up

there every 2.5 but there we go so if we want to see the grid

and we have this and now we have our x you know our labels x

you could actually put the graph and in one of the later units

to even automatically if I wanted to graph one line and have the

but you know you could always just type that in you know instead

just type that in as a string you know the equation of the line

again and in this case let's say the 2.5 in each direction is not

have x dot and we're going to set x ticks and set y ticks so

so it's a different sort of array I'm going to do num mp dot

very similar to the way we did x values I get my minimum my

I want to tick mark every two numbers so that's what we're going

you see again basing it off of those original variables if the

will update automatically so there we go so let's just take a look

tick marks and we see then now every two and maybe that's a little

and we see these now if I wanted it every one then I'm going to

this is just for the tick marks along the side but up here when we

going to just follow whatever tick marks we've assigned so now we

and now we get this sort of graph where I definitely can count the

let's take a look at the slope of each line let's see these and

so that way it doesn't get in the way of anything that's in the

all right there we go so we can keep this that's that's pretty

we have all these labels but we didn't see them anywhere so if you

add this other plt dot legend here we go and we can see these

each of these labels we can have you know this line we labeled

just wrote the word point but you could put anything and the other

some more steeper line so we could always do this and in the same

assigns it you'll see the matching you know which line are we

referencing so some good things that we can do with these we can

can change how often they come up we can have the grid or not we

all right and I encourage you just to get in and you know tinker

all some of the different ways you can you know match up different

different things and we will get into even more graphing but

systems of equations and some other more complex graphs so then

you know go a little bit deeper try a few more things with the

will get you to be able to display any function you have and you

way that you say yeah this is exactly how I wanted it to look

all right so yeah get in there and tinker with this and see see

so let's talk about slope how can I describe this line that's on

how it goes up and over so that's what slope is picturing these

that they go on some sort of angle but we're going to describe

much am I going up and how much am I going over up and over up and

two points and if I have the graph that I'm looking at that's

that's great I can just count you know looking at two nice points

makes it easier to count so I can say from this point to this

three and then over one so up three over one well then my slope is

you know from that to the next point I can go up three over one

plot a point etc and it would be the same slope anywhere on the

equation it's a perfectly straight line so that's great I can

you know just have things like this but if I have the points that

up that's the change in y because y tells me if I'm going up or

up I just counted three but I could also subtract taking a look at

or then how much am I going over and it's always over to the right

so from one then to two two minus one how much am I going over so

in this case it's positive because left to right it's going up if

then my second y value would be less and when I subtract then it'd

always going to the right but y would be positive if it's going up

and so if I had these and I can count but looking at this I know I

formula look like well I can write out the formula here slope m

m for slope because it's how much we move on the graph so somebody

caught on so we're gonna use m for slope how much we move on the

much am I going up so that's the change in y and I can subtract

a little subscripts you know y2 y1 and that indicates my second y

then the change in x would be very similar x2 minus x1 and again

x value minus the first one okay so now that I have this formula

and see so we'd have four minus one or minus one I'm subtracting

two minus one subtracting the x values and what do I get four

two minus one is one and three over one any whole number over one

a whole number I don't have to write over one so I can just write

see anytime you see a slope that just is a whole number it's that

so we have counting if I have the graph subtracting if I just have

you know I'm using that formula so this is great it's the same

slope for a linear equation so let's take a look at how we can

you to get into the mindset that anytime we have a formula you can

and you know there we go input the numbers and output in this case

look at the code and we'll see how we can do this now this part is

want to show you that you can connect any formula if you know here

y1 x2 y2 and put that into the slope formula slope equals y2 minus

and really then this is a useful formula and others you can just

and run it through the formula here we're just going to output the

there we go all you need is the two points next we're going to

but then let's take a look at how that comes together on the graph

all right so we've already figured out that the slope is three in

we have this two points and we want to figure out the equation of

linear and I have these two points that we figured out that the

to the next we went up three and over one or I just subtract four

and then four minus one is three two minus one is one so three

slope is three and we want the full equation of of this line now

imagine here that it would cross the y-axis somewhere and that's

the intercept is where does it cross so we have a nice formula

intercept form is y equals mx plus b now we call the slope m

one point to the next and we'll call the other one b because

so many things happen in this first quadrant where everything's

we begin on the y-axis and then move in that direction of course

different directions but so I have this y equals mx plus b now

x y comes from any of these x y points so I'll pick one of these

y-intercept it works out that way so if we take a look at this I'm

points how about I take this this first one here because it's one

easy to calculate it works for either one so when I plug in this

and then the slope which we just figured out was three and then

put that in parentheses so notice mx you know in my math notation

multiplying just because they're next to each other so I'll do the

you know m and then plug in for x and now b I don't know so that's

that I know one point so that's four things that I know one point

two things I know three of them I can figure out the fourth one

equals three plus b and remember our one step equation subtract

negative two equals b so now that I have this I figured out that

that b is negative two then I put it together leaving x y open

line will work in this equation and my final equation is y equals

we go we've figured out this slope intercept and so now I know

so that coordinate point would be zero negative two and we can see

it shows up like this because when x is zero whatever I plug in as

going to zero out so three x three times zero it zeroes out and

and then I have minus two so we see that that works out so this is

and we can when we see this now if we know the slope we can

intercept if I had this line I could generate that equation we're

straight line but if I had this but if I had this line I could

it begin at negative two oh I know that that's b and then I can

one so then I know that the slope is three and I can go the other

didn't have the graph I could say oh I'm going to begin at

point and then the slope is three I would go up three over one

plot a point and if I was doing this by hand you know paper and

points and get out the roar and connect the dots but what we're

going to do here is take a look at how we can write the code to

would need is two points and I'm going to do more than calculate

the full equation so we're going to look at how to take two points

equation so now we're going to put it all together and develop the

going to define our x1 y1 x2 y2 from two different points and

which now we're going to call m that's been the go-to variable for

moves so same formula but now it's going to be m equals y2 minus

then I'm going to define the y intercept which is b and that's

things happen in the first quadrant where everything's positive so

axis somewhere where x is zero so we'll connect this with the

know how if how do I if if I know a point and an x so an x y value

I solve for b a couple steps of algebra so then b is you know y1

b solve print out the equation so this will just print out the

that it'll display nice y equals three x plus four and I still it

as a float variable so three point zero it's not worth trying to

this is going to work for any number and then that way does not

to to make this work so there we go knowing these calculate the

then put it all together in the formula so then let's graph it

y points and I just kept these same points here so develop the

because we want to print it out so that we're going to print out

to go back to the graph that we were doing before define our x min

this line I can graph I can find two other points so I'm going to

in the first line y3 I'm going to take the x minimum and I'm going

plus b that's going to be my y value and then I'm going to take

that's going to be my other y value so now I have these other two

same as we were doing before and we have now I'm going to plot

and my x values are going to be x min x max and my y values will

going to plot this as a red line so if I part if I put an o in

separate red points but just r makes it a red line and now when we

there we go we calculate the equation and we graph the line good

I can have any other two points there we go there we go two three

we know this actually some of this would break but it might still

not a function there so there we go six and I'll make this one

eight just to show that we can just change these and all the rest

solve for m and b get our equation plot them here and when I run

so 1.25 x plus 0.5 and we see then it must cross the y-axis at 0.5

up 1.25 over 1 there we go all right so now we can graph and

do whatever you want with the two points you find

now that we've worked through the core skills in this unit let's

and I'm going to work through extra problems using the CoLab

you can apply these resources that you're building and use these

might come up in a textbook or in day-to-day life so we're going

extra problems here so we're looking at how we can apply some of

to real life situations and use this information to predict things

and real life or things that show up in a textbook that hopefully

so let's take a look at some examples here and we'll work through

a town's population increased at a constant rate in 2010 the

population had increased to 76,000 if this trend continues predict

want to notice that these are two x y coordinates time and

going to be on the x-axis almost all the time so if we look at

or 2012 76,000 so when we when we do this we have I could make

hey this is the beginning of this situation I'll call it time zero

these are nice even thousand so I just might call it 55 so time

is when we have the next year so that's two years later so two and

that we're recognizing these as you know x y coordinates that we

going to use that we're going to predict in 2016 so let's take a

hopefully you've already been putting together some things that

you know your co-lab notebooks so that you would already have this

know graph you've already done the imports this is the only thing

this for the two points and then we're going to calculate the

so x1 0 y1 is 55 x2 I would want that to be 2 and y2 is 76 so you

0 55 2012 76 000 and these this would then be the only thing you

because all the rest of this you'd have this you know already set

you know we're going to calculate the slope we're going to use

now we happen to know because we have x1 is 0 so we know that the

but that's okay we're going to print out the equation and then our

normally have this set at you know negative x min negative 10 x

maximum 10 so this is the other thing that we would change because

an x minimum of negative 10 so in time that would be 10 years ago

to worry about going in the past and making that negative 10 so

I'll just keep it at zero and we can you know look at it from

that's probably just as good you know we're looking at the most

and y minimum I'll also make that zero actually because y is

than zero would just be weird negative population they're not

anything all right or y maximum and I picked 150 because we

going to 76 and we're predicting a few years in the future so I

didn't seem like enough so we could always change these so all we

notice all the other things for the line that we're going to graph

printing out this line I'm still just using these x min and max

this line and we're just doing the line that way there's other

for how we're going to display this line this this works for

and we still keep our basic setup now here's where we're also add

have these and you could also comment these out and then use them

commented out about setting the x ticks and the y ticks we'll see

to change that and you might have the default x label as x values

can change these here time and population and I like I like seeing

situation so I'm going to put that in as true so we have these

could have your setup and have these all commented out that all

as needed and then there we go we're going to plot this linear

like when we run it so we figured out the equation so it's going

so we see this trend and we see it starts at 55 increasing about

so year two we see yeah I'll estimate that as 76 year four and

now this might not be exactly 120 we could we could plug that in

the equation you know what what is the value at six years but also

I didn't worry about changing the tick marks because the default

actually ends up being pretty good so there we go as a good enough

for you know population estimates you know you know the fact that

or something like that you know our rough estimate is going to be

and then we have our dice labels time and population the grid

we want to do this we can you know we can set up this equation and

answer all these questions there we go we see the trend and we can

that's about 120,000 all right now here's another one the number

cold dropped steadily by 50 each year since 2004 until 2010 okay

here so dropped steadily then we're talking about a subtraction

since 2004 is the first not first year we we mentioned this now

life yes they wouldn't be perfectly linear but a lot of times that

enough approximation and that's that's the idea for some of these

so now we have in 2004 875 people were inflicted and notice even

definitely see there's your xy value year and then afflicted and

so one of the things that we can do here well this is find the

things we can do is I can make 2004 times zero and then that's

so if we take a look at this 2004

I'll make that time zero and then this one I'm going to make 875

875 all right so now if it drops steadily by 50 each year all we

more point if this is linear so we can just do the subtract you

here then if we have year one then it drops 50 so then that's

that's all we need two points and we'll be able to do everything

the intercept and display the equation I might even well now the

anything so you know number of people afflicted you know okay how

of cases of the flu there we go so there we go time cases and you

shift enter or click this and we see this trend negative 50 was

negative 50 x plus 875 and a lot of times you would see this

and then subtract but that's okay now also this notice that you

to show anything why because it started 875 and you see 140

x minimum and maximum that's fine y minimum that's fine but y

maybe I'll call it like 900 so now we can see

the graph and sometimes you know you graph it and you say well why

look and see oh it's the window out of bounds here so there's 875

so there we go and we can use this to predict you know in in any

to get the exact amount we can you know for any any year we can

and then solve so that works and remember that even works within a

you know so the interesting thing is this plot dot show

um that needs to be last so if I do another print statement I

you know the same place where I print this I could print so

print this I could print so what if the question was how many you

the flu we might say oh look at this two years later you know we

subtraction so let's say like five years and you see given our

trend we can estimate it but we don't know exactly you know it's

can say oh about in about five five years how many people were

I want that exact number so I can go back to this and then x is

is in the print statement you can actually just do do the math and

print statement if you want but we can just do our math here

so we see you know our negative 50 times five plus 875 and that

be the value what would be the number of people afflicted after

have a more elaborate print statement to say something but since

question is now that that's 625 that that means something to me

that that might be you know could I estimate it exactly oh that's

can plug in the number and get it so we see you know some

we can answer some questions do some math for you know some

to this now all right all right linear function so here's another

afflicted so we want to know looking at this graph when is the y

be zero now rather than now one of the things you could expand the

until it gets to zero we'll come back and do that in a second but

is if you already also have your setup to solve an equation so you

this was the whole setup to solve an equation and print the

to do here we keep this equation variable the only thing we need

this case i'll put it here h 75 minus 50 x and remember it's

so that's by default set equal to zero so when we run it so the

that gave us a fraction but there you go so that's 17.5 so 17.5

so yeah now again fictional numbers i feel like you know in 17.5

going to be that nobody has the flu or something like that but we

and we can change the x maximum to 18 and then we can see this

according to this trend you know time zero and that would be you

case according to these numbers who knows it may in some

or 18 years that would be like 2022 and yeah i feel like that

nobody afflicted with the common cold all right but anyway you see

the these are realistic enough that we can use these to take a

and this is what we want to use this math for predicting trends

have so here's another one where they even give you the partial

all right so this is showing profit so this figure one is showing

in a given year x and they also do x represents years since 1980

zero is 1980 so then 20 so so 20 that would be 2000 and 30 that'd

we go now given these we want to find the linear function all

want to look at these i just want to find two nice points that i

estimating the points so i like this first one at year five it was

because it says y in thousands so therefore 10 000 would be 10

thousands then 10 000 would be 10 million in which case they could

but that's okay and then we have so five i'm going to shift it to

y value 10 so five ten and then let's find another nice point i

and we can plug in in these so i don't have that zero value and

at some point so five years then we're talking 10 million and then

then we were talking two four

okay so there we go we have that point five years 10 million 25

since we already see this graph since i made them 10 i really

a look at this and say oh yeah 12 is probably fine as far as a y

this to 30 or you know we can take it further so when i go to

going to be enough maybe i'll just make it a nice round 50 y

would the scale would be so much it would this would just be in

so maybe we'll just call that 15 or something like that all right

like the space in there we could just delete this and we have i'm

the equation so when we run it so here's our equation negative

and we see this downward trend over time and here's another one

change these tick marks maybe i could change this to the x ticks

is two enough yeah yeah maybe and then i can change the y ticks is

and we'll run it so we see we get much many more lines all right

we see we get much many more lines all right and with that we see

now just because these end up being rectangles more i might change

i just like that better and you know in doing this you can you

go there we go so with this then we can really see the trend

this downward trend down 0.3 so it should go you know over 10 it

here we go and then 38 years later then they would have no profit

that we're saying there we go so whatever this company is this is

and then and then there we go um

um 38 years later then they would have no more profit so who knows

you know our fictional company is you know something would have to

the prediction lines you know the the the simplest prediction is

line whatever trend would continue in a straight line unless

uh normal business normal day-to-day life things do change and

perfectly straight but this gives a often enough of a prediction

cases then we would change that to profit okay just just to make

so there we go another another situation and we can see where it

profit declining and that's enough years that you know 11.5

and then profit declining and yeah so this this company is

there you know they were in uh in sad shape all right and we have

okay so we see another very very similar we'll do we'll do one

2004 school population was 1700 by 2012 the population had grown

again assume assume the population is growing linearly all right

now how much did the population grow so this a is really just the

so 2500 minus 1700 is 800 so it grew that much and then the

anything per year the rate that's the slope and we can divide that

just plug these numbers in and you'll see that we can see the

time zero 2004 1700 2012 2500 and i might even leave the hundreds

you know more exact we know that the population would be

so 1700 and then y minimum and then we have 2500 so then after

yes then my y maximum on the oh i put in the wrong so there you go

1500 and so in 2012 so time zero and then in 2012 so that's eight

so eight years later then it was 2500

so here we go okay so we'll have that and

x minimum and then the x maximum so let's just make it a nice even

and we could always change it if we need to y maximum 2500 but

further so let's make it 3000 and see if that's going to be

there we go and do i want the ticks for right now see if you

as we saw it will still give you the it will still give you the

default whatever python wanted to calculate and where you can

want there we go and for school population yeah all right so we

and maybe that's you know depending on what you wanted to estimate

the grid might be fine and you know if you wanted to you could

i think that works out so growing by 100 every year and we can use

probably growth per year was 100 there we go 100 x so the slope is

and there's the equation for the population for two years after

so we see you know we've done a few of these i think i think we

population model but there's some other form there's some other uh

in other tech all these all these are coming from that uh textbook

trigonometry because that's a lot of good business applications in

that just the notation g equals f of p the amount of garbage g

population p so we can go a g equals f of p and that's how we

of population and what that tells us is population is the

the dependent and population very often is an independent variable

trends so we look at that and we see the same type of thing the

thousand and produced 13 tons of garbage each week express this

so now if we have this one of the things that we would guess is

population then no people would mean no garbage so in our y equals

zero and notice we because we only get exactly one point here

so if we wanted to set this up you know we want to look you know

python scripts to solve this but if we want to set this up we want

how we're going to set this up 40,000 is going to be the

just call it in thousands and then 13,000 is going to be the

just call it in thousands and then 13 tons same thing i'll just

and then we say well that's one point where do i get the second

situation would i think that it would start at zero zero and if

back and say so here's zero zero and then remember our x values

so we'll call that 40 and then that's going to be 13 because it's

and therefore the x minimum zero fine but x maximum 40 is probably

um 40 is probably not even enough because we start with that and

and we can expect that it increases so then let's say the x

it 100 and we'll see y minimum zero now 13 maybe make it like 50

enough of a y maximum so there we go and we can just write this in

so thousands of people tons of garbage

so we have this trend here there we go 40,000 generates 13 tons of

and we can see this trend here so there's the rate here 0.325

0.325 x so remember the x value is in thousands so every thousand

according to this fictional scenario makes 300.325 tons so that

about tons 2,000 pounds so that times 2,000 would be about 650

but then if that's thousands of people so that's each person

less than a pound of garbage maybe so again this is this is not

you know a couple numbers that seem realistic and then what's the

would read this as f of five equals two so notice that the

that five is the input and any input or independent variable

even though we call them p and g here so that input five and then

so we would expect then that that means that when we're talking

then the 5,000 people would produce two tons of garbage so that's

five is the input two is the output so in our x y scenario here

that's what that would mean 5,000 people would produce two tons of

all right and so let's go over here to 89 and another one you see

alphabet soup of of this as people use different notation notice

of i would read this as g of a so it's a function of this input a

dirt d is needed to need to cover a garden with a square feet so d

and we read like that g of a and the common notation you'll see in

f for function and then often we just go through the alphabet fgh

talk about in python and pretty much across computer science your

have a much better name so you know that's something that computer

but nonetheless you'll see this all the time you know g of a so

input that's the area and then given that area then how much dirt

square feet area requires 50 cubic yards of dirt so that 5,000

to do any calculations to this but i just want to show you that it

so that's just how we would write that given that input you know

first then therefore how much dirt we need and that that would

write that there's we don't need to do any math to that right now

one so 100 cubic yards or 100 square feet of garden requires one

all right and number and question 90 we'll see just another

function the number of ducks in the lake after t years after 1990

five is the time years after 1990 so in 1995 there were 30 ducks

years so in 2000 there were 40 ducks in the lake and given these

find the overall equation plot the points graph it you know make

the notation here and this is what that means all right now 91 the

just answering these questions it this won't matter but this is

but we can still answer these questions because each of these is

each of these is still going to be the same now notice h of t and

you know again another way of expressing this so as x y values x

the h of one is 200 so that x value is t for time in seconds so

after two seconds it's 350 feet up and again this is not linear

the same things that we've been doing actually won't work for this

meaning of each statement here and 92 the one to one is this is

x value like each x value and y value pair up like that neither of

parabola because we see x squared and we'll talk about parabolas

that there's actually positive x values and negative x values that

so that's why it's just not one to one but it's still a function

i don't want to talk too much about that we'll get to these these

want to mention that in case you're looking at that and saying

so in number 60 now this is also not linear and you see the height

something like this after t seconds here's the function h of t is

seconds here's the function h of t equals negative 16 t squared

so that's showing that you know it isn't linear in fact it's a

around comes back down okay so the domain anytime we talk about

so the domain is all the x values that are possible here and then

the range is all the y values and this actually has a limited

parabola that goes up and then comes back down there's a maximum y

after that y value there's no more like y values beyond that are

it'll just never get to those numbers then the domain

numbers but realistically in the context of the problem time zero

domain so you see mathematically these other numbers exist and

but realistically if i'm launching something it starts at time

can talk about t minus whatever and talk about negative values

you know and this is second so if it's like t minus 10 then it's

10 and that's 10 seconds until the launch but realistically we're

and then the other one would be it does go up and comes back down

the ground again and beyond that it's not really within the scope

look at how we can solve this just because even though that's

i want to find out what that t value is when does it come back

negative 16 t plus 96 t we can go into this where we're going to

t we're going to make it x just because i already have x as a

and make a t that's fine so negative 16 and again we're going to

x and that's always the this is the equation set equal to zero and

makes the zero now negative six would not be

plus 96 t oh because i put minus

so it only gives us zero what if i have

so i was doing this with things that only had one solution so

just printed the first one but this does have two solutions you

could get really fancy here and then actually just make another

finite set it's not zero six as a coordinate it's x equals zero or

this zero and i could have another print statement

there we go to show the first one and but this is just a matter of

output to look okay so i'm going to go ahead and i'm going to go

show the first one and but this is just a matter of you know how

and we have these you know we also have the you know which we

you know loop through this to get solution to get multiple

x equals zero it's on the ground and then x equals six so that's

that we could graph this and we see that it back on the ground at

bit ahead because we'll talk about quadratics later so yes we will

graphing this because we'll get into this with quadratics but this

at six at six seconds later it's back on the ground so some

turned and i'll tell you this that you know halfway through these

through it's halfway through it's uh at its maximum point so we

what is that maximum point and we can do that math negative 16

so halfway through that so that would be three would be x and you

that you know we set up the situation where i have x and then i

zero six it's back on the ground so three seconds later it goes

so realistically then the domain for this function as that's

you know mathematically it exists but related to this situation

values that mean anything for this are anywhere between zero and

because everything else doesn't really mean anything for this

this then if it keeps going down like it doesn't go back down into

are between zero and 144 so that's my domain between zero and six

range between zero and 144 so you know these are some things we

when does it equal zero and plug in some various numbers

all right so if we look at 61 just again reading reading is these

any code for this we have this situation the cost of making x

and the fixed cost is determined when zero items are produced so

zero so then if x is zero that 10 x drops out and then we're left

it's b for begin you know when you know before we produce anything

now maybe this is some sort of business and 500 is like rent and

even if they're there they didn't make anything they still had to

and then they pay ten dollars for whatever you know for whatever

so then the cost of making 25 items would be 10 times 25 we just

all right and supposing the maximum cost allowed is 1500 what is

have to set this equal to 1500 now if we take a look at this and

so if we have 1500 equals 10 x plus 500 and notice i'm not this

worried about doing anything with this yet but then what do i want

to zero so if i subtract 1500 from both sides then i get zero

because if i subtract 1500 from both sides and so that's set equal

that's where i could plug this in here so you see you can work out

python code so 10 and then here i do need 10 x minus 1000 so if

one i know there's only one solution so if i keep the other one

i know there's only one solution so if i keep the other one that's

so whatever it is we know that we could only make a hundred

because you know maybe they only have that budget you know

hundred of them their total expenses were 15 total expenses were

you know then we have to go out and sell whatever they are i don't

whatever they're whatever else they're making all right

okay so we and we see these same types of things we i just want to

that at the number 44 at the start of the trip the odometer on a

at the end of the trip 13.5 hours later the odometer read 22 125

there we go assuming the scale is miles so average speed we want

anything like a rate like that that slope and we have two values

21 395 and then time 13.5 and then the miles and we can go through

about graphing this or not it still will output the slope which is

the average speed and that's where average speed it would be a

we know especially over 13.5 hours the speed's going up and down

sometimes faster sometimes slower but we get that average speed we

they're not going to that exact speed the entire time all right

gas station to fill up his tank and he looked at his watch the

started pumping gas in the tank exactly 344 the tank was full now

minutes especially with these prices these days i would be

money but he noticed he had pumped 10.7 gallons so surprisingly

but what is the average rate of flow of the gasoline so again time

we want to call it minutes we can call it four minutes and then

to gallons per minute or and again same thing plug it in you know

because we already did that subtraction is just 10.7 divided by

you have these scripts and everything available but some of these

even before you look at those or you could break it down into

in four minutes and you know the rate of flow per second but

gallons per minute is probably fine so we see all these situations

how we can set these up and the things that you're building that

an equation even if you know the equation is something else and

algebra you know because that's it you want to get it equal to

from earlier that one step of algebra so then you have it equal to

so you'll have these things set up solve any equation or you know

rate all that so you know if all you needed was the rate you know

bonus but it'll still also answer that question for you so we want

ways that we can interpret these questions and solve solve the

of some of the keywords and hopefully you see how you can apply

that you come across you know look look for situations in in your

you can apply these all right so hopefully that was pretty useful

factoring in algebra then we're going to talk about dividing out

something that we're multiplying and therefore i'm looking for

am i multiplying so when i see a number like six then i can say oh

times three and here i'll put the dot for multiplying to separate

know when i when i look at a number you know like 15 then what are

three and it's useful to be able to think about numbers and what

helps us reduce fractions and simplify other other terms so for

two numbers here supposing i had a fraction that was six out of 15

reduce that fraction to lowest terms there's a better way to write

that they both are divisible by three that three is a factor of

divided by three is two and fifteen divided by three is five and

is five and so this is the that reduced fraction and i got this by

factors and i divided out those common factors so just like other

thing to the left side of the equal sign and the right then it's

if i do the same thing to the top as i do to the bottom but that

and dividing adding or subtracting doesn't doesn't work so if i

reduce my factors i can even keep going supposing i had something

and supposing i didn't recognize that 20 that six was a factor of

were both even so i could then divide by two six divided by two is

i could start reducing again recognizing that common factor or

oh three and twelve both divisible by three and then i could say

twelve divided by three is four so again i factored things out and

now a lot of times you know we see things and we don't we see

especially bigger numbers we don't right away think about the

do long division in a couple minutes i'll show you how to how to

to find the common factors that way even if you don't see them or

do i go through all these possible factors i'll show you how to

to do that find those common factors but this is one of the big

and this is more related to other algebra topics we might not need

code but if i have something like you know 4x plus 16 something

the four the 16 have a common factor which is four and i can

and we can divide it out like this way so this is another use of

because i see these common factors and i divide it out 4x divided

put x and then 16 divided by 4 is 4 so and again this has more to

applications but i want to show you this because this is also

this i see the common factor and i can divide it out and it looks

the 4 4 times x is 4x 4 times 4 is 16 and get back to that so

one of the other things we're going to do is talk about factoring

and that also i'm going to show you how to do this but then we're

you know how to find common square factors and then factor them

if i have a square root and sometimes we have square roots that

not perfect squares but i can factor them and sometimes that helps

cancel out with other things later so if i have let's say the

5 that's great square root of 24 almost 5 it'll be some long

but if i can factor that out recognizing these common factors and

just any factor i want a square factor so if i remember that 24 is

4 is a perfect square and then i can say so i want to identify

the ones that are square factors and four square so the square

and then since i took the square root of four i still have that

if i can't square root the whole number i can square root part of

sometimes then even if i can't factor out something like the

something like this and that's the simplified square root this

that's really useful now again bigger numbers things that you

this a perfect square we'll talk about how to write the code to

divide them out and even use the simpy library to display it in a

the essence of what we want to do with factory recognizing factors

even if you're going to write code the more you remember your

things will be it just makes things you know come together a

looking at numbers thinking about what factors made that number

factors that we could possibly divide out and so let's take a look

the code so here we're going to do the same thing we were doing

the common factors dividing them out and it's going to work a lot

because it'll go through and easily find all the factors for you

the modules operator so modulus that is the percent symbol so on

five key so what that does is that finds the remainder so i

five modules three it's like five divided by three but the whole

want the remainder five divided by three three goes in there once

that's what it'll print out the two and if we want we'll just

just to show you another one so 31 module is 10 so 31 divided by

matter it's a remainder of one so when we run it it'll output one

for finding factors because a remainder means that it's not enough

but if the remainder is zero then it's a factor and that's what

find factors so i just picked this number 12 and we're going to

in our loop so i i made this variable just called test factor so

one to the number plus one because i'm dividing so i have to make

at zero it'll give me an error and i want to include that number

one otherwise it won't do that last number so for test factor in

here if number modulus test factor is zero then it must be a

out the test factor so that's what this is going to do given my

through that and try each number and see if it's a factor and

it see it prints there we go one two three four six and twelve all

12 so how can i reduce fractions to lowest terms so let's take a

variables here my numerator let's call it 12 denominator 24 now we

to reduce to one half and i'm going to define a variable called

a factor and who knows for some things that might be the greatest

my numerator to find the greatest common factor all right so for

plus one because that's really it i want to you know i want to go

denominator beyond that the numerator it's not going to really

module so we'll do this line here if the numerator modulus test

modulus test factor is zero so for each of those if it's a factor

now that's going to be my variable factor and then once i have

so i'm going to do numerator divided by factor and i want to cast

going to look nicer when we do you know the sense of this is that

factors so this is going to look nicer when we display it so once

you know it could stay at one or it's going to get you know the

i find that numerator divided by factor cast as an integer and

divided by that factor cast as an integer i'll call it d and then

original numerator divided by denominator and then the reduced

this and we see that the original was 12 over 24 and the reduced

any fraction and you can even copy this code and use it within

or you can put this as its own function and deal with it that way

functions out of a lot of this code later on in the course so the

adding it let's just do this so we had the decimal to fraction

we had an input rather than just have the number here i have this

a decimal number to convert and remember then that input gets

string and we've done this before converted to a fraction so i go

input you know how how many digits minus one well convert it to an

an and minus one because that it that will come in as a string

so there we go then that becomes my exponent and then i'll still

which is n and then as a fraction my numerator is going to be the

10 to that exponent so what that's doing is taking that value and

so there we go so the so n has all those digits including the

this line for the numerator moves the decimal place and then the

exponent so now i have an integer for the numerator this will end

the denominator and now what we're going to do is just like we

that fraction so same thing factor equals one everything else i

finding the greatest factor dividing it out and there we go

original number that the person entered and then print the

enter a decimal number to convert so let's convert something like

and it took it and it reduced it to one eighth which 0.125 is one

and we can run it again just to see that if i have a point zero

there we go the decimal and then that's one fiftieth as a fraction

into other things that we've been doing all right now we're going

factoring out square roots so this is all good to reduce the

root something now this one if i just print math that square root

the square root of 25 here is five great but what if i have

now that doesn't work out nicely and by default it'll just give me

if i want to factor it and this is how we want to factor it i use

square root of 12 because underneath the radical 12 is four times

square so the square root of four is two and that comes out and

else i can do with that so that's the factored square root we want

and divide them out the square root of that there we go so here we

make it user interface but i just put it here n equals 12 and max

that's the one that will change and here's i put this as the key

get to you know if i have my loop where do i start where do i stop

floor of math dot sqrt of n so i'm going to take the square root

that's going to be some weird decimal number now i just did this

one i probably could have done math dot ceiling also but i just

root and whatever you know math dot floor so drop all the decimal

drop them all and then add one so that's going to be the upper

here so for maybe factor in range again starting at one zero would

if n modulus maybe factor squared so you see i'm running through

you know each of these is the square root factor so if i square n

factor squared if that's zero then now that maybe factor is now my

and we're going to cycle through all these you know divided out

some factor that's what's going to be on the outside so then i

my original number the square rooted factor square factor and

of these out here like this because we're going to build upon this

step but to show that square rooted factor square factor max

that's left over so for 12 since we had that example earlier we

rooted factor is 2 square factor is 4 and the integer is 3 even

was just text it actually comes out as a float so what we have is

on the outside you know 4 was a part of our in-between step and

rooted factor and then the integer that matches up with 2 coming

factor that's the greatest one and then 3 is the integer still in

to use simpy to make this look nice import math import simpy and

so now i have this there we go and same variables i had before

square rooting it but then you know find a one more to see where

the like the greatest square factor i could possibly have so then

factor and then square they're all starting out as one and i'm

so slightly different variable strategy so if maybe for maybe

if n modules may be factor squared then we have the max factor is

all right and then the other factor is n divided by max factor so

the factor that i divide out because i'm squaring it here and then

the radical is that original number divided by max factor so then

this look nice we cast them all as integers so the square root of

want the square root of it cast as an integer and then that's the

square root other factor cast it as an integer there we go it i

we need to cast it as an integer here other factor and then the

variable output and we see it's square root times simpy dot square

that output looks like in simpy and this presentation here we

uh that might not make it look as nice but this simpy output so i

and then when i output it here there we go it factored out as two

and if i put it back to my other 24 that we were talking about

work out and so it factors that to two root six because it's six

root of four is two and that's what we're doing the types of

through we just want to translate that into code you know i'm

of that number and which ones are square perfect squares so 24 i

that are perfect square and i find that it's four and then the

the simpy makes it nice so we can do this to factor square roots

this we can divide out common factors we can reduce fractions we

and these are a lot of the things that you know we often do by

now you have the code that you you can do this

you can do this now that we've worked through the core skills in

let's look through some extra problems and i'm going to work

extra problems using the colab notebook so you can see how you can

you're building and use that use these to solve problems that

in day-to-day life so we're going to go through some more extra

to do a walkthrough of the first certification in foundational

there's five of them we'll do three of them for this course and

but they were i also designed them to be standalone that you know

you could walk through and be able to do all this still though i'm

uh maybe you already did this and you want to check some things or

uh turn out as you expected or you're stuck somewhere so you can

in in this video and see so each of these steps lead you towards

we've built upon each thing so the first thing you're going to do

notebook this notebook is read-only and this is what gets shared

so you make a copy and that copy is going to be in your google

uh and it will be in a folder called colab notebooks by default

it in whatever folder you'd want it to be you want it to be in and

for this course you're building your resources so you're going to

drive uh especially if you have other colab notebooks or something

folder for everything uh in foundational math one you know or

to call it that so you make your own folder you're putting all

have so you walk through as you work through these you'll

but then you also have some code that you can reference so you

i already did this and then each of these i have as a different

click on the triangle here to expand the next step and with each

then up here you have a table of contents that you can go through

you you can jump to whatever part you want you know you can add

making each of these a section heading so you do need to acquire

this is actually self-contained you're going to write the code and

code that you wrote and see if it's correct or not so this cell

maybe for other things you might do in the future you might take

this other library as this raw.py file in github and really save

and then you'll be able to access it so this i have the note here

at the beginning of every session and the only reason i said may

away for two minutes and come back don't count that as another

they do time out the runtime will time out after 30 minutes of

typing that means no running any cell or even if you are really

hours so then you you know you might say wait i haven't i've been

time out they have that time out too all these are security

leave it open for you know indefinite amounts of time all right so

this library and you're going to run it and we'll take a look at

and as this is going yep it'll save the library locally and then

so it'll say requirement already satisfied even though it said

i do this i already it was already okay that's just what it prints

test pass go on to the next step then you know you're good so it

you know now we have this library available and each step will use

four yes i do have add subtract multiply divide i know you know

know how to do them in python but it kind of walks you through

the notebook you might be already familiar with it but so you know

so we go through just you know two variables and then you're going

with the comment change the next line now this was just add so

a plus b and i'm going to click run and you see two plus one is

passed go on to the next step so we go to the next one subtract

c minus d and i could hit run or i can hit shift enter which i'll

so this is just getting you familiar and then each of these

so you've run that first cell and then you have the test and

code and tell you one of these steps in a little bit i'll do one

and you'll see what what it'll tell you so here we go so we'll

multiply in python like a lot of things is the asterisk same key

the eight so just hit shift and it's above the eight there we go e

passed and then we divide now when we divide here g and h now they

happens to work out but if i had them as numbers that didn't work

defined as integers then i would have to do something else because

and if i divide it won't work out the answer will be a float but

this you know i just define g equals h equals and when i divide

time about whether they're integers or not it's just going to

so you can and a lot of things that you might work through here if

you might just have things where you define your input or your

actually just prompt for input and so in that case you don't have

when when you run it it will prompt for input and then that could

that you want that way you don't have to worry about changing the

a positive integer it actually could for what we're doing here it

just put that there and then we just cast it because the input

and then i need to cast it in this case as an integer to do some

all right so i need to do an integer or a float and then the only

following that model i have string b and a lot of these have you

there that won't break the code but you do need to change it all

going to be this and you absolutely could just copy this and paste

what if i just change this now that's really all you need to do

just for me i feel like you entered one i want to say enter

and i want to cast it as a string as i want to cast as an integer

that now so i don't need to tinker with this at all so i might not

to another or something like that that you know we'll see all

there we go so what i was saying is if i wanted to change you know

not be central even if it gave an error it shouldn't stop you all

casting it as an integer and adding it we're fine if i cast each

divide it would give me an error oh but i didn't want to worry

defining them as integers we're just adding we'll cross that

and we can get input and cast it on the same line so notice just

i'm going to do the same thing here i'm going to copy this whole

because that's really what i want to do

all right now here's one where i really don't need to do anything

change that to another now just the way i have it

see it actually just tested a bunch of these things because you

part of the line and you know i really didn't need to change that

your into your input should be this now my point here is this will

from going on to the next step so you know maybe that's it maybe

in there that like why why did it say it was never it looks

and maybe you're watching this video because that's where you were

from going on to the next step and in fact you know that's fine

what was different i'm not worried about it on to the next step

okay well maybe i just should have just said this you know you

check it and then see there we go and then and then you can make

is that this is supposed to be just you know a teaching tool check

move on but you know not to get you caught in something that you

moving on all right now as i was mentioning about dividing casting

number allows for decimal places and most of the time i'm just

anyway because just in case somewhere along the way i need the

situations where i might definitely need an integer certain simpy

that soon enough and counting but other other numbers that i'm

to cast it as a float okay so again prompt and then cast as a

the same thing and that's okay you need you know if you copy you

and need to know exactly where to put it but there we go and we

number enter a number and see now we will divide so i definitely

i didn't even say integer i just said a number so this is going to

enters in and then when we take a look then you see it's waiting

three enter number four and 0.75 which is perfectly fine because

good times all right so order of operations and yes python knows

so in case you forgot you know here's the acronym PEMDAS

uh parentheses exponents multiplication and division addition and

you can and then remember the python syntax for exponents is the

what i wanted you to do is test your knowledge of order of

about actually writing code here then you're going to put your

really think about the code see if you can do this in your head

and then you're going to put that here well you're going to put

when you run it python is going to calculate the actual answer and

you'll see all right so let's look at order of operations

and then multiplication and division as they come up so

uh four times two and then we have 14 divided by two so you can

multiplying and dividing all day so as they come up four times two

divided by two is seven so then we're going to have one plus eight

eight is nine minus seven is two so then we take two now we're

two to the third power is eight and so there you go put your

and there we go actual answer is eight your answer is eight code

okay so all these things and remember you can just like the other

put print statements you know all whatever math you want just

statement python will do that for you so those are some simple

okay so remainder remember when we divide something that does not

and the modulus operator looks like a percent sign but if i'm

it tells me just what the remainder is there we go so as i have

four is two but with but as a remainder so two and then two left

would give me two because that's the remainder or 14 divided by

so if i do eight a modulus b and again it's the percent sign on

key as the five but you have to hit shift so 14 divided by six now

by six is two with a remainder of two so that's what we're

printout there we go now actually we could we don't need to do

well supposing if i had 15 divided by six so six times two is 12

and so a modules b would be three if you wanted to you can test

this particular code is testing did you do it correctly here so if

try some different things whereas then if i go to 18 that works

be zero there there would be no remainder and we can use that to

statement here so if i have 20 modulus five so that's 20 divided

so 20 modules five the remainder would be zero so if that zero

a factor so you see we're using the double equals to test it so

we're going to walk through this and we're going to cast this

factoring really only works with integers if you know we can

you know if it's a factor there's going to be a nice integer in

enter an integer so input cast it as int and that's going to be

next one enter an integer to see if it's a factor and again cast

to be test factor because i don't know if it's a factor yet and in

number now remember whatever this is if it's true it's going to do

it will then it's going to do this but we're going to change the

modulus test factor and then now if that divided by that gives you

then we can say true it is a factor and then or else it's false

it's false so there we go and enter an integer so let's do one

and we can run it again and let's say make it 12 3 true so there

things and we can use that to test if something is a factor that

but we're going to take that and we're going to build that into

factors all right finding all the factors of a number so very

not asking for second input here we have the first input and then

so now i have my for loop for test factor in range so i'm going to

let's look at this backwards here i'm going to take this range and

there it would start at zero but i can't have that because that'll

right away so i'm going to start the range at one and then end the

because normally the range it won't do that last number so i have

whatever that number is i want to include that all right so now

it's going to loop through everything every number in this range

is going to be one and then the next num next time through the

so it will just loop through how for that many times yeah for

all right and this is what we're going to change the if statement

so number is our in is the original number we want and each of

want to do with it we want to say if number modulus test factor

equals zero just like we were doing before we're just kind of

so i get this number and i'm going to loop through all these

and i'm going to loop through all these factors from one up to

by test factor gives me a remainder of zero then it is a factor so

all right so we run it and what do we have let's say 12 12 is a

and you see it'll print them all out 1 2 3 4 6 and 12 and we can

supposing enter like 17 1 and 17 because it's prime so that's what

the factors here this type of thing we will revisit this at

to lowest terms okay now what about prime numbers so a prime

are one in itself so we're going to and then anything that's not

so if i have you know five is prime because the only way you can

five but six as one you can do one times six but you can also do

you have those other factors it's composite so here now we're

and again integers because we're looking at factors and i'm going

variable prime or comp say being prime and then same thing range

two because one will go into everything and that's not what i

does that won't tell me anything so i'm going to start at two up

again not including the number because this this this time it

didn't add the one there because again same reason i didn't have

we know that those are factors but i want to find out from two

if it's a factor and very similar then if number we're going to do

always go back if number divided by test factor equals zero there

variables here you always double check just in case the variable

different way of doing things but yes if that test factor works oh

so this would have given me an error

and and that's the thing with with reusing code yes we know we're

and paste because that helps you know if it's a whole block of

it is exactly the variables and everything that match up correctly

people do too much copy and paste and then they're lost because

so this one we call it test number so if number divided by test

of zero then that means that is a factor which means all it has to

means the number is composite and it's only prime starts out that

prime only if it goes through and we never get a factor here

so let's take a look if we have this and enter a positive integer

composite right because i can also do two times two and then let's

prime and that way you can check all kinds of other things you

is it prime is a composite there we go and you know now that you

change any of the code when you run this it'll you know it'll uh

interesting things reciprocals so writing it in math it's the

reciprocal of two-thirds is three over two you see it just flipped

the reciprocal of five is one-fifth because any whole number it's

denominator of zero or sorry sorry denominator of one so that'd be

it's one-fifth as the reciprocal and you can multiply a number by

there we go pretty good so and there we go zero has no reciprocal

so we can write the reciprocal whatever the number is we can just

there that's actually all you need to do to find the reciprocal

and some of these might be decimals but nonetheless this will find

and we see our output here enter a number and what if i have the

um five see point two but that's the decimal equivalent of

all right so just putting this in there showing you what you can

other things we can do with the input is supposing i wanted to

and are two numbers separated by a comma oh well now but i want to

so i can split it so this i'll store it as the variable nums still

and then here this is actually going to be an array in this case

to split it at the comma and then now this array sp has two

i'll cast that as a float and then i have sp1 so that's it you

by a comma split the input and then then cast each of them

and notice the code here just says cast it as a float so float

over here there we go and then what do we want to do and that the

the two numbers so that's why we cast them as a float divide the

by b all right so there we go split it cast it and in this case

okay so and our two numbers separated by a comma and let's see how

um this one will be a repeating decimal of eight comma nine

there you go i'd notice if you have just one number repeating the

that number over nine so point one repeating would be one ninth

two ninth and this one eight over nine so there we go okay now

is building up to you being able to factor square roots but let's

so factor multiplied by itself is a square and it is kind of like

sides would be the same or all sides would be the same so when i

they're the same and so is that number times itself and we would

the second power okay so all we're doing here is you know we're

getting the square so in this case it's going to take the input as

it there we go okay there we go number square all right five i'll

okay so let's go the other way square root function so anything to

it's squared but what if i knew the result and i wanted to square

of my square yard but i want to know what's the side side measure

first thing we're encountering here that is not built into python

library and this is where colab really helps you out because a lot

through some other things to install things you know in other

math and you don't have it behind the scenes it works you just

all right so a number to find the square root definitely going to

all right and how do we get the square root of that once we import

sqrt and we call it a number n

especially you know think even if things don't come up

open parentheses one two close parentheses one two we got it so

and we see here and remember to find the square root so we'll do

about 16 square roots four and we can do another one that doesn't

and it'll give you a bunch of decimal places

i remember yep so 4.5 yeah about there we go okay

okay other cool functions the floor function this is another one

so the floor function so we have a floor function which just drops

and there we go drops any decimals and sometimes called integer

so even if it's like 3.9 it's going to drop and just go down to 3

we have the ceiling function which as you can imagine if floor

is no matter how many what decimals are even if it's 0.1 it goes

round the one we use possibly most often round remember 0.5 and

anything less than 0.5 0.4 and down rounds down so we're going to

function here all right enter a number decimal place and we're

math dot floor function here math dot floor of n so enter a number

to call it 5.99999 and when i hit enter yep and that's the floor

drops down to five so there we go and remember supposing you went

you know i'm upside down ceiling function supposing i wanted to do

um 7.1 because that should round it up to eight and so that code

what these directions are it'll tell you oh you should include

end up trying different things it doesn't prevent you from going

either if you spelled it wrong or something or you did something

floor and go nine point three

right good drops it okay so we're going to put some of this

all right so a few things we've already done we're going to import

we're going to enter an integer to define the greatest square

i'll call that max factor and right now it'll be one and then the

stacking on these functions here uh the math dot floor of math

but then we're going to add one to it so we want to find the

number you enter it in we're going to square root it that's going

decimal places and so we could do the math dot floor and then add

now i realize that i could have just done math that ceiling and

for whatever reason i did this i like it so i kept it but there we

so that gives us the upper limit square root of that number and

number now what do we have now in my range from one to upper limit

that was the square fact the square root factor i square rooted it

that range so it's going to go through and maybe factor is going

be two it's going to be three you know up to this upper limit so

equals zero so that's it these these factors if i square them is

that out and if that is then max factor you see it's not going to

you know that's and that's the idea with some of these change one

think about what you're doing then we see yep i'm going through

and then finding square factors if n divided by maybe factor

of zero then max factor equals maybe factor so that's what we

so and again knowing that these were the square rooted so for our

we're just printed out squared

so we'll see this here so if i have something like 12

all the greatest square factor is four because two you see it

that and that's a factor of 12 three squared is nine does not work

here okay so here we're going to build upon that code and then

because when we look at factoring out square roots that's kind of

when you have a square root that doesn't work out perfectly but

that's what we want to do here so i still have get the same

all these variables equal to one because we'll change those square

all right and then take a look at what this is doing same thing

got that same upper limit and so still maybe factor is going

check for square factors just like we did before

there we go and in this case max factor equals maybe factor

squared there we go and then other factor is n the original number

there we go so what we have is the result is going to be that

this times this and we're just looking at how that divides out so

you don't need to change anything because i i thought that this

a little bit complex to try to follow like what what i was

through so i didn't didn't want to have you have to change

going on here how we're dividing that out and then when you run it

to factor maybe i'll do the same 12 here see 12 equals four times

let's say 50 50 equals 25 times two and again that first part is a

there we go the last four steps prepares you for this so factoring

of what we want 12 if we look at the square root of 12 is four

perfect square the square root of four comes out and it becomes

kind of building as we're building this code to do is this

i'm finding the greatest square factor and then that comes out

two two is on the outside now all right and in doing this we're

we have square root and for our final output we're going to import

going to give us a nice output at the end that looks like this so

that's the thing you know python outputs a certain way and then if

that looks more like math symbols that's what we have symbolic

similar input without the radical and our square root factor using

dividing these same type of thing where we're finding maybe factor

now in this case i kept it like this maybe factor squared so the

max factor squared and then we divide so now we do the output so i

yes and i realized that up here there's maybe factor squared and

rooting it again and the key is casting it as an integer because

point oh did not want to do that there yeah because this will be

okay so that's the square root and then remember other factor out

that max factor and if there's anything left over other that's

that as an integer and you know that's it i'm just taking this

as an integer and you see that you can do that with variables so

and now the output is square root which was this variable up here

square root is different than math that square root and using this

be my last i want to be that to be my last thing is this output so

this for the test that's why output the word output shows up here

but notice it's not a print statement it just says output and the

and it will display it so these types of simpy displays sometimes

okay so what do we have uh let's take a look at our um

let's take a look at 50 as we were talking about before

so and just because this came first that's why it says test passed

but 50 factors out to five root two

and if we run it again then we have you know go back to our 12 12

and we can see how we can factor out these square roots

so there we go this is now one of the things that you can do

and this would be a bonus we could do this later is we already

you could put all this in a function

you know all you know you don't need to the testing part but the

the testing part but the output you could put this all in a

for you so just like you have math that square dot sqrt to output

decimal you could actually make your own function that would do

you know like this so some interesting things that you can do and

you to see the build-up of this and how we can use these you know

to factoring a square root which is something that's you know

useful for some math you get into and then you can write code to

all right here's some rounding just interesting things so if i

whole number if i have something like this if i round comma and

me how many decimal places and if i have this round and it's a

big numbers so that will give me that many zeros so there we go so

millions the second number rounded the sixth decimal places so and

my print statements print round and the first number six zeros so

and then the second one print round

be and i wanted three decimal places so it'd be three

there we go and we can print so that there's my six decimal places

i don't want it around to the next integer it already is an

14 million 500 thousand rounds to 15 million and then the next one

all right there we go and fractions decimals and percents

percents so we could see here and we've done some things like this

here we're going to take the input as a string see i'm going to

the input as a string and as a string every string has this

string so i want to take the length of that string and then i need

now remember the person who enters this in is going to put a

i have to subtract one because i don't want that decimal place i

without that dot all right so that's going to be the exponent and

now my input digits i'm going to cast that as a float so the

10 to that exponent and i wrote it all out just to make it clear

anything with three decimal places it's going to be times 10 to

going to move the decimal place over and then my numerator is

and the denominator is that 10 to the third oh 10 to the exponent

oh 10 to the exponent there we go so there's my fraction and this

anytime we're converting fractions but now we're just writing code

so how many decimal places that's really what this is essentially

how many decimal places am i talking about and then that number

places 10 to that exponent and the denominator is 10 to that

because now that i have it as numerator denominator now i actually

go back to n if i want or i could put numerator times numerator

one but i could do n times 100 which it's always times 100 because

and so that's it that's the first two digits are there are your

the percent and we can run it so if i have somebody enters in

five there we go so the decimal is point one two five the fraction

over a thousand and the percent is twelve point five percent

so there we go converting any input fraction decimal percent

percent and this is where i was saying that you could you could

this you can define a function here where you can execute a block

so you know here's the function definition d e f and then the name

open and close parentheses that may take input but they don't have

four spaces one two three four so d e f name of the function colon

it can be as many lines as you want and it can have comments just

and then there you go here's one something outside the function

it's then called it it won't do anything until it's called here

so i could have this function definition and if i took out this

again so now it'll have this is outside the function then it'll

and then back outside the function

there you go now change the name and remember to call it so

there we go and then we want to run it then change the name to

because it can't spell function without fun

so if i just change the name i'm gonna call it fun and now i'm

so that's what is the function call is when i you know when i tell

now there you go same thing to show you how to define a function

and then here's one with input

so it can take input any input we're going to call an argument so

um now right here it's telling me that it's uh you know called

it's going to take this input which will store its variable name

here and it's going to print hello whatever the name is now here

at the end of your function skip a line at least one line so then

definitely not in the function and then the input and casting it

and we'll keep it that that way what is your name and notice

indicate a new line so now when we run the greeting you see it's

and i'm going to pass that into the function so now that function

going to be this variable and then it'll do the greeting so here

hello there we go and now now that you see this working this is

this because it's like remembering where these are

so you know greeting and then when we call it i'm going to change

there we go so we see you know where these variables are

function like this so then you see you know this variable how do

as an argument there and then in the function definition it's

something with it but the function of this should actually be the

yep see all good and these functions which is kind of cool you

input so we have this so i can have multiple input in this

you know do something with it well we're just going to add them

for any formula you have you see like i just made this add but you

formula you have make a function out of it so notice here you know

as three and then it's going to expect three inputs a b c and then

c and now down here i have to call it with all three variables so

existed before but then it just stayed there floating not doing

here you know now that we have have this i'm going to call the

because the function is expecting three inputs and then it's going

so we see enter a number and i'll enter a number

and another number six another number seven oh because there you

hey fun with functions now all the this function and the ones we

statement as a way to show output and that's good but we could

value so notice this function here then it just returns that that

and what that does is it returns it right where you call the

right where you call the function so notice this if we're going to

you know there's our input it's the float i don't know stored as

what i print is your number multiplied equals and then right here

it's going to be that function i call that function with variable

and it's just going to take it times two

all right so notice some of these you run it and then we're going

it again so enter a number all right there you go 16 your number

so all we're doing here is changing the return statement multiply

everything else we actually can keep the same there you go enter a

enter a number 16 and it's 48 so that's the return statement is

that's where the number shows up your number multiplied equals and

returns that so that's pretty useful sometimes because we want

where do i want this output to be right there and so we put the

so that's the return statements good useful statement here so we

things we have all these functions we you know we're looking at

solving some things you can even just solve for x and there we go

now notice here we're importing simpy again or for our symbolic

it even further from simpy we're going to import symbols and from

import solve so all these now i'll just define it here x equals

in single quotes so that's telling us that we're going to use x as

and the way that simpy solve works is the equation has to be equal

so at what i'm going to do all right well we'll see

here's the output solve for x and i even have this here zero

enter it in now what this does simpy has this built in solve and

be right here so that solve we know that behind the scenes it has

equation using the variable x and this is just like a subtle thing

another time but this returns a finite set and for this we're

answer that we want equation zero so enter an equation and then

answer there we go all right enter an equation and so if i have

minus eight so we can see that x must be four

four and i'll put it there now if i didn't have this

i'll do the i'll do the exact same thing two times x minus eight

it just the way it displays it because it's a finite set now that

to possibly if i put in something that had more than one answer i

too if i have let's see i'll run it again so supposing i have now

even if it doesn't work out to a nice integer but i'm going to

you see negative three and three it does have two answers and

function x minus four so there we go

if you didn't get a syntax error you're ready for the project

so different things you can do so this is where we now can make

and this is where i leave it open to you how you could do this you

like you could take one of these functions here like the factoring

yeah so you could take this factoring the square root and copy all

all right so as an example of something something you could do and

this now what's interesting is you as defining the function here

and we definitely want input and i'm going to make it n

and i'm going to make it n so we could do this that

it factors the square root and you could make each of these then

indent everything now since we're factoring square and we don't

delete that and here we go use you know we can keep all these

and this is the thing with a function

is you do want everything to be and you can take out things like

there we go then this has to be indented

and this has to be indented there we go and all of these divide up

and we see that we have output okay

all right and this part we don't need you know

and what do we want to do we want to

yes this stuff does not have a test because this is so what you

square root and then down here later we could try to call it

square root and you know uh let's see 20

and oh maybe I have to return the output

it's actually kind of funny that just popped up but there we go

so you see the output but because it's in a function we have to

all right so this is you know this is not the part of the function

and these are some things you can do because then you can you know

just define them in your own you know colab code cell with some

you know you can make them all functions but that's the idea and

if you wanted to where you just have um

which one you know what function should you run

should you run you could do a table of contents there too and then

you're building this multi-function calculator you're putting all

a colab notebook that you're going to be able to use and it's

calculator and then as of making this video I can't tell you

but hold on to it and then you'll see where to put it together for

getting the credit for passing this part of the certification and

all right so hopefully this was useful you know how to work

how to put all this together into this certification project and

there's more than one way to do all this so find find a way that

making your own calculator all right and we'll go on to the next

so with graphing a system of equations we're looking at really

on the same x y axis and we want to see where do these two lines

two lines cross that's going to be the solution to our system of

have two equations and this one's y equals three x plus ten and

so let's say I don't know it's a place you like to go let's say

so four dollars times x which is every time you go then y would be

have a deal oh well if you pay ten dollar membership fee then your

so there you have the the decision like okay well with this

but when would these be the same price you know how many times

the same price so we want you know that's just one example of the

so i put these both in y equals because we're going to get into

but we're looking at just how this graph plays out here so if i

so without numbers here we're just going to estimate let's say 10

how we plot begin at 10 and then up three over over one and let's

like this and let's just say that's the line and then we have y

it begins at zero but it's a steeper slope you know be at zero

say you know it looks something like this and then they cross

intersect that's going to be the solution to our system of

there are numbers here and it was you know somewhere around 10 so

that that would be like 10 you know 40 something like that you

there so you see where these cross and depending on what you're

clearer to see might be harder to see but we can then always test

that's where that's where these lines intersect we can plug that

so if x is 10 so it would be three times 10 plus 10 and then y

that work three times 10 is 30 plus 10 is 40 yes that does work

you know four times 10 and y is 40 and then that works so that's

were doing this you know old school plotting a bunch of points i

y table and plot a bunch of points i was using graph paper and

and the reason why i didn't worry about getting this you know

drawn to scale is because i want to just you want you to see you

together two lines on the same graph my question is where do these

we're going to look at the code how can i set up a graph how can i

and you know for whatever the answer might be however you know

might be how we can find the solution each time so now let's take

you remember graphing one equation on a on a cartesian plane this

different one of the big things though is we're going to use the

so we'll talk about that more in a second but the first thing we

import numpy now you know this plt that makes it a lot shorter

dot pi plot numpy we shorten it you know it's just a little bit

you know the common abbreviation so we're going to do that and

throughout throughout this code so like before we are going to

our range x min x max y min y max there we go and one of the

instead of creating a loop like we were doing before what this

of x values and the first thing we need to do then is to find how

i'll take these x values you know notice x max minus x min so how

and that should give me an integer number but i'm going to say 10

that's how many points i want that's actually going to be for

need that many they it displays very nicely for anything with a

probably plenty and for some more complicated graphs you might do

me the number of points i have and just again based on these so if

this updates automatically and then here's where i'm going to

so using that library np dot linspace and it gives it takes the

and then how many points so there we go it's just going through

and and how many points in between and i'm just storing that as x

as my x values hopefully this is looking familiar setting up you

for x and y and here's where we're going to plot line one so in

two times x there we go and what it does is that will take each of

needing to create a loop it'll go through each of these x values

dot plot x and then y1 so here we go i have this and the second

y2 equals and this one i took it beyond linear i just made x

it up a little bit and so there we go that's my second y value and

plot here x the x values and then y2 so there we go to find the x

when we run this there we have it we have our each axis that first

light blue line and y2 was the parabola now that's a pretty good

had fewer points let's just say even two times this yeah that's

still going to see this become choppy yeah wasn't that bad but if

one times this then it really becomes a lot choppier because i'm

and there's a lot more going on here but you see how that linear

out just fine but this one yeah so that's why just as a default

we go and then to graph anything else all we have to do is change

oh okay well how about if i make this like negative negative 3x

three times x there we go if you write it by hand you know doing

you forget the multiplying so that's the key python syntax you

um and let's take a look at this and maybe we'll make this one um

there we go just to the third power how about that so there we go

and this one here cubic and we'll run this

you know even this yep the curve looks nice

so there we go all you have to do is change y1 and y2 and you

two but a lot of times we're going to do linear equations and have

questions where do they intersect so there we go numpy and use

the core skills in this unit let's look through some extra

extra problems using the colab notebook so you can see how you can

you're building and use these to solve problems that might come up

life so we're going to go through some more extra problems here

few other things you can do with graph with graphing and one of

lines or just in general above and below so if you have an

below the line we see we don't need to import anything different

the graphs you've been doing set up x min and maximum how many

for your array and we've been doing this before here's the line

now this one if you're doing an inequality you could also add the

a dashed line but this might be hard to see this line but here's

so we're going to say from x at still the same x values and then

two y values from y1 up to 10 and you could define that as a

already set our maximum or you could actually say up to y max like

this one we call it you know we want to make this red so that

give you a default but you know you can define these and that

go the fill between same x value and then you have the range of y

to work and we'll do a few lines here we'll make make some art so

familiar define the line and then plot there we go but then we're

between that x and y2 and y1 so you see you can define it even you

up to the top this one i'm going to go up to the other line and

line and here we go so fill line three very similar and there we

different face color and this one's going to be green where am i

between this one that we've defined up here y3 and y2 so that's so

are just going to be a band and here we go i can do this one

notice all these are just simple linear equations to show you this

two so what happens when we run this so we see we get the first

and then from there to there from there to there and there we go

let's say all right we see this red we see this yellow so if the

there i went up to y max it would just overlap and i know you

thinking hey you might have might turn orange you know yellow

it's the order of things that it graphs this that was red but then

it overwrote the red and if i wanted to i still call it a line one

down here so now this is yellow and then this whole thing would be

and remember because the yellow started down lower so the red will

but it didn't go down this far so the red didn't cover this so we

you know the yellow line was x plus three and then the red line

had that little gap that you know when the red went on it didn't

right but then we can you know y1 and now this one i definitely

y1 wasn't defined yet so you see it gave me that little error

make this official before we go on to the next thing

and we can see so since i've moved a few things around before we

let's change something else here i was saying that we need we

the red line and the yellow line the dashed line and it might be

can always hope here so single quotes and then i'm going to make

thing with this one the other argument single quotes and i'll make

two would be dashed lines instead of a solid line it's hard to see

there we go and because these you know with the shading sometimes

the exact line i mean mathematically you might just be in the

line but the way it shows up on the graph you know it's not you

it's nice but sometimes you might say oh okay if i'm making some

want or need that extra line so there we go so there's some

draw lines shade above and below and you can be you can get

of all sorts of things and this is essentially you know what

you know defining each individual pixel but defining the lines

equations and then that way you can scale things because let's say

change the dimensions of my graph i could scale this line with the

so that's the advantage vector graphics you can make larger things

size because it's all defined on equations anyway so there we go

with that other cool things you can create an interactive graph

interactive we're going to set up some sliders this one's called

across some python code that defines sliders that actually uses

works too but here's what we're going to import there we go in

to get us to be able to adjust these sliders and see the effect on

so there we go and then this interactive widget so i'm going to

is going to happen within this function so i'm going to define

a simple slope intercept so i'm going to define the function of m

for zoom so notice all of your dimensions here are based on that

and then we still have the points and the plot that's my y value

takes these inputs and here's the slider it's really the most of

is right here on this line so interactive plot now i can make that

just decided to call it that and interactive open parentheses so

am i going to run and i made it f and the main reason i made it f

of code was not that long but there you go what function am i

some ranges here m will go from negative 99 b will go from

so notice the zoom i could make it really small and x min max you

it's still going to be square but i could zoom it really into just

or i could zoom it out to 100 and i could change that number as i

we have is this and when we run that interactive plot it's going

sliders and then run this function so let's see how that works so

puts everything right in the middle

so my slope is zero and b is zero so you might be able to see that

this blue line is on there because it's a slope of zero zero what

see look at that now i have a slope of three and as i move up b

different place it begins right there now we see it starts right

bring this down and you see i can zoom in

pretty cool and see this is what i zoom if i'm so zoomed in one

so that's off the graph now i'd have to move that back now you

and these are some cool things you can do with your graph you can

even if you you know you're graphing all kinds of things you just

this is a nice way to zoom and you know you might see some things

where you can zoom and here's how you can write the code you can

graph that zooms in and out and you can adjust some other things

cool things you can do and i just want to you know it's a good way

slope and intercept as they change how does the graph change you

and therefore across the y-axis down here and i can change the

all the way so now i'm really zoomed out and it it looks like this

because it's so tiny it only crosses at negative three and my

go some interesting things you can do with zooming and you see

interactive plot and then your function all the graphing happens

okay so here's another interesting way to graph we can actually

to lead you to working with data science so let's just get a

else created this library this mediostat library for getting just

um here's you know the weather information that they take time

temperature maximum temperature precipitation snow wind direction

the maximum wind gusts air pressure and the amount of sun so there

solar panels you can calculate the hours of sun at a particular

wind turbines you can calculate wind speed at different places and

this one you do need to install this and so we're going to run pip

install this and so we're going to run pip install and i just made

it's a its own separate code block and there we go we just see

that installed this so there's a lot of things already built into

we've imported some things but didn't need to install anything

therefore it's not already built in so some of those newer

really pretty straightforward pip install mediostat done so

we can import it date time that was our i was already at python

do need to import that and then now we still need to import this

this data so that's why we import that and then the install

these i've included the link here you know for all the

all right so here's how it works start we'll define these

time year month day year month day so i decided let's do like the

so we have uh uh created a point now the point you create this is

so they don't have really built in like type in the name of a

you want to have an idea like let you know where you live you can

and longitude of where you are you can find latitude longitude you

you want a particular city somewhere in the city so the example

look this up you know somebody uh did this for british columbia so

philadelphia so there we go so here's a point in the middle of the

we can just get the data here so from that point so there we go

um there's my data point philly and start and end so there we go

you know start date end date location and then collect the data it

this does work using the same variable data equals this data

it works out and this one i just said hey let's just plot the

it looks very similar data dot plot and then down here we have plt

we need to do it actually all the rest of it the dimensions and

labels graph happen automatically so there we go we just have y

these three things so when i run it we see the average temperature

so all these things we know in a previous video we were talking

legend how to label the axis and everything this already does all

it'll just graph what you have here all right so there we go and

celsius so there you go if you're in philly you're like hey it

time but yes this is celsius so notice it's going way above that

and then we can just change this that we can have temperature

that and maybe we'll do one of the other ones precipitation prcp

and we see so average temperature and then precipitation

there we go pretty cool and we can see and of course then you know

here snow now what you can see is all right if there's average

and then precipitation then you'd expect hey maybe there's going

so there we go snow is in green and you'd think look at that

and there's precipitation and below average temperatures but no

average temperatures but no snow or below zero temperatures and

happened but just there we go here beginning january some snow

we go that's it so interesting things you can do and you know even

this on your you know having the colab app on your phone you could

something you know related to uh you know write write another line

today's date or yesterday's date and sort you know use that as the

app and click on click on something run it and you know see the

wind up till now if that's interesting you could do some other

things we can do with graphing and you see just getting a little

python is so good for data science because a lot of people make

a few lines of code and get the get the data you need and

powerful like that and you see the the graphing that we end up

library so pretty good some some cool things you can do with

theme that you know next we'll look at solving equations and

which ends up being tricky but we'll show you how to do it okay so

with it you know set up some you know some you know weather

okay now we're going to look at how to solve a system of equations

and you can still picture the graph the idea that that's that's

equate these are two equations two functions that you could graph

to get the solution and sometimes for solutions that it might not

then you know maybe it's you know not in a nice integer solving

time so we're going to look at how to do this just on paper or on

code on how to do this and set it up to be able to solve anything

so if i have these two equations one of the ways to solve this and

is if we notice if this equals y and this equals y then they can

ten equals four x so we see that if that equals y and that equals

three x plus ten equals four x and that's a little bit of the

so now we can solve these what can i do to solve for x i can

and then i get ten equals x now once i solve that x is ten well i

i need my y value and i can plug this in to either equation so i

to plug it into here so y equals four x then i could take that and

and that gives me y equals 40

and then that's how we would get the solution we can set these

and then solve solve usually for x and then plug that in either

get our y value now this is very similar to how this is going to

one of the things that makes uh sim the simpy library and solving

bit better is this works out nicely when they're equal to y and

equations that were not equal to y or something like that with the

and we would do the same thing with this is the idea that i would

equal to zero so i would end up subtracting y here so i would get

and i subtract y from both sides equals zero and that's the

when i solve it with the code and then the same thing here you

and then with each of them set equal to zero then you will see how

simpy library but that's that's kind of the method that python

advantages of this is they could be in whatever form they want

side of the equal sign just subtract it over and then now you have

you can solve it so you know and there's many other ways to solve

that's kind of the couple ways i want to show you you know this

set these equations equal to each other solve for one variable and

and then we'll take a look at it now so let's take a look at the

simplest way to solve a system of equations now this is especially

so you want to set each equation equal to zero but as far as

you know it doesn't have to be simplified just as long as it

use simpy our symbolic math library and i'm just importing

so that's the asterisk there and we're going to define x and y as

there we go x y symbols and now here's the equation set equal to

and the second one here then i just defined it as first and second

solution so this is the lin solve is the is the formula from simpy

the extra brackets first comma second so it takes this as

equations and then the next argument comma x y in an additional

symbols so linear solve these two equations because after each set

them equal to each other and use these symbols and when you run

comes out as a finite set now again the simplest way is there we

one line that puts it all together you can just change you know

run it and you know you see your answer it's pretty clear it's a

three supposing i want it to look a little bit nicer so still

and these symbols still have the same i'm going to use the same

now here where i have this solution here that's the same line but

variable is going to be a finite set but there's things i can do

another variable x solution here and i'm going to take this

behind the scenes it's a two-dimensional array so i want the first

also a two-dimensional array solution dot args zero one so having

a coordinate pair so there you all print the parentheses x

closing parentheses and then when i run this there we go it will

three in parentheses and i can do other things with that now there

want to see the graph now this gets tricky because we can't graph

that we would otherwise because going through loops doesn't really

through numpy doesn't really work with simpy but simpy has its own

we go it doesn't work with the others but it has its own so from

import plot and so this is the simpy plot and slightly different

here var x y here then this one is similar to what we were doing

that looks pretty similar to what we were just doing same with the

and so this part i'm still going to go through the same thing you

solution get the x solution the y solution and print so this is

what we just did before but what the other things that we can do

that these are equations set equal to zero now simpy syntax for

if i want to get it ready to factor so my y value for the first

so in simpy we're going to define this as an equation first and

that's that first equation and i'm just spelling it out that it

need that so now y first is that set equal to zero and then y1 i'm

which solves that first equation you see y first comma y so i'm

spell it out that it's set equal to zero and then do my algebra

and solve for y and then store it as this variable y1 i'm going to

equation so set it as an equation second the second equation is

i'm going to solve it y second comma y i'm going to solve it for y

i can print them because that might be interesting so it's just

y equals that i will display exactly comma and then here's the

zero again because you know the index of the array so we have

but we were talking about graphing this so let's do that next but

the solution i will display these you know y equals equations and

i didn't just factor it to see that as y equals though that is

that but to plot the solution here see now i'm going to just

as a symbol and then take a look at this i'm going to plot but

is just plot within simp i plot and what am i plotting this y

so there we go so when we factored it into y equals that helped us

but put this into here that this is what i'm plotting these two y

the x value notice also a little bit different the way we display

to 10 i could you know make these other variables but that's going

so then when we run this i see the solution i have it as y equals

and then i see the graph and we see where the solution is negative

so we can do this for you know any other ones here there we go

it negative 2x you know i don't know minus two and maybe this one

x and i don't know like plus eight change it up a little bit and

we'll solve it print out the solution factor it show the y equals

and now we have this new one two six is the solution here are my

two times x plus two and y equals x plus four so we see these two

the two times x has a steeper slope so that's got to be this one

and then x plus four the slope is only one so that's got to be

and we see that they intersect here at two six

all right so we see how we can solve and we can solve and graph

linear equations for equations beyond linear we'll cross that

to be in the upcoming sections now that we've worked through the

now that we've worked through the core skills in this unit let's

and i'm going to work through extra problems using the colab

apply these resources that you're building and use that use these

come up in a textbook or in day-to-day life so we're going to go

here so here again is where we're going to put together all the

learning into your calculator so that you'll have these as a

where you should already have proportions set up so the text to

to enter it in and solve your proportion and remember we made this

that there we go and we already did this converting decimals to

code there and that's also section heading solve for x so each

we're building and then we're going to take like the final product

one if you just have a an equation you want to solve for x pretty

there and factoring so we can solve or we could factor you know

something like that we have we have that and each of these you

because you just have this the solving solving for x we did put a

it's going to prompt for the input and then you see that the

whereas the factoring one it's just this equation that you know

simpy dot factor okay so that's also a section heading then

solving for a variable so not just set solve for x but even if

rewrite it you know like y equals a over b or something like that

numbers this will do that for you and we defined a lot of our

it's not going to be an error to put them in there and not use

what variables you use you can always edit that part and then

the involving numbers or a bunch of letters and variables you know

equal sign and then we have this then we're just going to define

pick what variable you want to solve for set it as an equation

going to solve and this loop runs through all possibilities for s

we have that that we can solve for a variable like again it's

therefore in the table of contents so we're going to add the other

intercept from two points because this in itself is a very common

going to define the two points there we go x you know x and y and

solve for b we're going to get m and b and having those two we're

and then we're going to use that for the graph and you can keep

know you could always change this x min negative 10 and we could

could always change them to 10 in every direction if you wanted to

these you know for the graph and the setup the details could

these and you see for things that we might use we might want to

there so you don't have to really think about you know the exact

this you remember it but it's just there and it's commented out so

and then that way here we just plot the function as a red line so

of these you have a lot of things that are there by default that

you could solve whatever you wanted to solve and graph it so we

section heading then we can double click and remember it's this

because it's in the text not in the code now that became a section

of contents so we have that now the section heading goes all the

bottom so that one we can minimize now this one if we minimize it

document here so besides that i might just want to graph something

this the other code as you're building it remember the slope

then other ways to customize the graph came from some of the extra

in week four where we have you know again all the different things

labels as well as setting the tick marks and this one you know we

here you know and added a remember we customized it and added this

this for yours we're going to put all these as well as the zoom

so this function then does all the graphing and i'm going to skip

the slider remember that interactive plot we're going to run this

we're going to have is to zoom and that's going to be our

hundred that's good and then taking that so whatever that

what we have for our x min and x max and then once we have those

for our np dot linspace to graph and there we go and i just put

two lines as by default you know maybe you want to have two lines

see where they where they intersect so this graphs two different

a linear equation as long as you're using python syntax whatever

i commented this out but again you could have it if you want to

and i just happen to have this one fill between same x values so

y max so that would shade above and it would make it green and the

values from y2 down to y min and make it blue so that would shade

these here in case you need them we're making these default

okay i do want to shade above or below uncomment it and then we

can change now for this one i kept this the tick marks in there

lot of this is based on the zoom so then how many ticks do i want

difference the how you know from x max to x min divided by 20 i

to be a good number to display nicely so we're going to get that

and then cast it as an integer so the number of ticks has to be an

then that way no matter what the zoom is you'll see that we can

grid so this will give us a few things that we can graph you know

given these equations that we have in here a lot of times i like

than just like make that line blank so if you run the code it

something so here we have this graph and we see the default zoom

but i could move this back down and you see everything adjusts the

the graph zooms in and then you can see you know or we could zoom

so yeah pretty good i like kind of for this particular graph i

and then you can see you know about where they where they cross so

want to graph something and there there we have that and there we

and you could have the whole every all the directions don't have

could have that and you could put an html break right after that

words underneath that would not become a part of the heading

but for this we can just put the heading here

here we go i remember there's really no entering we just put it on

somewhere else and now that's also a part of the table of contents

solve and graph a system and this is nice it brings it all

you can put in as long as you have python syntax whatever you have

and second equation and i even have an extra one here just to be

this will even work if you have a square root just to have

the linsolve before to solve uh using simpy the linsolve to solve

is actually another one non-linsolve i like this one because it

in there it can be linear or non-linear it works out so just

mean that you have to have you know something different if one of

but there we go the comments just so you know here's where you

and then you already have the code that you would have copied this

eight uh yeah from from uh the recent week to solve and graph you

tinker with them a little bit you know as we went through and

uh blocks of code you know take the last one that brings it all

over here you know that's what you're doing you're building your

the system of equations and then this will work solve and graph

i just want to show you some of these

so even if it's non-linear and you see it'll give you the solution

solutions notice because we had the uh the loop the four loop here

solutions so it'll have the two solutions then also show the two y

graph them so that's pretty complete that's a good good one you

thing in your toolbox double click and let's make this one a

and i've added one more thing here and we could still make this a

notice this code isn't complete but i put one thing in here they

so you certainly you would recognize that that's import so that

you see this you would also it would be something you already

but notice you have this dot save fig and you're going to give it

notice you have this dot save fig and you're going to give it a

to use notice that'll come before the plt dot show and that

i've tried this if you put this after it actually will save

and what's interesting is this download can come after so that's

that'd be fine too but it can come after because it doesn't

and this is a way to download this to your computer or your phone

because sometimes you might have have something that you wanted to

this and you would put if you wanted to up here you import that

you know you would just do save fig and files download so there we

have all of this and one of the things you can do is in the view

all right and now you have all these in your table of contents and

if you need to there you go you have all the sections collapsed

for x do i need to solve for variable you know i i like i like

and down save graph and download it

there you go because that's what it does is it downloads an image

so already we have so much of the core algebra that you're able to

the code for it and now you have this at the ready so one of the

to quadratics and more complicated graphs we kind of hinted at

if you haven't seen a lot of these and you say oh wow okay now

getting interesting so we'll get into quadratics parabolas you

curve to them and things beyond that so we'll add more to this all

to being able being able to solve a lot of algebra problems in

so when working with word problems not just here's a math formula

problems or some people call them story problems that's when a lot

translating that into a math equation that we can solve so what i

keywords that you can find in the word problems in the sentences

to use and this is not all of them but this is a lot of the main

with this these are the things you're looking for and then after

we'll go through a bunch of examples in the code and how to use

into math equations but let's take a look at these keywords so how

some of the words are going to be if i just actually see plus some

more so if i have something and then i got more how much do i have

money i had increased it might even just be and you know i have

cards how many cards do i have received because a lot of times

up something goes up and a lot of times calculating elevation up

it might even just come up and say added to sum also and sum means

might see some of we might see these you know the sum of three and

plus four so these are some of the ones if you're looking and you

adding okay and a lot of this it's not so exact that you could

find one of these keywords and then automatically translate it to

use a little bit of your human intuition there but it it's very

numbers what which one of these words are around there and that's

of the path operation so let's take a look at subtraction so

minus less anytime i have less than i'm subtracting or it

decreased so i subtract gave because a lot of times things if i

it becomes added to what i have or if i gave then it becomes

down and that also goes again with looking at elevation it might

subtracted and then anytime it says the word difference or i know

between two numbers that's subtraction you know we just find the

be a negative difference might be a positive might be absolute

or if i want to find the difference then i'm going to subtract all

so how do i know that i'm multiplying obviously things are

we want to look for that general trend things are increasing and

since things are increasing you know it's multiplying or adding so

it would be multiplying by just the word by because if i'm

three by five and then i've won't you know also then just seeing

means the area would be three times five of so just the word of is

of something you have the percent and then you multiply times

factor because two things you multiply together are factors times

you know it might just come out and say times and even if it

dimensions and say what's the area and anytime i'm calculating the

so there we go again not this is not all the numbers that lead to

of the main one all right so divide out of so notice of would be

so you know four out of five that would be a fraction four divided

per same actually per pretty much means out of but that's another

of miles per hour i would divide miles divided by hours or

hours quotient we don't see this word that often but that quotient

means that i'm dividing and actually that's the answer when i

you know you know kilometers per hour or something like that but

be some sort of dividing something divided by something else

related to time but it's not always so anytime i have a rate i

and again it might just come out and say divided by so you know

put the word each here because that actually could go for multiply

of it you know if i have if i have 20 students in the class and i

each how many pencils do i need so then that's multiplying because

but if i have if i know how many pencils i have and then you know

and i you know how many can i how many can each of them get you

20 and then i have 20 students how many can each of them get i'm

we have to get the sense of it and dividing like subtraction if i

to be a smaller number then i know it's dividing or subtracting

further you know what's the sense of it am i is it something you

it out among different things or is it just a subtraction okay so

things and the one that carries over a lot is or some version of

equals so that's where we get the equal sign coming into our

then i can translate you know this is was will be something like

so looking at these keywords you know we get the numbers from the

these keywords and we can put together an equation and so i wanted

in the code what we're going to do is we're going to look at a lot

and translating them into math equations and then of course

we want to be able to translate this you know you're not always in

the math problem it's going to be you know some words and we have

problem so let's take a look at how to do this let's let's apply

into math problems all right so let's take a look at the code so

practice some of the math we've been talking about with the linear

especially and some of the python developing equations finding

let's put all this together and use that to solve some problems

can see that y equals mx plus b pattern for a linear equation show

ways you know sometimes we just change out the letters a little

that pattern so if we take a look at number one and this all comes

i have listed as textbook two algebra and trig so you know there

find these because we're just going to pick a few of these and i

you know what we can do with these so if we take a look terry is

terry's elevation e of t that's how i'd write that that's the

um instead of just y equals we'll call it e of t elevation as a

in feet after t seconds is given by this so e of t equals 3000

instead of x we're using t because it's time and we have 3000

e of t that's the function and so we see the slow the intercept

she terry begins at 3000 feet up and then goes down the slope at

that that is the slope the slope of our equation on the slope and

is going down there we go so we can see the slope we can see where

um and there we go just uh finding that slope we just look at it

now we have jessica walking home from a friend's house after two

12 minutes after leaving she's 0.9 miles from home what's her rate

miles per hour well right here we have distances in miles but the

so if the question asks for miles per hour we definitely see also

have a time and a distance and then another time and distance so

distance and having these two points we can go to our calculator

something that you've already created you know remember we might

calculator and then hopefully you know then you would have created

for the different things that we want to be able to do and we can

equation from two points because even though the that question

and everything we're going to do more but the first thing it's

for us and hopefully you know that's it you know we have these

make use of it so you would have had all this and then we would

is 1.4 but that x value it's two minutes and we have to remember

it's two minutes and there's 60 minutes in an hour so it's two out

this if you want but you don't even have to python will take care

steps and the y value 1.4 miles so the second x value 12 minutes

an hour so it's 12 out of 60 again that reduces but you don't you

0.9 so having all this you can just put in these four values here

we'll print that out so you see the equation and then we'll also

what the question is but this will give you an idea of some things

this and all we needed to do is recognize what these values were

run it all right so this i absolutely would take this slope as

python calculates things it might be this 0.9999 it's three and if

you know you could always go back and put a round function in in

with this you could always round it to two decimal place to one

but either way we see that remember our first point was two

zero at two minutes she was 1.4 miles from home but at time zero

1.49 or 1.5 it might round so we get our values here and you know

function in there but i'm not worried about that right now and

but it's still ran occasionally that happens just the the way it

some things you know we get a warning you know some value maybe

what we were working on gives us this warning but it's still ran

and then we also have this graphed now if we have you know the x

so this is her time from home

so this y value notice it seems like this graph is completely

only goes down to net 9.6 we want it down to zero and then we have

know what's that all about so remember mathematically a lot you

these values exist you know we make this equation here negative 3x

but that whole equation exists so we can actually change the graph

you know later on in this document is you know we can graph you

but we can just change the graph here and our x minimum if x is

because negative time you know she was still at her friend's house

the walking and the x maximum i mean we can really zoom in now

going three miles an hour and she only had half a mile to go like

small i don't know i'll maybe i'll just reduce it to five for this

um is zero and the y maximum i don't even need that to be 10 i can

it five but we don't even need to that much we can really zoom in

short distance all right so now when we have this now we have

you know this is the only part we needed so we could look at this

a half miles away and so therefore you know she's walking about

about a half hour to get home and there we go we see you know the

whole equation we see the graph and then we can graph the the trip

know change these you know if you really needed to explain this

always put you know x values you could change that to time y value

there we go so we have these calculators that you know we're

about it here a lot now but if you had this at the ready you might

just you know see this oh i see my two values plug them in click

you know move on in a few seconds here so some things we can do

similar and maybe we'll do number three you know a lot quicker a

marina sailing directly at it at 10 miles per hour so now we

because it's a hundred miles away so that's where we begin at 10

the distance from the marina then it's a negative 10 100 minus 10

that's the equation 100 minus 10 x and we could always go to

and see you know just the trend how long does how long will it

marina there you go 10 miles an hour you might even be able to do

10 hours to get there and that's the thing we have the setup for

difficult but sometimes you can just calculate it yourself all

these and remember a linear function we'll get into the other

beyond that but a linear function has no exponents so there we go

linear seven is linear eight is not because we have x squared nine

squared nine is linear ten is not we have x squared eleven we have

not even a function like the ones we've been doing so that's

function like the ones we've been doing and certainly not linear

denominator then that would not be linear but with just a five in

so there we go and increasing or decreasing we would just look at

positive it's increasing if the slope is negative it's decreasing

coefficient so slopes four it's increasing 15 slope is five it's

and 16 the slope is negative two so it's decreasing 17 it's

that and just like we were doing with these other problems you

could just plug these in two four four ten there you go run it and

equation and see the graph so a lot of these you know once you

becomes a lot easier there you go same thing a line if we have two

in there all right now let's take a look at 37 so these two lines

of equations are they parallel perpendicular or neither well one

we can look at one of our other notebook sections and we can graph

a look at this 4x minus 7y equals 10 so let's take a look at this

how about this graph lines and zoom in or out remember you would

too and we defined our function so that we can zoom in or out and

i could just graph this now also so one of the things we can do to

if you wanted to is put it and this would be something that you

of these in the y equals version because they're not in y equals

to graph you see it's it's going to be y equals and they are going

well let's go to another one solve and graph a system because this

equal to y in fact we just make it equal to zero so i actually

four four times x minus seven times y and the only thing we need

that equals ten subtract ten from both sides you see much quicker

second one seven x plus four y equals one just subtract one from

have here so it underlines these but the moments we run it'll be

well because as long as you have to set equal to zero just

over and remember you would have already made this hopefully and

your second set equal to zero and remember we're solve then we

solution and then we're going to also convert it to y equals here

and then graph it print the y equals version

and when we run it we see we have the solution and it's not even

than the question asked but if we already had the code you know we

and yes these are two very weird fractions that you know it would

were calculating this by hand we get the solution and so we also

or perpendicular and we see where they intersect at this value

almost negative one little bit off and then a fraction there and

four times x over seven minus ten over seven so we see this one

and this one has a slope of negative seven over four oh then they

what this graph shows so let's try to change this because the

even though we're multiplying by x that coefficient is four over

seven over four and that's what makes them perpendicular the

seven negative seven over four that if you multiplied them

change the scale on this we can probably it'll probably look more

so this just naturally scales it this way all right so we have but

in this same window the y value goes up to 15

so the you know it perpendicular this intersection would be at a

and the you know we just can't see that right angle because of the

all right so that work i would work for any of these now you know

just add y to both sides you know subtract 12 on the first one and

and we can use that same thing same with 39 maybe we'll do another

yeah if we have this let's just do yeah we'll do one more here

so if we go back to this then i'll make the first equation then

y plus x minus 12 because that i would have to equal zero three

have to put the three times y whereas algebra you won't see that

so three three y plus x minus 12 equals zero and then on the

y so on the second equation we'll have eight x

eight x eight times x plus one plus y and see i don't even for

you know for other ones you don't have to worry about factoring it

zero we will get the solution and then we're going to show the y

so yep these do have a solution where a point where they intersect

perpendicular because the slope is negative one third you know one

slope is negative eight so a steep slope at negative eight and

not as steep so they will meet at this point there we go so pretty

system of equations solving it factoring it graphing it remember

you don't build these calculators for yourself then you have it

some other things and so we see you know the similar things lines

in any of these you could build for yourself you know a notebook

do this for two for two point uh two lines or you could actually

and you could use the slope intercept equation for the one line

later and then you could use the slope intercept equation again

it in so that's what we have here like this one you know you have

always could then take it and you know copy that and write it in

and do the other one copy it as line two so that you know that

do if you wanted to you know take take something like that and

one but nonetheless you know right now as i have it it was it was

get the equation and you could if you wanted to copy it into the

so all these yep same thing two points you get the equation two

and there we go just looking at that equation you know parallel

perpendicular then you could multiply the slopes together to get

and neither is anything else as long as they have different slopes

things you can calculate and i'll leave this to you if you want to

practice some of these see some of these you don't have to have

or you don't have to have the the note jupiter notebook or co-lab

pattern that slope intercept equation here what's the y-intercept

for this particular one you can actually count you know oh wow

five over four so the slope would be negative five over four so it

y equals negative five over four x plus five or if you wanted to

and plug you know go back to the notebook plug them in and see it

see that that'll work for any of these you know having this you

so b is one and then from that to the next nice point it looks

three there you go there's your slope so you can you know work on

so some of these again any of these you want to see a graph

we could just plug any of these into that other notebook section

the graph all right sketch a line

good and again you can practice all these if you want the tables

so really just remember x and then whatever you want to call it g

and so if i want the equation then what we want is we can just

so that's you know x is zero y is five and the next one x is five

and we can get that equation but then also if you wanted to you

and see if it's the same slope so that's how we would determine if

always just plot any of these points and and see what it comes out

just plot these points as their own array and see if it looks like

another interesting way to do it and there we go so so yeah let me

that on on one of these and so if the x values are 0 5 10 and 15

in one of these where you're going to graph so if we have

yeah here and we'll even keep the zoom in so one of these where

we have all this for the zoom but what you can do is you could

all right so we could get x equals and make it you know 0 comma 5

and we can just make that the array 10 15 and then

and then you could now we could just comment out all of this

and take this and this first one we could put you know comment out

and we could have y1 and put your y values 5 negative 10 negative

negative 25 negative 40 so remember that you know that that numpy

of values and we could just put our own array and if you wanted to

you know make them make them dots i'll say ro so we'll make them

for the points so we could have the x array have the y array and

there we go we just commented all these out and

so we have this looks to be that they're in line and we can change

you know yep it definitely looks like they're in a line

so we could you know there we go we could say that yep that's that

works that is linear so some some of these you might just notice

in in that table but also i wanted to show you for this or for

in in that table but also i wanted to show you for this or for you

to do for that little question but also just showing you what you

more complicated sets because all of these that you know this this

and just a few values just to show you but remember this is this

toward data science if you want to continue following this path

four values you might have like you know four thousand and so we

do this we'll have other ways rather than having to you know write

those values if there was like a thousand of them and then store

other y values but then you know can i do this can i just take

what it looks like now these are the things we want to be able to

some some things you know and and it answers a very simple

because otherwise you know they might get some of these that the

could not say that they're they all line up nicely so that's it

know missing values of a linear function and remember you can

a table of values too if you wanted to so you know those are some

we would have to put the slope formula so remember the slope

looking at the code you know find the equation of the line goes to

at the code the y values b plus one and then the other y values b

up one you see the y value went from b to b plus one but then look

didn't change so a minus a is zero so we have it's actually not a

x value x equals a so these are some of the things just you know

the slope and you could always do some of this with you could

if you look at the the other one that we were doing here solve and

have to comment out the graph probably wouldn't do too well but

you know you if you define more variables you could still take

again the solution might not be uh really possible but you could

actually solve and this solves it for zero but you could actually

for y or whatever value you want so simpy will do that for you so

you could have a make another section in your notebook that just

factoring all right so there we go and so we have some other ones

now this we have at noon the barista knows she has twenty dollars

average of fifty cents more from each customer so remember that's

customers tipping more some tipping less and then it averages out

because we would right away see the equation we begin with 20 and

because we know she's getting something more 50 cents from each

so we see that equation then we you know right away we have that

and we can do something with that we could like graph it we could

this particular question we just says n more

so it would be 20 plus 0.5 n you know we just change that variable

she can plug in a number for n figure out you know how many tips

and this 115 these are some of the things that you know you can

a gym membership with two personal training sessions and a gym

training sessions cost 125 while gym membership with five personal

what's the cost per session so we take it as the membership itself

and then you could have like you know each of these so really we

many training sessions and then the y value would be the cost and

we don't it's not asking us to graph we absolutely could go back

and then five sessions was 260 so we could go back to this and

you know slope intercept from two points and we could say you know

cost 125 you know how many sessions and then dollar amount and

would be then 260 and oh now this we need to change this the x

fine for five but let's take a look at this and let's see what we

go up to 10 y minimum zero is probably fine but the y maximum and

like you know 500 we're going to go beyond that but that's fine

can see that we're going to go beyond that we're going to go

that but that's fine and when we see the graph so we see the

itself must be 35 dollars and each session must be 45 dollars and

there we go you know and we see where we see the cost 45 you know

and then we see the increase okay if somebody wants to plan you

how many training sessions do i want maybe this is in a month the

it might be about eight in a month and can calculate oh there we

that'd be four hundred dollars or you know do i want to just try

and not get any training sessions and you know there we go then

at the zero value and that's what we want we want to you know use

some different things you know the cost like 116 here so the

there's a linear relationship between the number of shirts it can

price p that it can charge for sure so there you go selling and

and this is you know things that people want to find out because

what price you know they definitely start out like can't you know

have to make money i can't lose money but then beyond that we see

different prices are possible so somebody says that finds this

from somewhere else some other people selling shirts oh okay i can

if the price is thirty dollars but three thousand shirts if the

so there we go again that classic y equals mx plus b but we just

change the letters p of n you know the price based on n shirt n

there we go we can we can look at this you know n shirts now

i might take that the price is the independent variable because

you can buy and so we might you know do something like that and

units here we'll get into other things where we can then based on

we can find that that you know that equation for price and then

it a step further and say what would be my profit because just as

so a thousand shirts at thirty dollars so it'd be thirty thousand

shirts thirty dollars each but three thousand shirts at twenty two

thousand is forty four uh uh sixty six thousand dollars so just

selling more shirts but the price that it had to lower or reduce

much different and so it sells three thousand shirts so instead of

sixty six thousand dollars look at that almost uh more than twice

less and some people who don't look at the math of this would say

you know reducing it eight dollars is that going to double my

the revenue we don't know the cost of making the shirts but it

just lowering that a little bit because it reached so many people

and so that's what people are looking at you know and in again in

math behind you know how to how to get this how to find that

value and you know what you know what are we looking at here how

you know ideal shirt price so there we go there we go all these

equation we're trying to maximize things if we look at number 18 a

relationship between the number of bean stalks she uh she plants

that each plant produces so 30 stalks each plant yield you know

so 30 stocks each plant yields 30 ounces of beans but then 34

beans you know they're all competing for some of the same

you know they're all competing for some of the same resources so

equation and then we can try to find so what would be the maximum

what would maximize the bean yield so you know it'd be really

um yes so notice this uh and the yield would be 30 ounces that

30 stocks times 30 ounces see then 30 times 30 so that's 900 or

and i'm not going to do that in my head but we can see some

these in some of the other business applications later on so some

population and like number 120 drawing linearly we can write an

another one where we often do this all the time converting

and we have these two values zero degrees celsius is 32 degrees

100 degrees celsius is when water boils and the corresponding temp

can do is we could just take these now we could also use this to

but we could also take this you know slope intercept from an

there we go still better sub equation from two points and

we can say all right so we have zero and then 32

and celsius would be 100 and fahrenheit would be 212

here we go and see the see the graph

now the graph in this case okay and i'm not even worried about

the main thing we wanted was this equation here 1.8 x minus 32

fraction that's nine over five so there we go that's and remember

the x value was celsius so that's how we can convert celsius to

and so if we look at the increase you know 1.8 is almost two so

degree increases almost two degrees fahrenheit increase so we can

so if we take a look at this you know we you could always you can

another function for yourself that you could you know you could

celsius to fahrenheit fahrenheit to celsius you know that make

you know put a heading to it because remember as soon as you have

because we're not in the code but if it's in there then it becomes

of contents so you can always make some of these things for

want to give you some of these practice problems you know if some

want to combine some of the things you know you have room to

so there we go and then once you have that equation you can find

that function of 28 degrees what's that become in celsius or

uh that's a good one because negative 40 celsius is negative 40

they actually agree it's got to be really cold but when it's that

having a thermometer would probably agree too yes it's cold so

some of some of these and hopefully this is you know find this

interpret some of the word problems and use some of the notebooks

solve problems answer questions and yeah you can continue you know

know make make it your calculator anytime you have different

might be useful make a function pick a formula another notebook

on to the next thing now that we've worked through the core skills

some extra problems and i'm going to work through extra problems

see how you can apply these resources that you're building and use

come up in a textbook or in day-to-day life so we're going to go

here for these extra problems we're going to look at just ways to

you have different situations word problems how can we decode

then use python to solve them so just here's some examples and all

first textbook you have the link so let's look at number 10 on

canadians work 39.5 hours per week so if the typical adult

what's the percentage of hours in a single week left over for

so we need a few things here 30 so we need the number of hours in

even just going through your python colab notebook we have all

i'm going to click on solve and graph which we're not doing yet

the blank you know line of code here and even in the print

first of all we can have the number of hours per week and that

and then if we want to look at the number of hours that canadians

from that and python knows the word of operation so i can just put

you really don't need it 39.5 so there we go uh 24 hours a day

and then we actually can also subtract eight hours a day eight

times seven so we have how many total hours in the week and then

the sleep hours again all this in in python shift enter and we

72.5 hours is you know and it works out look at look at that you

of time to go out and do things and we can do all this just know

step you can break this up into different steps and put variables

this all right and let's take a look at another one that number 14

for the first half of 20 2009 we're down 46.733 percent we'll talk

second from 2008 when 1500 new homes were started so from 2008 to

would be that times one or times 100 which would be times 1.00 so

that times one but then since it went down we want to think one

percent it's the 100 minus this 46.733 and then the original was

so we can just go right back to this and we have 1500 new homes

46.733 as a decimal is going to be 0.46733 and you see that we

so one or you know 100 i mean it's not going to mess up the code

but so you know that's 100 so if it was that 1500 times one that

year and since it went down it's times one minus this rate you

and these are number of homes so we're going to round it down and

you know that 0.005 is you know i don't know that's what five

know they put a few boards there or something on the lot said i

going to call it 799 and that's how we get this times you know one

went down and in later weeks we're going to talk a lot more about

so this is not the only time you're going to see this and in that

these other percent increase or decrease but on page 73 i want to

so if 999 changed to 1049 so the amount of change we would just

and we see you know we can do that you can do that in your head

but then the percent changes we take that difference over the

so if it started at 999 that's the that's the difference so that's

and again we can look at percent change so that 50 cents 0.50

so that's we get that difference divided by the original number

we're going to multiply it by 100 because it's the first two

the next number is a zero so it's really a five percent change

it to sometimes people like to put that into the formula times 100

there we go as a percent I mean if it's just for your own

make it fancier if you're doing this to explain you know costs and

business proposal yeah it's nice to have the extra bit you know

people can see oh this is exactly what you're talking about for

put it all in the print statement and you know see see the output

1999 lowered by 10 percent so we're looking at these keywords

we need that formula one minus the rate again might be new to

we're going to talk a lot more about it in later weeks but 1999

we're not decreasing by an amount so we're not subtracting

one minus three and I can even you know convert this as I'll just

10 remember 0.10 so I put that in there with the extra zeros but

get this one minus three if it didn't change it would be that

one but it's times one minus three and then that's the new number

any practical connection it was just what's the number if it's a

to two decimal places it's 17.99 so that'd be something like you

1999 and it was on 10 off so you end up paying 17.99 and knowing

you have calculator so you can do this you know right there on

this app you can set up different things for percent increase and

in the store oh what would the actual price be and at that point

we go and here we have this what about when increased by 40 is

before I even get to what we're going to calculate I'm going to

because I don't want the code I just want to show you the math of

I'm going to call x increased by 40 percent so if it increased

and then that would equal that new amount

so you know the the beginning amount or you know x or we can call

times one plus the rate equals the new amount and I just put this

just looking at this formula and so if I have this

then the question is what amount number six here what amount when

so that would be x times and when it's one plus three that makes

point four I put the zero if I wanted to equals 3500 because one

and then in doing this we remember solving for x I'm multiplying

the print statement I'm doing this 3500 divided by 1.4 and we see

you could write your math in here some notes for yourself the

plug in the numbers so then we see what's the actual math that

oh dividing by 1.4 so we'll do that and that's the one thing that

others are comments there you go 2500 so we have some of these all

I actually want to skip over to this baseball one I skipped over

are more straightforward now let's take a look at making use of

this notebook for yourself let's make use of the system of

15 the local baseball team sells tickets with two price zones

to third are priced at twenty dollars per ticket all the other

dollars a ticket and last night's game 5332 fans were in

worth 71,000 how many tickets were in each zone so let's call x

those the twenty dollar tickets and let's call y the ones that are

things we want we want to know how many tickets in each zone so

so twenty dollars for the good tickets ten dollars for the lower

seventy one thousand seven fifty so if we go to our solvent graph

you might have your table of contents jump right to it solvent

and y and you can even write some more stuff in here and you know

a look at this if our two equations are twenty dollars times x

times y how many of those tickets did we sell equals that total

sure we get it exact seventy one thousand seven fifty so now we

twenty dollars times this many tickets the good tickets plus ten

that are further out equals this how much this much money that but

i want to solve this i need another equation so what about that

three hundred thirty two fans were in attendance so x and y were

section so x plus y must equal five thousand three hundred thirty

in the way we have this set up here we can just set this equal to

equation and we're just going to put this twenty times x plus ten

x plus ten times y and then equal to zero i just subtract that

and then that would be that equal to zero

all right and the second equation set equal to zero so x plus y so

five three three two that would make that set equal to zero and

other things here if you want to factor and then just setting up

way to set it up because remember we already set this up to solve

and the graph you know just might help visualize it and when we

see the solution here and 1843 and 3489 so that's how many of each

because when we add these up then that's going to be five three

we know how many of each each ticket so 1843 people sat you know

and 3489 as we expect that's there's more room out there in the

that's how many people sat out there and now and now we can figure

and we see the graph just to see the comparison here

so pretty cool we can we can figure out given this information

how many people sat in each place and who knows maybe if it wasn't

breakdown all right and you know some other applications we'll

increase maybe we'll take a look at this one here so number 15

for her trail mix recipe let's talk about but the by weight her

30 percent cheerios 20 peanuts and she wants to make two kilograms

just calculate how much of each she's mixing so pretzels cheerios

we could just go right to right to here and before i even get to a

we could just define some variables in python we like to use the

pretzel weight maybe we'll do pretzel uh pretzel there we go

times two because it was 50 percent pretzels and she wants to make

going to be 30 percent and peanuts 20 percent so cheerios equals

uh peanuts equals 0.2 times two now we have now we're going to we

each of these and you can put comments in there for yourself if

uh talk about the cost so there we go if we have the the cost so

9.999 and so if we have this our cost equals pretzel times 9.99

kilograms at 9.99 a kilogram plus and we'll look at cheerios 6.99

so now we have the total cost here and that should be then the

fine but the question then asks what's the average cost per 100

so 100 grams compared to two kilograms is divided by 20 and we

to also bring it up to the proportion earlier on in this notebook

kilograms 100 grams and so it's we can just put this in the print

because that's the cost for 100 grams there we go so how many of

but then the final question uh we need to divide it by 20 we only

100 grams and there we go 80 cents

which makes sense because if we take a look at the total cost for

yeah when we when we're buying this full two kilograms you know

16 dollars worth of trail mix i mean you know hopefully two

it'll last a while so she just wants to make 100 grams so that's

100 grams you know so that people can eat it before it goes stale

there you go 80 cents all right not bad so some other things that

making use of these calculators that we have we can take a look at

options for things like green energy and so i i put in here you

one thing you know one thing and it was i think i did one solar

you know you want to buy this you know live live off the grid or

for somewhere so here's a solar panel charges up a battery and the

and then the output is 110 watt hours now 111 so you can certainly

possible that's just you know that output i normally would then

111 watts for an hour and 111 watts that should be enough to

charge a phone few light bulbs also so you know that's something

make use of 111 watts like i said laptop charge phone power a few

and then that way so what what this would be is so the uh

hours because we use so much and that's like 15 cents i think is

so 111 watts that's you know 10 of that a little bit less so yes

you know this 111 watt hours and i was thinking you use 111 watts

a 1.5 cents okay now beyond the savings if you're also doing this

grid or you know uh you want to run run some stuff like out in the

really plug it in well that's the value of this but let's just say

electricity cost so you know that 245 dollars and if you run this

all you're making full use of all this for an hour then it'd be 16

um rough estimate you know doing a little bit of rounding here

you know a little bit less than three years and if you do that you

in a thousand days which for a lot of investments that's not that

and then after that free electricity you know you paid for this

you're saving this money for the thousand thousand days and then

already saved enough money to pay for it and you get the free

another comparison so this wind turbine i find all these nice you

isn't it you know it's kind of interesting so uh did that not

wind turbine that can handle like low winds and it's rated for you

the idea is that if it's low wind it might be a whole lot less and

you know it can say 2.5 meters per second low wind speed which

and even at that point it'll turn it'll generate some electricity

like this and this one was 270 dollars 400 watts but low low wind

okay so let's just say you can generate 40 watts for 24 hours

not dependent on the sun you know you could have wind blowing

kilowatt per day 40 watts it'd be like you know 960 watts we'll

just an estimate anyway so that one kilowatt hour you know on

so if you have it somewhere where it's low wind there we go that

gives you how many days 1800 days so this one would be if it was

five years to pay for itself and then after that you know free

not so bad as a return on your investment but i just happen to

a windy area and let's say you can produce a constant 400 watts

places you know especially if you're near like a body of water

wide river you could definitely get a consistent wind and if

and it would only take about half a year to pay for itself because

watts that's 9.6 kilowatt hours per day or dollar 44 and then just

divided by a dollar 44 187 days you know just over half a year and

space hey by three of them and because you're making you know

uh three of them is probably enough to power you know somebody's

you know you're you're you're saying you would be saving all that

of the three of them and power the whole house yeah about six

that it would be that about that same yeah so really interesting

you can do these calculations i just wrote them here but you can

co-lab and do these different calculations graph different things

that that that's the idea with having all all you know all the

fingertips you can count you can calculate things and figure out

spend this much money and then you know generate this for

say it's not a priority right now and you know that's it but you

numbers so that's that's really some of the some of the uses of of

and you know also making use of you know the co-lab

uh graphing some different things or solving some problems so

and hopefully this gives you ideas of other things you can do once

notebooks and you have the ability to calculate and graph some

use of it to see what you can do to make money or save yourself

the next unit then so let's talk about quadratics so the word quad

why would we have you know quad square but then it's and we call

two so where does the square come from the square in quadratics

so where does the square come from the square in quadratics comes

to the original algebra back with al-khorizmi so the if i have a

each side i'm just going to call them x but then the area of that

be x squared and that's where that's really where this notion of

actually tied to physical area the area of a square but there's a

and this comes up so often that we get beyond even the the actual

this for a lot of other things so we look at the our equation y

graph this and when we get into the code we're going to get into

look at some of those applications too but when i'm going when i

quadratic it's going to be a parabola and you know it'll probably

than that but it'll be a parabola so that means you know and if we

values for x give me you know i square them and give me some y

when i square that negative times the negative makes it a positive

around and so you know we keep going up and then these two ends go

but yeah the the negative x values get squared so then the y

i still can get negative y values because if in my graph i

subtract you know five or something like that so there's

squared are less than five so when i subtract i get a negative y

but you see the the general trend of the parabola here and they

but if you have if you have a negative x value

you so any negative x value it looks like a frown so our

be negative so let's yeah let's take a look at this if i had any

quadratic and this is the general form so if i have the general

y equals ax squared plus bx plus c and that covers all the

any quadratic equation so a would be my coefficient for the x

be my coefficient for the x term and then c would be the constant

a has a number then i can have a quadratic b and c can be zero

oh well there's no x term b is zero c was zero and then the other

five so then c would be minus five b would still be zero so there

any quadratic and as long as a has a number to it then it's a

then it's some some other type so if we take a look at this and

down like a frown a is positive opens up like a smile and the

is they're all symmetrical so the vertex is the point at the

how it turns up of where the gravel turns around so since it's

is going to be this definite formula negative b over 2a so what i

i can plug in my four my values for a and b and get the x value

the x value i would plug that x value in to get the y value and

that would be the point that if i had my parabola that would be

it turns around and also just looking at that x value remember

divide the parabola nicely so we're going to look at doing this in

and then finding the vertex finding a lot of other things so we're

be able to do that so i'm not going to worry about all the

you is one other thing that we're also going to look at the code

this formula is a lot of times when i graph this if i were to

my x y and you know let's say i have a parabola there i might also

where does the parabola cross the x-axis and so when it crosses

because you know on the axis i'm not going up or down at all so if

would talk about solving for x but if i have something like this

i have one variable which is x how would i solve it we approach it

than we would the other the other ones because when i square root

to happen as a part of the process when i square root something

positive or negative possibilities because if i had if i had the

four and then the square root that it's positive four or negative

positive two because if i had two squared that equals four but if

that would also equal positive four so if i'm going the other way

square rooting yes it could be positive two or it could have been

finding the roots for a quadratic you know setting this equal to

involve square rooting which gives me a possibility of a positive

where we have you know our two possible answers and so that's how

quadratics you know the way they cross they're going to cross the

to find the roots i want to find out those two values and the

so again knowing a b and c the quadratic formula

all right so and notice it's that

that vertex plus or minus something and then that's how we use

b squared minus four a c all over two a again so this is the

because i have plus or minus it will give it will give me two

the code we would run through this once for plus get that answer

there we go but also since i'm square rooting something we're just

the real numbers here and if i have a negative value under this

solution so you know we could get into imaginary numbers and you

why should we let that stop us but for our purposes we're not

we're going to say that if it's negative then there's no real

just stop there so in doing this i would often test this we call

discriminant and i would often test this you know b squared minus

actually that can be zero which is fine because that'd be plus or

it would have one root it would just touch the x-axis that's fine

means i have no roots and i would have a parabola that just never

that's perfectly fine it's possible and in that case i would test

then that would save me some time you know not having to do all

this old school and just factored it all out you know right right

it takes a while we're going to look at how to make this a lot

we're going to put these formulas in and then we're going to find

oh you know maybe we'll even graph them too so this is the formula

they're both had the same denominator so i could i could make this

you know do the adding or subtracting in the numerator all over

and mathematically that works out you know it's another way to

this is the quadratic formula as we would often see it just like

all over 2a and so for as complex as this could be there's only a

i get a b and c formula for the vertex formula for the roots we

we'll you know then we can look at some of the applications but

the essence of looking at quadratics here and so now let's take a

um well you know how to solve these formulas and everything so

so solving a quadratic equation is much easier with code because

talking about that we would work out by hand we're just going to

so you can just define a b and c once and then run those values

and output your answer roots vertex and whatever else you need so

that work out uh nicely a b and c and what do we want to do let's

so we'll print out what our function is then we'll calculate the

variable and that's the negative b over 2a that's going to

no order of operations but this 2a does need to be in parentheses

so we get that x value and then we're going to take that x value

for the y value a times and my x value vx and then square it plus

so i get i plug all that in get the y value and just like before

there we go we just print the equation now we have the vertex as a

and we'll go on to the roots and the roots remember the quadratic

but it's actually d is for determinant here that's the part that's

because if that's negative then i won't have any real roots and so

under the square root b squared minus 4ac and then i'll test it

zero because if it's not then i'm going to print out no real roots

than or equal to then we're going to calculate the roots and

quadratic formula so since i already figured out d i'll make use

so we're going to run through this twice for the first root

and then all that's over 2a and notice parentheses for the

and then the second root negative b minus the square root of d and

so i'll print this out calculate the roots print it out or you

and there we go so i actually already ran this and what do we have

x2 plus 5x plus 6 and then we have the vertex is this negative 2.5

and then the roots x equals 2 negative 2 and x equals negative 3

so there we go if all you need to do is solve it it's nice to have

just change a b and c and run it again so if we want to graph it

do everything we were doing plus the graphing part remember our

numpy i'm also going to import math for some things that we'll do

i just pick different numbers here and i'm going to print out a b

i'm going to print this out calculate the vertex print that out

so notice i didn't do the roots yet and we just have x min x max

else we were doing for graphing define the x values set up the

parabola i just called it y1 even though i probably won't have a

ax squared plus bx plus c there we go i'm going to graph that

so i'm going to plot x and y1 then i'm going to plot the vertex as

here for the the plotting parabola the vertex point and then

it's going to be a line and here i'm defining two arrays each of

that's going to be a red dot so then we're going to find and plot

doing with calculating the roots from before but i'm going to add

and once i get the plot i'm going to plot root one root one zero

you know the array of my x values and then the y values and this

so that's why it's good so it's going to be a green circle so i'm

i'm also going to print out the roots because i want to see that

it'll have your equation here there you go x squared plus four x

and give you the roots now these roots are irrational you know

wacky decimal we could you know do quite a few more lines of code

wacky radical answer but we're not going to worry about that now

if you're writing this down somewhere you might round it to a

but here we go we plot the vertex as a red dot the roots as green

the x-axis and then we graph the parabola and so i can redo this

that we were doing before one five there we go six and that's all

and it'll run through and calculate all this there we go x squared

and here's the vertex negative two point five negative point two

two and negative two and negative three there we go now there we

where it crosses the x-axis and see the y value of that vertex is

two five so it's just below it but just barely and there we go so

we can graph it we can get our roots in our vertex and for most

quadratics that's what you want to do so let's take a look at how

graph so i'm going to show you some sliders here and that's going

other notebooks so how a b and c so the first thing is we're going

map plot live in line and we're going to write that first and then

widgets and interactive the same and then the same other things

so the way that these widgets work is we're going to put all the

we're going to connect again with functions here i still defined

later on i wanted that to be a short name for that line but you'll

this f but it's the function of a b and c i'm going to take those

now you could always make this say quadratic and you know do

with the quadratic roots vertex we could actually take everything

with the plotting and with solving and put all that in this

it more than just that if i might say quadratic and take the

you need so kind of cool that you can do that but here we're just

so my function there we go x min x max all those i'm going to

remember a b and c would be the input so we would have those and

plot the the parabola plot uh the vertex as a point plot the roots

there we go this all of this here is really just extra for the

title of the graph i did that instead of printing it out which i

could have done but i wanted to show you we could set it as the

needs to be a string so here i'm going to take a b and c and

i'm going to take all of this and i was you know this doesn't

but i was thinking of that h1 for the heading so i'm just going to

all these that exact y equals combined with sa that string and i'm

as h1 but if i just leave it at that it'll still display in a

like extra quotes and i don't like that so i'm going to just find

it's just blank with quotes and that's going to be a string and

for w in h1 you know w like for word in h1 and then h2 equals h2

these two strings put it all together as h1 but then i still

this way combining these now h2 is a nice unified string without a

you know again it's a lot to just it all leads up to the sake of

as the title okay and you know we'll see hopefully you know it

that was you say oh that was worth it all right so here's where

this interactive plot now that's just the variable that i chose

really happening this interactive function so if we take a look at

so that's where the name of your function goes and that's where

i wanted this to be short so i just that's why i called it f so it

that function and then it defines a with a range from one to nine

to nine and c also with a range from negative nine to nine a i

was zero that's actually not a quadratic and i want to illustrate

at one so we just define this interactive plot and then notice

then if i just say interactive plot it's going to print it's going

so we'll see this interactive how that makes the sliders here

there we go and it default shows them in the middle and see this

that when it starts out you see the title of the graph y equals

zero you see that's it just and if i go if i make that down to one

basic parabola you know x squared and i don't have to put the plus

going to write that every time and then we can see what happens if

i can move c and it'll adjust the graph it'll adjust the title

of the graph there we go and then we can see what happens if i

b you see it moves it along here and now

it got to this point that now where it is i do have roots in the

and then if i have i can see where b as i move that around

affects this whole graph and i can see then how a affects the

so we see some different things that we can do and all these are

you can tinker with these but this is all you know it was just i

that you can set up sliders and in this case to display the

display the graph see the roots in the vertex

you see and i move that down so we display the graph see the roots

with sliders you can see how a b and c affect the graph so knowing

this code at your disposal you can really see you know most

do with quadratics you know find the roots in the vertex see the

you know do some other things so there we go this and some

um now that we've worked through the core skills in this unit

and i'm going to work through extra problems using the colab

apply these resources that you're building and use that use these

come up in a textbook or in day-to-day life so we're going to go

here so here we're going to walk through the foundational math

have not already done it please give this a try on your own this

done that and you're stuck or if you just had a question about one

hopefully answer all those and just like the other certifications

and put this in your uh in your google drive so you know file save

and this one we're going to focus on building cartesian graphs so

deal with graphing we're going to look at some different functions

so you make a copy in your google drive it might be in your google

you can put it wherever you want and now hopefully you know i

you're now working from your copy and we need to acquire the

write just like the certification one the code do you write when

it'll run through these tests to see if the code is what it's

some feedback so this first cell you don't need to do anything and

what you need to do here for future reference in case you want to

you know you include in your colab notebook so this custom library

code is going to get that and save it here in a work in a local

need to run this for each new session as i mentioned before it'll

because of an activity after 30 minutes at the most or even if

out so each new session run this and it'll go through and then

to run yep and see just a few seconds code test passed so there we

so let's take a look at some coordinates here cartesian

renee decart because that's part of his name cart and you know

in life when your last name becomes an adjective so he among other

coordinate system that we're familiar with with zero zero in the

horizontally positive to the right negative to the left and y

down so then each x y coordinate tells you you know how much

going up and we can plot all these points our go-to library is

and as we were talking about importing your own personal library

to do that in a little bit different ways a few more lines but a

uh i say commonly you know reasonably used i've been around a

google colab already did that install part so you just need to

we have that nice benefit you know things just work in the google

and we're importing this library yeah that's a mouthful so we're

that way every time i reference it plt.plt. i don't have to

okay so this is going to be the first thing is the basic just

you probably could change these variables but i don't this is

in the documentation for matplot library so i keep these and it

assigning two different variables to this but it's you could have

going to worry about that right here but you could have multiple

could have like nine little graphs in it if you wanted or

would put numbers in there but we're not worried about that now

exists and then plt.show that's always going to be our last thing

you run it and there's our basic graph notice we didn't add

so there's not nothing going on here and zero zero instead of

bottom left because we're just showing the positive quadrant here

values up to one positive y values up to one and that's it and

test and yes code test passed if you click the button correctly

so we'll go to step two all right so now we have the standard

axis and we're still going to go through import this establish

something here so this gives us the size of the graph and i just

this would be better we're going to modify this as we go on but

that'll give us the size of the graph 10 in every direction so

so you see it does give us 10 in the direction and if you follow

the zero zero exists in the middle here even if you don't see the

positive to the right up to 10 negative to the left down to

in the middle positive up to 10 negative down to negative 10 so

here somewhere all right and this one was one of these that i ask

rerun so it's run this to see what it looks like then change the

so we go through here and we can just move this 20 20 in each

now we notice from this even the one where we only went up to one

change but we see the lines on the outside do and the tick marks

in each direction there we go code test passed so some more graph

i was saying we're going to modify this this is really what we

plt.axis so what were those numbers it really was an array of x

as just one group notice the parentheses and then the open

was that defines the window size so that's why it's plt.axis so i

sense to define those as variables up here so this line here the

and then it's really easy for you to then change the window size

numbers so there we go and this one is not one oh yeah this one is

notice it should still do the same thing there we go and then

20 you know and again we see 20 in each direction

so this is going to be our go-to way to set up the graph and as we

those axis lines that you might be making and we're going to

so we have those axis lines that you might be missing i know i was

want to display these well we have our basic graph and we have the

were doing this on your own that the b that we're going to see

so we have this dimensions that we're building upon and these

subplots the window size so now we have the plt.plot and that's

points now in this case we have this array here from x min to x

what this is is a an array of the x values and then the y values

so we have it starts from x min zero and then goes to x max zero

and rather than plot to those two points it plots a line from that

and with this the line is going to be blue if you don't put

python will pick the color of the line for you and then for the

then we're going to plot zero zero y min y max so what is this

the x value of zero to the y value of y min so from zero y min to

draw that line and again i'm forcing it into be this this space

space or not doesn't matter so there we go so these two plot

plot statements define whether you know define that it's going to

and i even have the comments here so then you see blue x-axis blue

y-axis so that's it we're going to show this plot and

we also see again the the value of defining everything up here

of these the size of the graph and then where these lines go to

this up top so if i change the dimensions of this then everything

the axis lines will still go to the ends so we run it and then

so you can make it whatever you want

i just happen to like blue so there we go changing these so now

to plot a point like we saw that we did that into define a line

well see as we continue to build this define these values here and

just going to end up keeping all of this as the window size and

it blue so how do i plot exactly one point so notice this array of

only one in there and y values so it's five four now here's what

it a point in fact a circular dot so if i have just b with nothing

if i actually just had r with nothing that would make it a line

too so that would be weird but ro makes it a point and we'll see

so keeping all that the same and just plotting this one point

one point there we go one point and it is five at five four over

now we could put the grid lines in but that'll be another story so

so the directions run it then change the location to negative five

one okay so then we just change this to negative five and change

there we go and when we run it

all right so plotting a point we indicate that it's a point this

plotting a line we have other ways to indicate that

so how would i plot several points

well let's take a look all right so a ray is to plot each point

and we can still define it as an array here i'm just going to

the array here an array of x values and an array of y values so if

we go down here so notice all that's still the same and here i can

because you know x values and y values and i did not use exactly x

so we're okay i mean you really could name these arrays whatever

but there we go we have what am i plotting the array of x values

and then here i have red points for each of them now right here i

so we can add so that's the assignment for this step add two

one one and two five okay so the x value would be one and the y

so i have point four two and then point one one and the next one

y value would be five and there we go all right and changing those

don't have to change this code plot that array and we see three

because each of them i made a red point so let's mix it up let's

okay so that's it hopefully you are noticing the subtle difference

each plot statements takes an array of x values an array of y

you what you're plotting so the default plot is a line if you

it would give you a line and python would pick the color rbg

also work as just letters and beyond that there's actually a few

write in the word for the color give it a try you know i'm not

but yeah give it a try and so these are the colors so then we have

ro the o indicates a dot rs you could plot a square and then r

would be a triangle so that if you have different things you're

dots or squares or triangles and then of course that combined with

plot a lot of different things on one graph so notice we've

and i'll call this one line x line y point x point y so yeah not

and there we go so we have these we'll keep all these defining the

and let's plot so if i wanted to plot so the directions are a lot

squares so if i plot here so line x comma line y and we want it to

we go and point x point y and it's a green square gs

so you see if you've been working through the course hopefully

coming at this you know relatively new hopefully it's seeming

more you do this the more you just get familiar with with graphing

be able to do this right off the top of your head so we graph

and we will see so there's the red line from that point to that

here so there we go so we see how we can graph a line and

okay so here's a scatter plot game and i call this making a

but i kind of made the game so but we could take a look so this is

want to look through this code how can i plot a random point and

try to guess the point i made the i made this uh you know a

and we'll define this once and here's what we're doing each time

if we are we're going to get an x point which is going to be a

this is going to be good for different games or different things

a whole library of randomness and random dot rand int and it gives

max from x min to x max and actually the way most ranges work in

x max is x max it will won't go to that it'll go to the number

so there we go so you see again if i change these dimensions it

that so now if x point is the random integer then i'm going to

store this as my y value and you see essentially it is an array

you know but then and i might have been able to skip this step but

you know what what this is and now we go here i'm going to apply

uh gives the grid now this is we you know we had their axis lines

that'll help because the game is going to be here's the point you

so that's what this is yes and we're going to prompt for input

red point and that's going to be an x y with a comma so we're

and that's going to be an array with two elements so then

yes array zero so the first element cast as an integer that's

and the second element is the y value cast as an integer that's

and this is part of why i like to break it up here because i have

this defines it as an array but this is the point that we wanted

point and y guess equals y point then that's going to increment

and there we go all this in the loop and then if statements in the

and you know in range zero to three we really because we're

need that zero the range will start at zero by default so there we

it's going to go through it three times because it'll be zero one

again python ranges if it's range three it won't do three it'll do

it so we go through and guess it three times and then print out

was way at the top initialize it at zero and we'll see see what

with the grid lines we can see so zero and look at these four five

point would be zero five and here we have zero negative three zero

why i made it eight in every direction because it actually the

bit better so what's the coordinate of this point so looks like

three so that would be three negative three all right and there we

it'd be a good way to test your understanding of coordinates

all right and step nine so now if i want to graph a whole linear

define these two points and i can draw a line between them but

the whole linear equation so numpy is the library we're going to

import numpy as np now that one i mean numpy is not that big of a

just seems to be a common thing so you will see np dot this or np

functions all right and then we have the linspace function so that

an array of values when we're talking about functions we can you

this gives us that array and we're going to plot the function with

so if we take a look at this all these so after we import numpy

same setup for the graph and then linspace

we give this now i purposefully made it a little bit less you

we'll get to some other things that we can do with this that'll

purposefully a little bit less than the maximum values here on the

argument so it goes from it's the x values is what we want for

negative nine to nine and in that there are 36 points so believe

actually going to be enough which is you know four points you know

that's actually going to be enough and notice then what we're

and you see that's the array of x values numpy gives us that array

given this i can just put this equation in here as that's the y

through this array of x values that y value is going to it

what it's going to plot so then when we run this we see that it

this graph there we go two x minus three and yep looks like it

and looks like about up two over one etc okay so there we go um

um and we have this run it and then change it to graph negative x

the same now in this case since this was cut off here we didn't

then change this to negative x plus three now we see the the

here because it cut cut off there so what can we do here well we

already defined x min and x max we could actually make this go

this come up in the next things we do all right so we can create

so in this case we've been talking about linear equations let's

slider so we're going to import all these things here now you know

matplot library for the graph for the plot and then we have we're

and then the widgets so we can make all the graphing happen in a

there we go function of m and b and if you're if you're working

in this course this might be familiar and then we'll see just you

different about what we're doing here with this function so if i

everything happens in the function defining the x values y the

y the maximum defining all this and in this case we're going to

we already know that this is going to be a slope intercept

title all right and then we have there we go np.lin space and we

time i also rewrote the numbers but you could always make this

else here so what are we going to do here we're going to plot x

because these m and b values are going to come from slider and

take that so here's our interactive plot i mean i could have made

but the function is interactive and then it takes f what's what

it a range and b equals and we give it a range and we run this

to plot now notice it default goes to zero so as soon as i

with the slope of two line with the slope of four and then b is

up and i can move it down and we see that i can adjust my graph so

that we can adjust a graph all right and then if i'm graphing a

graphing system of equations that then hopefully this is a good

a look at here's one way to graph a system of equations all right

these now here's where we make it a lot better so how many points

it also based on this x max minus x min so that's going to give me

the negative so that's going to give me you know this range and

enough times 10 that might be plenty for even more complicated

have that you know there's the range and then times two and then

here we are defining this linspace from x min to x max and then

so we see again we're building these i have this comment in there

pretty soon then you won't even need that and now we again we

our complete uh our base of what we're importing and then the

so then i can plot two lines you know define y1 in terms of x

and what what am i plotting x and then y1 there we go and in line

linear this this plot works you know x squared minus three and

and then y2 and we can see these so there we go all right change

so y2 we're going to make it negative x minus three so we're just

negative x minus three and we see there we go and with these you

want these lines to be but without that python gives me the colors

know it'll go through a few colors and then we'll go through a few

light blue then orange you know it'll go through a few colors and

two points so systems of equations you know so there we go that

two equations and you can graph more if you want systems of

so the simpy library and we probably don't need to import

example so the import asterisk is going to import everything from

to define these symbols so now i'm going to here i'm going to

going to use to solve so i could define these separately and

do but what do i have i have lin solve and then notice open

for lin solve and then what do i have i have this array of these

zero so that's what you know it would be this two times x plus y

and then the other one x minus two times y plus seven equals zero

up if i want to solve these two it would be those set equal to

variables x and y and what happens when i run it it gives me this

as a point that's the solution there we go so one way to solve

and if we change these to 2x plus y minus 15 equals zero so 2x

change that and then 3x so remember three times x you know we'll

have to remember to put the multiplying symbol in there minus y so

over and that's it just 3x minus y equals zero so there we go and

so there you go one way one of the ways to solve a system of

equations solving a system of equations and you may have already

code to solve and graph and output you know make the output even

if you haven't already gotten to that you will

so now we can get the solutions as coordinates so as we saw there

but given that set we can convert that into an x y coordinates so

i split out first and second as two different as a way to put

solution i don't have to change this code then solve first second

sometimes that's a little bit easier

now we know that that solution is going to be a finite set so this

i'll call this variable x solution is solution dot args zero zero

solution dot args zero one and so there we go you don't have to

ways that that's organized especially since you know you think

one x y solution but there's a lot going on here so now i have

so now i have these as x solution y solution and i'm going to

and just for kicks i'll also print it out as a coordinate pair

so see zero zero i mean it works out

because x plus y and x minus y so they would cross a zero zero

so there we go again another way to then elevate that and solve

just showing you different different ways that you can parse these

now we can even get the system from user input so i'm going to

you know this remember to use python syntax and then notice how

to zero because i'm going to prompt for the input here first and

and the second equation and i'm not even casting these as anything

linsolve works out like that there you go first second and then

let's do this as long as as long as two things set equal to zero

plus four minus y so if you think about things in terms of y

put a minus y here so three x plus four minus y okay

and i just hit enter and it jumped down so now the second equation

all right so what if i have five times x

plus five here we go and notice the space minus y

here we go hit enter it jumps down and there we go we have an x

all right and then solving graph which

which in in this whole course the solving graph we have the

you'll see as we build and you know you get more and more familiar

ways to solve and graph then you'll see that we can get more and

here we're going to import matplot library and simpy and numpy so

so this one here we're just doing this with linear equations and

notice i'm only asking for m and b separated by a comma so i'll

i'm going to split that and make that an array and from that i'm

the first one and that must be m and then the next one that must

this input i wanted it separated by a comma so then we're going to

input is a string splitting the string makes the array and then

array the first element element zero i'm going to cast it as a

the second element i'm going to cast it as a float and that's

the same thing for the second one m and b and then everything's

split it at the comma makes that array and then that array the

second element cast as a float b2 and now that i have these

i can solve it so here we're using simpy to solve it because i'm

i'm going to give it this variable first is this because i know it

mx plus b so it'd be m1 times x plus b1 and then minus y that

and then linsolve first second and then i'm going to just in case

just in case there's a weird solution i want to take that

to three decimal places so i'll take that solution cast as a float

places for many things you know i might not need it but i give

comes up it works so there we go solution i'm going to find these

um three decimal places round them and now i have these the x

so then to make sure the window includes it rather than give it to

my normal window and this is a float so if i cast as an integer

and then i can add 20 subtract 20 you know i gave it each of these

solutions i cast it as an integer and i know that my graph is

and then given these you know points how many points do i want and

and then graph x because i already used x in other places so graph

for my x values so there we go min max how many points

and here's my y1 is m1 times graph x plus b1 and everything here

graph x plus b2 so then when we set up the graph and line one

and the point so now i'm going to plot the point which is x

okay so that way whatever equations we have and then print out the

so if we take a look at this x solution y solution we run this

this first equation all right so let's see let's call it three

one oh because i didn't put the comma

so three five and negative one

so we have these and we see these points this is the solution even

as not a nice integer decimal numbers but then we see the graph

they intersect and these are kind of the things that we want to

good way to plot and graph where if you know that it's linear then

and we can do that for other types of types of functions you know

when we get to all the parent graphs and all the different types

you see that this is one way we can prompt for you know if it's

coefficients and then that works out to solve use those

simpy to solve and use them again in that plot library to graph

display the answer all right pretty good

so the next thing we have is quadratic functions when it's not a

anymore not not a nice straight line so anything that involves x

is a quadratic so x squared because it could be the area of a

the side but i know that the area would be that side times itself

there there's going to be a parabola and here's the simplest

plus bx plus c so b and c could be zero they can be any number

because if a was zero then i don't have a quadratic

so then you know there's the simplest one so now let's take a look

back to our graphing setting up our points you know range our

up the graph and i'm just going to have a basic y equals x squared

squared and if we run this that's the most basic parabola so a in

a in that case would be one and you see it does touch it zero zero

so now quadratic function abc's let's take a look at abc and let's

and we're going to use these widgets so

so this one here all right you have a and b but we forgot c and we

put the values on you know in a different way but just did it this

now notice here it's our interactive that we want ab and c so

f and we see function here of ab but i want ab and c so

that's that's going to be your that'd be your assignment add c to

so there we go out here comma c equals and then i'll do the same

nine okay so there we go so now i have this interactive i'm going

for a one for b one for c and then when we define the function abc

and the plot is going to be ax squared plus bx and then we have to

so now when we run it now it's all zeros that doesn't look like

as soon as a is something now we have a parabola whether it be one

what does a do to the parabola you see as a gets bigger the

and zero we don't have a parabola but negative values it flips it

negative like a frown and there we go so let's let's have a

i'm going to skip over b for a second what does c do notice the

and c just moves it up or down

and could be zero that's fine it just touches it zero zero so

down the y-axis and then b is the most interesting it moves it

little bit now watch what happens when we move it so that was a

but now when b if b is zero it's symmetrical on the y-axis but

positive b value stretches on this side so there we go some

the a b and c and what they each do to the graph

so this bottom part or if it's flipped over that'd be the top part

that's the vertex so that's the point where the parabola turns

you the formula here negative b over 2a so the parabola turns

negative b over 2a so then given those inputs

then we're going to create the parabola now this i even have the

because certainly i can print x with the two asterisks for the

um just another fancier way to output this and you'll see it'll

it'll look like an exponent so supposing then we just prompt for

we know we know we're doing a parabola so we just prompt for these

then what would be the vertex well the x value was negative b over

going to do negative b divided by and the 2a both all that needs

need the parentheses so negative b over 2a because if i don't

do an order of operations now if i have that value then given that

x value so we have a times and remember the general form of

i have i have vx is my x value squared plus b x so that'd be vx

vertex plus c so there we go so i use this a b and c to get the x

and then i'm going to plug that in to get the y value of the

and then x minimum so for whatever we enter we'll just make sure

so counting whatever these values are and you know there you go

take it as an integer and that's 10 minus 10 there we go and it's

so notice just that x value just that y value as a red point and

i probably could have done this based on x min x max but just

the lin space just giving us that enough before and after it and

we plot x y without the third argument by default it's going to be

the color so let's see let's take a look so it's going to show us

that little two that was the whole idea to get it to be that fancy

one a is one and b is let's say negative negative five and c is

so we see the vertex and the vertex is 2.5 negative 0.25 and we

we see the vertex plotted so pretty cool and that's it just same

those coefficients and run it through those formulas we can plot

we did the linear equation plotted for m and b we can do a

so you know we could we could do this again and see some different

at how to apply these parabola qualities to projectile motion and

through the air is a projectile and they all actually make the

pretty interesting it can't not be a parabola so as we look at

the graph of this that we're going to look at is going to be for

straight up in the air so in this case the a value is going to be

because it accounts for gravity and that would be in meters per

the a value stays the same the b value represents the initial

on how hard you throw the ball up in the air or launch something

velocity is b and then the c value is the initial height so c

the ground you know like kicking kicking a ball and if it was on

the ball is a parabola and then in that case c would be zero the

we have our classic quadratic ax squared plus bx plus c and in

seconds and then y is the height at any given time so let's take a

what we were just doing but a little bit different so a stays

prompt for that if it's a projectile that's going to stay the same

i'm going to prompt for it and cast it as a float because remember

so rather than just say oh what's b i'll prompt it as initial

now once we have those you can and i already did copy the vx and

the equations for the vertex are exactly the same so you know once

same formulas vx and vy then if i'm going to graph it i'm going to

always 10 in every direction just seems like a good starting point

to be concerned about positive values here so i made x min

negative just so that you can see the axis same with y min made a

can see the axis but any negative values won't apply here now

up here so how do i get these well the x value i'll tell you this

so the x value of the vertex is exactly right down the middle

the middle sometimes we call it the line of symmetry so it's

we know we're dealing with all positive values here so whatever

i'm going to double it so that's why i have two times vx so there

it i should be at the end but i just added 20 just to be sure we

maybe a little bit more it just gives a good perspective on the

x value double it i decided to add 20 but then we have to cast it

ones came in as float values and your dimensions here have to be

sit with the y value i'll take that vy and yep that's the vertex

point of our projectile and i'm just going to add 10 to it i

you can see see this and it's not right at the edge also casting

need to be integers so this should look familiar we don't need to

changed this to four just to give you know four times this just

graph isn't choppy two might even have been enough but there you

to the graph and there we go we define the x values same function

then that that that will work again we need these to be integers

value as our quadratic ax squared plus bx plus c so now we're

been doing plot and then we're also going to plot you know a red

so all this is going to be very similar but you'll see the

then the graph the scale will be a little bit different so when we

uh let's call it 40 and initial height two so this could be

up in the air you know you could probably throw it at 40 meters

by the time you release it yeah it's about two meters off the

see these and you could always round these i didn't for this but

but we get the value of the vertex now what this means in time is

you throw the ball up in the air it will hit its maximum height of

83.6 meters so there we go if you could throw a ball you know that

reason i have the negatives you see the blue for the for the axis

a ball at 40 meters per second then uh straight up and you know

you know 83 meters in the air and then time wise you know four set

where it's going to hit its highest point and then a little more

ground so there we go we can see what our parabola would be and

different things relating to parabolas whether it be a ball thrown

yeah there we go so see i i just broke this out as you know see if

see just shifts the graph and here we're going to import all these

than put a slider just decided to make this and i made it a

take a look uh our x values and then we plot we decided to make

and if we take a look so what do we want in in this range

i'm going to actually didn't use c itself as a as a variable

this range then i made it negative x squared plus c and then what

plot it you know to find the x values ones up here and here's the

and this just i did that just for the sake of setting the title

so you'll see how you'll see how that comes up in the title and

and this little this little display here you'll see how that that

the notice i don't have the defining x min max y min y max this uh

just make the graph fit and each time so here's a python sleep

just in seconds sometimes sleep methods are in milliseconds this

so i wanted to display that and then wait a half a second then

and then wait so you'll see it'll display this and then clear the

so all those different c values we can see it grow and we can run

so all those different c values and notice the the y-axis changed

the larger graph so we can see that value all i have to do is you

and it's the same shape it just moves up there we go

okay so we found the vertex but the quadratic formula since we're

equations what does this do this gives you the roots now if it was

now if it was a projectile that would be when it hits the ground

think about it you know ground level is the x-axis and then you

then you know like roots of anything growing you know wants to go

roots they are the x-intercepts because it's where the line

the zeros because y equals zero so you'll see it come up with all

if i know a b and c here's the formula if i were to write it not

what x value makes y zero that's the question so that x value

the square root of b squared minus four ac and all of that over

minus because somewhere along the way deriving this you do a

you have two possible answers so that and that's what it gives you

but also notice this is the common way you write it but this it's

so we often write it like this you know all over one denominator

vertex you recognize that formula plus this and then the vertex

another just another way to write it whether you split it into two

together over one denominator but when you write it this way then

plus or minus something and again we were saying that x value of

so it would make sense that that value right in the middle and

value minus something you know that's where we get our two roots

you know they're symmetrical all right so given a b and c we have

i have again here i'm just going to import math because i need

and i have it printed out like this where i just have the comment

prints two as an exponent i think that's just a little bit nicer

found how to print that out nicer so you're going to print this

all right ax squared plus bx plus c equals zero and given that

b and c so there we go now here this is where you're going to

we'll define these variables here they're zero but we're going to

a little bit later because i want to define these variables that

statement but i want to run it through this if statement here now

minus four times a times c because remember that's what's

and if that was negative then your answer is imaginary because you

number now certainly you know some mathematicians you know years

that stop us if they're not real but for our purposes here we

the square root we're not going to go into imaginary numbers so

is less than zero then we're going to print no real roots and this

this stuff that's underneath the radical here and that's the

minus four ac if that was less than zero then there's no real

zero then this plus or minus doesn't mean anything because plus

so if that equals zero then there would be one root and then any

two roots and i think that still comes out the most common it's

these other options come out often enough no real roots what that

is a parabola that does not touch or cross the x-axis so you could

just parabola and it opens up and crosses early you know at some

you so that's the discriminant and knowing that tells us you know

than zero then there's no real roots it if it if we only have one

only one root we're just going to print that out twice and not

to translate this this quadratic formula into this well we're

going to do the very similar thing just one's going to be plus in

minus so let's try this so we have negative b let's do the plus

c all right now all of that remember was in the numerator so i'm

all this as the numerator and then divided by our vertex

okay so order of operations all this stuff python will know but

denominator i'll put everything in the numerator in parentheses

denominator in parentheses and then the second root i can really

and paste it here but the difference is it's going to be minus

there we go so now a lot of things you can do that you know in

we already did the vertex and that and finding the roots these are

you need to do with a quadratic formula so having this there we go

so let's just try numbers that work out pretty nicely one and we

try that look at that the roots are negative two and three good

nice integers but this will still work if you have you know like

you see it does have roots definite real roots but since it

and there's actually like a more complicated square root answer

this that could be a bonus bonus thing to do to output that square

so there we go the roots are this and this and you could always

expecting you know that you might have some weird irrational roots

three or four decimal places and i'll leave it up to you if you

all together and have something where whatever the application you

get the roots in the vertex and graph it you know as well as

so you know just to show you you know we can put these two things

interesting things you can try so sometimes the graph i like the

i'm much more interested in just a table of values so in this case

i made this linear so here we go we have this now notice i didn't

a fig dot or a comma x i just have x equals so i'm not going to

here the graph i just want this because this plt dot subplot in

going to give us a table and access off right because i just want

we're going to we're going to give it a title we're going to run

and notice how this works i can define my columns and i could use

it cols for columns so i can define my column headings and then

tuple of values so in this case in the first row x is zero and y

and then what i can do is in this case since i have this for a in

it from one to ten here's what we do rows dot append and then i'm

values you see so x is my a value whatever that's going to end up

then two then three and then two a you know just something simple

is how you can get a table of values rather than graph it you can

row so you set the columns set the first row and then append each

x dot set title i've already defined it up there and then take a

and it takes the argument cell text equals rows column label

know i feel like that's uh that location's you know okay this goes

just close the parentheses there just to show you so it's still

what we're showing is a table instead of a graph and you see how

both so you know that would be that would always be something you

table on the graph but who knows maybe you want one or the other

data science this is like an introduction there's some things you

or things like that but as you get into you know in the next math

when we get into the stats course we're going to look at like all

so this is just one way and you can we'll see even more ways but

there we go and you could always make this quadratic or whatever

again just for simplicity to illustrate the table itself i wanted

equals 3x plus 2 and where do i need to put this well this is just

correct thing so that's going to be the title and here i this is

a is my x value so it would be three times a plus two there we go

it you know and zero zero five ah now zero zero actually should

now one of the things you can do is you could keep this and then

and remember for the range at zero you don't need to put anything

or you could just say oh i know what this initial value is going

so you know a couple different options there

so there we go we have our table okay so different ways to display

have a nice an interesting uh mini project here so we have the

projectiles so we could have you know we could we could you could

um so what this would be is remember knowing where you're you're

randomize the height and location of a wall and you have a toy

so you can you know the game would be determined what initial

would get the rocket over the wall and you could take it even

path or you could you know there's some other things you could do

to write but i'm going to show you here's one thing you know one

so we imported all this stuff and actually a lot of the things i

uh one of the earlier steps where you're graphing a projectile so

and i'm also going to import random and import math so here i just

so this is the what it will print out it just has to land so many

randomize also like h for the height of the wall i just made it

so here's this random integer between one and a hundred and then

rocket has to land this many meters away and then we'll go through

so input initial velocity and instead of it it's zero zero it

you know just because but it really could be zero and then same

the vertex here and the x min what i would do is x max now i just

i know that the wall is going to be you know anywhere between

i just left it as that so and the y maximum i left it so that you

projectile goes but we could always go back to what we were doing

and y maximum the same thing where we get this that where we get x

we could always copy these because that's what we want to do here

and that was that was the idea i want you to you know make use of

these these later steps that's really what what it is i want you

have and then maybe modify it a little bit there we go we could

because wherever that wall ends up being or the the finish line

print everything and that'll be fine there you go path to the

the lot the line for the distance and i'll just make it a red line

i just have it go all the way up to the top here location my two x

y maximum so that's just going to be a line if you wanted to also

you know h equals and we'll do the same random integer

where do i want this to be i want this to be i don't want um

a hundred's probably plenty you could even make it a thousand

um you'd be surprised you know these rockets um

so many meters away and there we go so if h would be the height

the height of the wall would just be h so that's the slight thing

times the wall the scale of us the wall end up being so tiny but

that there so there we go so the line and plot the parabola now

pick just one of them and as it turns out because of the way you

b minus is the root that we want that's the one that's going to

that's the one that's going to be greater than the parabola so

quadratic formula again and the way it displayed i wanted to round

places so here we have if root two is greater than or equal to

distance i wanted to print out the distance and then tell you

much here we go all right so this is it there is actually no test

here we go a rocket has to clear a wall 85 meters away initial

my initial velocity 300 and then we see that initial velocity is

then we're still we didn't clear it so missed it by that much and

things you can do with this so 19 meters away i'm still going to

velocity of 250 let's make sure we clear it you see and we see the

did clear it now some of the other things you can do is you could

that distance take that as your x value of your vertex and you

always make a parabola that perfectly that perfectly clears this

interesting or you could modify this game to say that it has to

um not necessarily that distance or more but maybe you say all

exactly 50 meters away and you don't want to miss so that would be

you could you could actually write the code and i'll leave that to

as for something you could write the code to calculate the

so there we go now for this you know upward motion one of the ways

you know whatever the distance half of that and then that's your x

of the of the vertex and you can use that to calculate you can you

some formulas and as i said we'll get even better formulas as we

we will do things to adjust the angle and such to get it to be

so showing you how to make a projectile game i think it's pretty

you know that you could and you could even make it you know like

have to how many ones do you get correct but there you go the

the this is going to be more like the open-ended part all the

you you should put this together in a function that will graph or

a table solve a system of equations now if you've been working

you might have some of this already put together and that's where

here is you know hopefully you've built up to this so you know you

i know how to do this and i have some of this stuff in place so

organize it use some of the code that you've already made you know

different functions because then you get to the certification

display you know this is what you want to do build the calculator

and a table for any y equals input cool and you can make that

very clear in the code that you know here's where the equation is

graphing graph two equations and plot the point of intersection so

pretty big you know graphing the two solving and graphing and

in the vertex so that'd be some things you know we've been doing

do some things with quadratics enter a b and c see the graph plot

maybe also print them out you know solving and graphing systems of

showing the table and then you have you know these projectile

game you have the scatter plot game you can even modify that

you know something else that might be interesting so there you go

working on you put it all together to build this graphing

certification project and by now you're even deeper you know what

we're about to then have some more twists and see some more

so what do we mean by parent graphs and we'd have heard this in

child graphs have everything that the parent graphs have plus some

additions and so these paragraphs are the basic patterns that

so let's take a look at this the one the most basic would be y

just call c here and so that would be this blue line here just a

y equals three or y equals four or something like that it could

nothing else and whatever the constant is it's always going to be

call it a parent graph and whatever the constant is that graph's

you know flat no slope you know nothing else going on just y

and that's why it's a parent graph because that's like the basic

things like y equals mx plus b well the paragraph for that is y

b is zero and just that basic it's not y equals a constant because

right the paragraph is just y equals x and then we can get like y

or anything like that we can have other things so this wouldn't

things you know i have a slope or i could have a different

this so that's the parent graph another one i wanted to show here

of this works better than somehow sometimes how it plays out with

function and so it's this step function some people call it step

greatest integer but the floor is when you're not rounding you're

it's 0.9 doesn't matter we're just dropping all the decimals so

these decimals it's zero until i get to one now it's one and then

until i get to two so that's it there there are no in between

and then at the next integer it jumps up so again that's a parent

the ceiling function where no matter what it is i go up to the

they're all very similar looking graphs just shifted a little bit

a parent graph we talked about x squared as a parabola and i

even the axes for these because i want to get the idea of this

it's kind of like half this parabola bends down now the reason why

as you get into like x to the fourth x to the fifth all the even

graph look more like this all the odd ones look more like this but

going on in that graph so remember x squared could have ax squared

had other things going on it's still a parabola but it's a little

maybe it's like a little bit narrower a little bit wider or it

still be a parabola the x to the third if i end up with x to the

then that those curves get a little bit deeper

or you know that's you know again paragraph and then beyond that

and we'll take a look at some of this in the code too and the same

so x to the fourth is going to look more like this maybe a little

but if i had x to the fourth plus a bunch of other things in there

something else times x you know all these other things that could

most you know x to the fourth could turn around one two three

always perfectly symmetrical but if all the if all the exponents

symmetrical kind of cool it works out that way so these other

we could have other things going on and i didn't want to draw a

talk mostly about it when we get to the code because that way we

way python displays the graph is going to be much better than my

parent graphs these are the basic patterns the simplest pattern of

and then we can look at what what else we can change or add to to

that affects the graph so let's take a look at the code so here

parent graphs and the code to create them and they're all going to

setup we've had so far for setting up any plot mat plot library

our points set up the graph so this simplest one is a just a

equaled any constant number notice x is not even in there just y

this in the same way that i'm plotting the x-axis i'm just going

minimum to the maximum and then y is just always going to be five

a red line and that's what it looks like it's a flat line there's

that whole part of the equation no x value and y is just always

y equals you know seven point five or y equals two or you know

no slope so that's a constant graph

and then the linear graphs we were talking about linear graphs but

the simplest linear graph y equals x so anything else i do with

in this case it's y equals one x so the slope is one plus zero so

so again the simplest and then anything else i do that is an

so there's our simplest and notice everything else i do to set

my x value and then the y value is x so there we go x x and

the slope is one and it just goes on the diagonal like this there

so every point here you know three three four four five five etc

so the quadratics we were looking at quadratics this is this basic

y equals x squared so before i do anything else to it that's just

and here we're plotting this my x values and then the y value is x

so our parabola so in this case you know x squared so a is one b

and then any other thing we do to the x squared you know add other

uh three x plus four or something like that or some you know

is uh an offshoot of this paragraph so there we go and i can make

and we see the negative a flips it upside down so we have you know

like a smile so i'll go back to the smile there we go again the

x squared and then a cubic graph y equals x to the third so the

it's like the area of a square if i know the sides the cubic x to

volume of a cube if i know the sides so setting up the graph here

about later just starting out with the basic x to the third and we

so some you know some of these you know i could sketch it out

kind of looks like a parabola with the left side flipped down but

simplest cubic graph so what i wanted to show you is also

have other things so plot the x values and supposing i had like x

squared minus three x plus four you see i can have all these other

and this is how it affects the cubic graph you see it gets these

kind of flattened out before but x to the third is going to turn

we go we see these possibilities here all right so again you know

this is you know a more complex graph the parent graph still is x

so we go x to the fourth and yes they have a name too quartic so x

even exponents so we see i'm just plotting x to the fourth all the

this so x to the fourth if i also had like x to the sixth notice

need more points to make this look smoother but you get the idea

beyond what i can graphs what i can draw so i like showing showing

even exponent graphs uh we might be able to do x to the eighth and

that looks kind of horrible so x to the fourth is probably the

other than that then we have to change you know how many points

it a lot smoother but that's where it kind of looks like a problem

around zero and all of the even exponent parent graphs will look

x to the fourth plus other things and then the graph would look

go up and down as many as three times i could have as many as

you thought we were going to run out of names but nope x to the

quintic all right so x to the fifth and when we look at this there

looks a lot like x to the third so all the odd exponent graphs are

but again just a little bit more flattened out and if it's just

other things then the graph could be all all over and it could

but there we go so all the even graphs are like x to the second or

the y-axis and kind of looks like a flattened parabola all the odd

the third the fifth all the odd numbers look like this you know

then with you know as they get more complex with other things

beyond the parent graphs then we have all they can actually have

value graphs and there's a lot of patterns with absolute value

as they would be for x squared graphs the way the vertex can move

value graph is just this absolute value of x but like the other

something here inside the absolute value or outside i can add or

so this is how i would write absolute value with the two vertical

parentheses and here's how i would put this in code abs and notice

anything that's part of the core python library so that's the

then it looks like this because absolute value means it's just the

negative doesn't matter it's just the number so it would be like

would be in the positive direction but y equals negative x so for

y value is positive because it's absolute value so that's why

the y equals x graph it comes to a point and makes a v so there we

whatever x is it's going to end up being positive when you square

all it just doesn't do anything except make it positive and so

similar patterns to some of these but there you go perfectly

that's the absolute value graph and then i can have square root

how i write the square root if i was writing it out and if i'm

now there's math dot square root and there you know and other ways

going to use numpy dot square root because we're already using the

else so since i'm already using numpy for everything else that i'm

want to use np dot square root so the the comment we'll talk about

square root graph and we can start talking a little bit about

is the inverse of x squared and that's why i also included this

it you see now just the square root it's just going to be positive

negative part then that's not a function these x values have two

them together i see that it looks like a parabola on its side and

the inverse of x squared because i want to do the opposite i

helps us see it but the full graph the true square root graph is

function all these x values have one y value and then there is

beyond this all right so there we go so there's the square root

is with a rational exponent now overall for this graphing python

so i'm going to do this here and i'm going to show you that this

x to the one half exponent but if you try this for any pretty much

half it's just not going to work it's just going to give you an

there you go that's the same that same graph so x to the one half

graphing it you know it works out nicely the denominator is your

so square root denominators two or if i had a cube root the

to write it like this here x to the one third so that's the cube

times itself three times gets me x so like the cube root of eight

two times two is eight so we're getting the cube root and i don't

exponent here because numpy just like there's a square root

so that gets the cube root and this one i can have negative values

up three times i can have negative negative negative so that works

the cube root of x there we go so somewhere around here i have

you know there we go and then we get the cube root so the all

the other ones i could have other things going on in this equation

that would shift it up you know i can move it around but you know

paragraphs these types of shapes and then other graphs are just

all right so the floor function now there's the floor notice it

value but the it does have feet at the bottom that go underneath

to show you the floor the uh other function that's similar that's

and the floor function is the one where you drop all the decimals

like 5.9 and it doesn't round you just drop it and the answer is

just drops the decimals and just takes the whole number other

integer function or the integer part function same idea there's

matter what the decimal goes up to the next one and we have round

and to plot the floor function now i want to show you this this is

the function np.floor but there's a reason why i didn't do that

in a loop to graph this as a bunch of points because i feel like

function a lot better it does give these steps and sometimes on

a step function so it's the same value until you get to the next

and it's that same value until you get to the next integer

so all these are just sudden jumps once it gets to the next

change because if it was 3.1 the floor is 3 if it was 3.5 the

values if the floor is 3 and then when it gets to 4 boom then it

went through all of that is that the actual floor function i feel

show you you see it gives these values but these vertical lines

because it's not continuous like that and so it's just what you

but that's what the floor function looks like in numpy but truly

it looks like so that's why i wanted to do the loop and show you

floor function exponential functions so now we go from things like

is the exponent so what happens when the base doesn't change but

so that's two to the first to the second to the third well we can

value is going to get really big really quickly

and so here to the exponent of x so we have we see it gets really

and then when we think about our exponent rules when x is zero

so all these exponential functions cross the y-axis at zero one

flips it to the denominator so two to the negative first is one

negative is one fourth and so the negative exponents for negative

get smaller and smaller so it has that curve but it's not at all

that just gets smaller and smaller but actually it will never

where you can keep zooming in you'll see that it just gets smaller

zero and then over on the right it just has that upward curve and

is never perfectly vertical it's just really steep and so that's

functions we use this a lot of times in percent increase compound

but there we go that's a lot of the exponential functions here

all right so there we go so these are all the key parent graphs

all the other graphs you're going to look at in algebra are going

graphs oh it's like this graph but then with some some other

uh later on when we get into or another course in trigonometry the

and one of the big differences is they keep repeating these are

thing and then that's it but we'll get to those graphs but for

the parent graphs so when you look at a graph and you want to

be that generated that graph you you say well which parent graph

like this so it could be you know an exponential oh it has that

different one so you know that's why you want to get a sense of

develop equations to fit different curves later on and we'll get

so there we go and you will have this as a resource too to consult

different parent graphs now that we've worked through the core

through some extra problems and i'm going to work through extra

so you can see how you can apply these resources that you're

use these to solve problems that might come up in a textbook or in

go through some more extra problems here so now that we see these

little twist here let's look at the these sliders we can add to

that's it that they're meant to be modified so that first one y

value just it's that horizontal line if we add a slider now notice

that you know we're going to import this in line and it's a widget

our graphing is going to be in this function that we redefine here

f because i wanted a shorter function name f of and then it's

in this case we're going to also also going to zoom so all these

slider so you can zoom in and out and we still have these points

here within the function set the plot and notice all this same

all this and so mathematically you know if i if i wasn't writing

y equals a but we're going to put it here i'm going to have two y

perfectly horizontal line we kind of have to do it this way and

horizontal line just like we did with each axis you know the x

y values y one and y two which are both the same which is whatever

slider you know run this function f a goes from negative nine to

we run this so there we go and as i increase the slider so we see

in a little bit but that's what we get like y equals some constant

that parent graph it's still the same look that horizontal line

graph to the child graph it just it just changes where it is see

then the zoom that becomes even more interesting when i have our

now i have y equals mx plus b what happens when i use sliders and

m b and i'm you know i like the zoom in there so i'll just keep

so we have everything for the zoom now here and these next few we

numpy linspace and that's the array that i want to graph for my x

i have y equals mx plus b so look at that m and b are still

is going to affect it and then that function is going to take that

slider change the m value change the b and i still have the zoom

so it's right here because the slope is zero but what happens as

now i start to see my graph and i can see the slope and then as b

we can see it the y intercept moves up and down there we go and

you know now we have some more that we can see when we zoom in

one i didn't put the grid on there but just to show you that as

you know that's really just that's the paragraph y equals x

i can change the slider and see the negative values and from that

two values makes it its own graph but it still is that linear

graph still has those core features but then we modify it a little

i can change the slope i can change where it crosses the x-axis

illustrate this so we have now three variables for our quadratic

and so my function i'll take a b and c and zoom in

so then my y value is just this standard quadratic i remember x

a is one and then just x squared that would be the parent graph

parabola but then when i have these three it still has that core

parabola but then i change and we can see what a b and c do to

i'm going to adjust a b and c and as well as the zoom and so we

middle so let's just put a now that's one but we can use these

to the graph when a increases you see it gets narrower and then

parabola anymore but then as it's negative it flips it down there

affect you know we'll bring it back up to positive like a smile

jump to c next because that's the easiest it just same shape it

negative same shape and it moves it down and see each time i move

again and then recalibrates it with these new values and then b i

interesting because i move it as b increases it dips down to the

and then i can bring it back and then as b decreases it dips down

it dips down to the right so just some interesting things that you

this would be x squared minus 5x minus 5 that'd be the equation

minus 6 and we can zoom in and see you know i i didn't put the

you could always do that too so we see these sliders now the

want you to notice with absolute value how similar those patterns

value because remember whatever x is if you square it it's

makes it positive but the core absolute value function so you see

in three different values here from my absolute value parent graph

because i have a times the absolute value and then inside the

that's where the b shows up and that shifts it left or right and

so we see that plus c shifting it up or down and let's take a look

got nothing as soon as i change it now i have my absolute value

this is linear each line is linear but then it comes to a point

were negative the absolute value makes it positive so we see some

as a increases a times the absolute value this gets narrower just

and c keeps the same shape but just moves it directly up or moves

we have b this is this just moves it exactly side to side so

the parabola there's no other deeper curve to it it just moves it

you know c was more it's still it still moves it side to side

it still moves it side to side so b is negative eight so the if we

then we see that that is the x value

value so it's at negative eight and then because it's b it's x

is the is the formula so b is negative eight then that means

makes it a positive that's x plus b in the inside the absolute

and then c is three then we go up three so we can see again you

and we see the basic absolute value and then what each of these

similar to the quadratic but that has its own its own things and

the most useful i feel like for a lot of algebra you know more

functions can have a slider too

and this one it gives a runtime warning which i just left in there

the you can't take the square root of a negative number so you

zoom values that i could you know at some point i get a negative

compute it won't show up but it actually that doesn't break this

but then that's it so here if i take a look at b and i make b zero

so again everything zeroes out where is it but if a is anything

and we can zoom in a little bit see this curve a little bit better

and then we see b it moves it but remember it's because like the

so when b increases it's really x minus three moves it three to

and then c also you know moves it up and again we're looking at

where that square root function starts whereas if b was zero and c

we start at the origin zero zero so we see you know your basic

these different values you can have a times the square root and

b and then i still have plus c outside that can shift it up or

parent graphs all these key things and we start hopefully you're

how this comes up and you can have these sliders with all

now all four sliders and we can see you know x to the third now

as this to show this is what happens you know a increases this you

bit narrower but b is even zero c is negative d is even zero

set up you know that's where you get you know some deeper dips in

so and it's usually some mix of positive and negative coefficients

dips and you can do the same thing x to the fourth and you can

x to the fourth would also have a times x to the fourth and then b

c times x squared d times x to the first and then the constant e

the zero so there we go and having these you know we can adjust

always symmetrical but i picked one that actually could make it

notice where the zeros are because x to the fourth a is the

b is the coefficient i made that zero x squared c is the

first i mean that zero it's symmetrical because all of the even

the odd exponents zero out so these even functions happen only if

and notice i can have an e value because e is x to the zero and

so there we go so some things you can see when we have sliders you

floor function this is just kind of cool and so what i did here is

but i made this floor function i did this

to make these a bunch of dots because i like how that displays the

that's really like this is really the floor function and i made

the points instead of lines and you know they're big enough that

between but that's really this is really the floor function as we

know if a increases it just gets steeper and a could be negative

but it still has that step look to it so and then c it's almost i

but that's it so this is the floor function we have the different

have that's the parent function just the floor function but then

parent function would be yes i would have a times this so i keep

nothing else going on that would be the parent function and then

these you know that taking it beyond the parent functions you can

kinds of other things for you know there we go some some other

oh this was the uh exponential function we'll end it on this here

my a coefficient and then b to the exponent of something cx minus

nothing going on because a is zero but if a is at least one but

i have uh b has to be a value it has to be one won't do it it has

it has to be greater than one and then c it's c times x this won't

the exponent zero out so again it would be a constant that has to

and there we go we can zoom in a little bit

a little bit there we go and you know d that could shift it and e

doing before but you see these there's a few of these that needed

show something so we see you know again general parent graph of

the things we can do to shift it so applying some of the formulas

in algebra we can use these for business applications like demand

would be how many would i be able to sell so imagine if we're

and i find out that you know an average price of like six hundred

will get me you know i might sell like 84 million computers so we

you know this might be the case and then maybe i find some some

oh well you know what our average price is going to be triple it's

the neighborhood of 1800 hundred dollars but then maybe they'll

and then we look at this and we say okay well what should my price

we can find this we can use these two as you know an x value for

how many sold and we can do the slope formula and we can get an

being linear and i can take a look at what the demand might be so

84 minus 8 over 600 minus 1800 so remember our slope formula from

if we have 76 over and then the the denominator happens to be

get that into an exact decimal that would be my slope and you know

something like negative 0.06 so then i could get that into a y

um you know somewhere around there negative you know 0.06 x plus

what would be my y intercept i could i could go through that same

the code i'll show you that we can just bring in these formulas

a couple points getting our slope intercept form and then i would

there we go and yeah it's probably you know i'm gonna just rough

so let's let's just say that that would be it and then that would

as the price gets higher you know we would graph this as you know

and we would see that as the price gets higher then the number

you know less and less and at some point actually the why are some

and at some point you know we the price would be so high that

that as the general estimate now let's just say that's the

be the demand equation and a lot of times economists are looking

um business owners too what would be my ideal price point well

i do to figure out my revenue well my revenue how many sold would

as i was saying is defined as the x value so at every point i

the demand and in this case that price is always going to be x and

times negative 0.06 x plus 50 and you see that would be my y value

quick distributive property we recognize that this is

a quadratic with a negative a value so that means it's going to

and so when we look at this then

if i have you know the price but then instead of the number sold

i want to have revenue so of course if the price is zero i have no

there's some other price that is so high that nobody buys it so i

of this graph is going to be a parabola that opens down so then i

parabola that opens down so then i see that there is some middle

and that's kind of the thing that we want to look for you know

the slope formulas we're using the formulas we've developed to

this would always be true revenue is price times demand so then we

quadratic and we can graph these we can find out what these what

so pretty interesting then we could also apply this to profit so

we could do this and then subtract subtract the total cost we

some of the applications that we want to get to that you know

based on some data you know what should my price point be to

if i already established a price point what would you know can i

are things people put in their business plans so we see these

and i when we look at the code i have two other examples so we can

together that you you know you can use your math skills you can

to make some business predictions and these are some of the good

of of all this math and writing code so let's take a look at the

to give you an another example or two to show how you can write

and for this example i'm just saying we're selling t-shirts and

they seem kind of realistic but you could define the price here

here so if we define the price and in this case we're going to say

dollars each and then we have the demand as our next variable and

giving away t-shirts maybe you could give away 50 you know this is

not really going to give them away but then your demand decreases

time the price increases increases now in this case again if the

away 50 right here the price is five and then so two times the

be 40 but you see how it just cascades we just define the price

about demand and then revenue is going to be price times demand

standard formula the price you're selling it four times how many

we've you know once we've defined these then we have these other

on them and then the total cost so this is another one that you

uh made it so i said four times the demand so if we have four you

it cost four dollars to make each t-shirt all right and i'm sure

but i just said four dollars so that's where the total cost would

you know however many we sold and then that's the total cost so

minus total cost makes sense what's the rev how much money do we

and how you see how we have all these we can just define these

formula for for profit revenue minus cost and this is a standard

demand and some of these then we could develop these equations if

data so if you're making things to sell you would definitely have

how much does it cost to make but there you go all these then if

demand equation we can calculate all these and we would output you

and revenue because five times forty and then the total cost we

dollars and then having these this is where people can do some

oh well let's say supposing i decided to make the price ten

and we take a look at this the price increases the demand

but the revenue increases and so given that cost we still have a

of more than double so the price of each doubled i went from five

profit more than doubled it more than quadrupled so just

here and how does this happen well it might be it might be helpful

we're going to graph these things import matplot library and i'm

going to remind you here that the price is x and the demand is y

that the demand later is going to be the revenue now this is where

you would have you know if i have at what at a certain price what

and i had in mind here you know that this was you know i don't

you're buying two dollars 46 so it would it might not be t-shirts

but let's just say you know i don't know bottle of water and you

maybe you hype it up and tell people it's just this amazing bottle

or if it's really hot that day and water's cold you know you still

something else that you you might have but there we go so we can

one so if it was two dollars then the demand is 46 if it's ten

to 30 as an example if just having some of this data from some

these formulas we did back with linear equations because this is

demand or x and y so i'm going to calculate the slope based on

intercept and then this becomes a y equals mx plus b equation so

you know some examples of some data that we have that we can we

that's going to be my demand equation and i'm going to go through

now notice this x minimum and y minimum i'm going to make zero

aren't going to really have any practical meaning i'll make the x

there we go the number of points that's all fine so after i define

i'm going to do a couple other things i'm going to do the x label

and then i'm going to give my graph a title so notice both these

there we go and then for this one line one i'm going to have so

equation y equals m the demand equals m x plus b and so we figured

and so i can use those but i'm going to use x because that's the

defined so we have our demand equation and then i'm going to plot

label is still going to be price but for this one y label is going

to plot x and then the demand i commented this out we'll get to

so here we go so i have my graph and then we see yes if i was

give away 50 the demand decreases as the price increases so at

um you know whatever i'm selling here that started out as two

something you know some people still want those if i have 25

nobody wants it and then somewhere in there we have the ideal now

linear in real life but this is a good estimate so given these

equation and as we were saying before i can build upon that and i

revenue equals x time demand because as long as i keep that

and then so now i'm going to plot my x value and then revenue as y

my y label to revenue so we see that we have a way to do this and

of course i'm going to make no revenue if the price is zero and

going to sell none so i'll make no revenue but this is how the

interesting that i can have some you know low prices and i think

best i might sell the most but the revenue actually there's some

that gives me my maximum revenue and you see it looks like now if

and at 25 these parabolas are symmetrical so it's exactly 1250

or if people like round numbers you know 12 or 13 so there we go

use all of this and we can see that the you know the reason why

where the demand would be zero so this is probably the more the

one but having given getting some research of given prices demand

a couple values here then i could plot this out and i can plan oh

research that i've heard what should i make my price you know and

that people can do you know you have some new product how can you

all right and there you also then if you're making a business plan

download or export these images you know you've you've written the

the graph as an image to include in your business plan so there we

examples of how we can make use of these you know formulas the

generate some graph or some image and some data that we can really

okay so let's apply some of these graphs to some other economic

python your math to plan predict you know look at trends in

profit and if we take on that marginal cost revenue or profit if

when making and selling one more product and for some of our

equations and for that the marginal cost revenue is the result of

and for that the marginal if cost is linear the marginal cost is

wherever you are but sometimes we have costs or we saw that

revenue that are not linear and therefore we have some times when

from a cost revenue profit perspective to make one more and

sometimes you know oh that one more is only going to get me a tiny

and cost sometimes it goes the other way you know setting up

be a really high cost but now it's all set up so making two three

you know might make that worth it you know things like newspapers

you know there's a big setup for the printer and so the marginal

it now there's definitely you know other methods printers but for

big printers like a newspaper marginal cost it'd be a whole lot

going to print thousands and thousands so the marginal cost after

a hundred thousand newspapers printing one more is almost nothing

things you know what what's it cost to make one more and you know

and profit the same thing and so if we have revenue you know and

comes up but you know toward the end of the day people say oh what

make one more sale you know maybe it's a store should i stay open

tired to make one more sale that might be you know a lot more cost

little bit of revenue not worth it and so they say okay these are

so these are some deeper analyses as we get into things beyond

we can do even more than that makes that even more interesting but

know as we talk about this the marginal is that at any given point

to make one more all right and let's look at some other things

and here again i'm going to give some simple numbers and make it

supply and demand graph it's not going to you know and in real

linear there might even be a little curve to it but even at that

it might be you know a little bit of a bumpy line but we have

definitions are at supply how much are people making you know

i'm looking for and the demand how much will do people want how

so you know if you're looking at you know a movie the movie

theater the supplies there's only so many seats you know only so

a play or a concert always so many seats only so many showings so

and then the demand could increase if it's something really good

so a lot of times we see that you know that affects the price and

look at supply demand and price and we're going to graph supply

we'll see some similarities some familiar things here for the

values as old demand and new demand and i just gave them this

x values as old demand and new demand and i just gave them this

it reminds you that it's an x value so we have old demand and new

the demand increase and then i have old price and new price so now

we'll see this on the graph but the price is also going to be the

i'm going to have supply old supply and new supply i made that you

as i was mentioning you could potentially get data from a table

data you could have one table that has you know a particular price

and you know make all these numpy notice i did not import numpy

putting a few variables here and you know to illustrate the graph

get that data from a table and you know make each of these numpy

that would be really interesting too and that's something that

course so given these and then i have a basic graph set up adding

going to make the x label quantity just general quantity and the y

and i have this title which i already graphed but we'll get to

and i'm going to plot this as a red line just because all right so

i'm just making this just like i do it with the axis i'm going to

and these two y values and make you know make that a red line so

not going to even put an equation to it i'm just graphing this

to this point and i'm going to do the same thing for supply as a

just going to use this data that i had these numbers that i came

um and this graph we can take a look at this again it's simplified

of supply and demand so if i made green supply red demand then we

quantity so the green that would be quantity supplied and the red

supplied and the red that would be quantity demanded and then on

so graphing them both on price we take a look at this and we see

then the quantity demanded is going to be low and a lot of times

um an independent variable on the x axis and we think cause and

correlations so i'm not saying necessarily one causes the other

so uh at the when uh x is let's say one and y is 14 so we look at

low quantity of demand when the price is high and then as the

increase as the price decreases so that's what happens oh you know

you know $14 no i don't i don't think i want it but as the price

you know the demand increases more people would want this now

this they have a different incentive they want to make money so

the and when the price and when the price is low then nobody

into this so you see we have low demand but we also have low when

is low here then nobody wants to get into this what you know why

if i'm you know whatever i'm selling is not going to be you know

the green line the quantity supplied will increase as the price

industries that you know oh you know why why are people you know

because they all see they can make money oh this is the hot item

you know uh make money so that's what we have that the supply

increases so there we go so we have these two opposite trends and

core of economics because as you know i get different data this is

as an example but these graphs could shift you know maybe there's

this that maybe you know instead of uh the supply increasing like

causes it to increase a little bit steeper or maybe you know if we

supplied is low here for this price but maybe you know for

level was even at a higher price and this whole thing shifts so

of economists are trying to do and then if this is quantity across

where these lines intersect and then that would be the point of

saying is the quantity here supplied and the quantity demanded

that's ideal you know you know let's say you have a bakery and you

morning and then at the end of the day you have it all sold

on the left you know and we'll say let's say that you know for our

at six but let's say a little bit before that so it's still to the

so the demand's not really there the price is increasing but not

not really there and we still have a gap between demand and supply

line is higher there's more of a demand but not enough of a supply

side of that we have more of a supply but not as much demand and

be on the right that would be the one that's dangerous for anybody

because you know they're getting into this trying to make money

where somebody's happy oh good the price is going up i'll make

the demand drops off so now they really have to scramble for

the price here or on the left for the people who want this product

supply and that's part of what keeps the price higher because the

supply well then the price increases you know things like you know

about this stuff all day anytime somebody wants to too many people

oh well then you know the supply increases too many and we go to

sell this you know then other people notice that you're oh you're

and then the demand kind of goes down you know what's wrong with

and on the left then we have the price is too high because there's

and you know that sometimes causes oh there's you know there's not

sell that you know this must be good and then that kind of

bit so you see this one simple graph describes all these

know our goal all around is you know finding these points of

you know therefore everybody you know making whatever products are

for these products are finding them you know and you know fair

prices we're all trying to get to that but we see all these things

and it's not just one direction of cause and effect it's multiple

as just general correlations all things happening together so we

say we say the trend is like as the price goes down how does how

it goes the other way too as the price goes up how does that

that is elasticity so elasticity is how quickly does uh demand

and we can look at this there's the formula percent change in

and this is again all these economic formulas i'm just going to

that when you have these formulas you can use this you know get

then use it to calculate these formulas to answer these questions

how elastic is that price because if things change if the price

changes demand then we can say it's a very elastic like it like an

bounces that you know it stretches and changes easily or things

it does not change demand you know things like uh gasoline you

price goes up we don't like to pay it but if the price went up a

car so that demand would not be very elastic the price could

and you know we can so we can see that if the percent change in

know some significant number but then the percent change in demand

calculate that you know the total value of that fraction is going

um and we're going to say that's not very elastic but maybe some

you know a concert ticket you know there's ones i've seen where oh

i'd like to go see this but the price is too much no i won't go

oh i heard the price went down that would absolutely change the

i i won't go oh the price went down now i want to go and that

that aren't essential and we would say that that would be very

and you know down the road you could always combine all these into

just going to show you like just old demand new demand old price

calculate the percent change so remember then that's you know the

there's their difference divided by the old price and it's going

to a percent so you see again and we want to get in the habit like

you know colab that oh i have a formula i want to use you know

and then now all we have to do is change these values up here and

that we need so price change and that's the ends up being the

with demand the new demand minus the old demand and that's the

and that gives you a percent it will give you a decimal that you

elasticity number is demand change divided by price change so what

print that number and here's some of the analysis so again

what's the cutoff point you can always just put these in you know

statement and so if the number is greater than one then i'm going

and notice this is you know else if so i just continue this on you

one then it's unitary or proportional so sometimes you know we we

increase i think that that proportional increase increases demand

and then if it's less than one then that's inelastic that those

you know again it the demand hardly changes based on price and i

and so you see this is how you can you know i i like to when i do

tag on one more thing at the end you know for whatever reason

capture everything if it's it's either greater than one equals one

really but capture one other like grab bag you know what you know

very often we'll have if and lf without you know exhausting every

easily just put an else there at the end so i have some sample

and it would print out so we get this negative 2.2 and then demand

so as that example and we can do the same thing with supply so

um as price change because people say oh hey i want to this is a

into this or then people say oh nope it's uh you know this is

so in uh it you know in the united states kind of recently the

market uh was hot and that's supply like okay not everybody

house or try to be a realtor or try to sell some property but when

more people decide to get into it like oh maybe i will sell this

that if the price is good and people see oh opportunities to make

so price elasticity of supply answers that question how how

to changes in price so very similar formula percent change in

and just a few values to sample and very similar formula you know

divided by the old price and uh new supply minus old supply

and we're going to go through the same thing elasticity number and

greater than one it's elastic which remember as a fraction greater

that this percent change in supply is much greater than the change

and we have to do percent change because if we just do you know

number of things supplied that really doesn't always give us a

in general are low cost so a change of a dollar might be a whole

high cost like you know a change for a car that a price change of

at all so same thing with supply you know what what's what's the

relevant to the previous supply so that's why we do the percent

you know a certain percent but supply change is a lot more related

oh wow that's elastic people are noticing these trends and we can

analysis either supplies elastic unitary or any elastic and i just

there we go this number and it's inelastic because it's this tiny

everything you know supply cannot always be you know elastic and

and given the old prices and old supply you could always take

something like this so that therefore you could graph and if you

other numbers so notice this i mean even though these numbers

bit of research that you just had a few values here you can get

you know six things calculate you know the graph and elasticity of

and really just get a handle on you know where where you are in

potential changes you know is it you know these graphs i kind of

proportional but you know very often it's going to be bumpier and

graph and what's what's what's going on if you really wanted to

there's other plugins for other uh data especially things like

use this uh to calculate elasticity for various stock prices you

um and a lot of these it'll work within colab there's a lot of

that are a lot of things that'll work within your google drive and

to update things like stock prices maybe like every 15 minutes at

going to be enough to give some sort of analysis you know for an

different things you know that's kind of what we want to take this

formulas and build upon these so that you know you can uh build

at some reasonably regular level have all the formulas and

you know have this output at some sort of report for you let's see

time you know pretty much real time let's run through some

thinking about investing in this what's it looking like oh it's

elastic so you know you really have to take notice of the price or

i see this trend maybe you know a supply of something that maybe i

all these things we want to you know use this information to uh

want to invest in yeah and you can do this all right here in colab

drive so uh we'll get in some more some some more math now and you

journey so let's talk about calculating interest and usually

bank or in some investment or sometimes it's going to be money

am i paying so the simple interest formula is exactly that simple

it's good for you know not a lot of money if it's long term but

the less accurate it's going to be and we'll see why that is but

rate times time it gives you the interest so if i was borrowing

you know i have borrowed you know a hundred dollars and the rate

0.05 so the rate's always a percent converted to a decimal so the

and as a decimal it'd be 0.05 and then the time would be always in

i borrowed a hundred dollars for two years then what in what extra

so there we go that would be another ten dollars

so what i would then say is okay well you know given this rate if

and it's going to take me two years to pay it off you know maybe

years to pay off a hundred dollars maybe calculating interest

if you know i could calculate oh how much extra would that cost me

the total amount that i would pay i would take that interest and i

principle because i have to pay that hundred dollars back plus the

take a look at this you know if time is one like let's say one

compounding that's the notion of compounding that i'm taking that

the principle so let's say the time was one year and so

so this would be times point zero five times one then the interest

and then what am i going to do i'm going to i'm going to add that

you know the principle plus the interest

um is is going to be five dollars now the reason why we compound

a year so then i have that new principle plus the interest but

interest that's my new amount and then that would then generate

generate another five percent in interest so you see that becomes

percent gives me my new amount of interest and then i'd have to

that you know times that point oh five there we go and then i take

new new bit so that could potentially be a lot you know every time

add it to the principle again and then recalculate it but let's

what's really happening here so if i have this the original

and i probably don't even need the parentheses here so the

plus but remember the interest was principle times rate times

so we're doing a little bit of substitution there so the

but the interest remember that was this formula so i'll just put

factor out p so you see the things we were doing before with

then i factor out p times one plus rt but supposing t is one if t

which you know times one i really don't have to write that and

so we see a nice algebraic way to write this where i have the

then also adding it now we're not going to stop there what makes

because that is now my principle calculating the interest for that

new amount what happens when i have to calculate it for the next

original principle i did principle is what i started out with and

the rate so if i had to do it again for the next year it would be

because that's what i did to get the principle plus the interest

the rate and when we think about this in another way it really

really makes sense because we're saying that that principle times

saying you know this original principle 100 or whatever it is if

you know then that's 1.05 so you know what do i have i have a

five percent and then doing that you know that 105 percent every

that that principle times one plus the rate because if that's what

how can i simple that with an exponent and that's where time comes

time because every year then i'm doing that times one plus the

now we get from simple which is nice principle times rate times

and again good for a short term but you see on long term this is

compound interest because that accounts for every time it goes a

but then that becomes a part of that new principle and then that's

and if this was my money in the bank that's great that's good news

that i'm borrowing then the bank is going to be wanting to do this

that they get all the money that they that that they deserve you

sure that i pay that extra interest so that's where we get the

interest and then we get to this so then that's the you know

gives you you know your new amount so there we go a new amount and

we're going to talk about that new amount we're going to call it

a maybe as a variable a for amount but then it'll be annuity which

that we end up with so there we go the connection with simple and

this a little bit further when we get into the code so let's take

look at the code so we're looking at simple interest as the well

because it's good for short term and for some things that might be

but we just get principle times rate times time equals the amount

again short term because we're going to do some other things to it

and that's what we'll get to here so the principle if we just put

dollars let's say i borrowed money from the bank a thousand

always the percent the annual rate converted to a decimal and then

borrowed a thousand dollars from the bank they said okay it's five

years i'll pay it back here's the formula interest equals prt and

a hundred and fifty dollars in interest so i'd pay back that

but right away then that tells me the cost of borrowing that i

so that's going to cost me 150 dollars so then i want to find out

the new amount is just that principle plus the interest and there

the interest just like we did before and then this new amount is

so there we go and we add that that hundred and fifty plus the

now i have that new amount okay so most of the time we're going to

and then adding it to the new amount it's very rare that i'm

just the amount and not worrying about interest so here's what we

amount is that new amount is principle plus the interest and if

principle plus for one year that would be p times r times t which

it so that would be the new amount principle plus the interest so

if i divide that out i get p times one plus r and if i do the

this so p times one is is p and p times r is p r now this is going

out this p because now i have that new amount just that original p

and the rate is always going to be some decimal that it's going to

one one plus point oh five is one point oh five you can do that in

like a step and then this is what you're going to enter in the

one step that you automatically would do in your head and you're

calculator for these so that was what you would enter in the

now again this is for one year and 50 so if we go back to that

year just to show you that this works one year that'd be fifty

year then then the new amount would be 1050 so that what we're

to the principal that's compounding and compounding is exactly

principal and then calculating the percent interest from this new

happens over a few years so if my original amount is p so then the

r but now that's my new amount so so again the next year so p is

after one year i multiply times one plus r and after another year

and we'll just do this one more and again p after one year after

then that's just the same as making this an exponent p times one

because i'm multiplying this factor one plus r three times and now

that instead of having to go back and every time calculate the

then do it again you know in some loop i have one formula that

to do is add the exponent for how many years so now the annuity

r exponent t so that's 1157 now it's not an error that this is

was 1150 when i did this by hand i just you know calculated the

and then added it but that's after three years that was 1150 and

good for short term but the longer it goes the less accurate it is

i'm compounding it every year notice that's every year that i'm

next year so remember after after one year it was 1050 so then for

that five percent interest on not on a thousand but on 1050 and on

percent interest on that new amount so that's the compounding it's

the hill and accumulates more snow and gets bigger and bigger as

every time we're compounding because i'm still taking that rate

larger amount so compounded i get 1157 just that same amount over

seven dollars just because the formula was better so time is

every year but we also have a formula supposing compounding

us this more complicated formula but that's okay we see how it

of r we're doing r over n because this is happening n times per

annual rate so i have to divide it by how many times i'm

every year was the exponent the exponent becomes nt how many times

so then i'm compounding however many times as an example here i

common monthly we know recalculate compounding 12 times a year so

these parentheses gets smaller but then the exponents larger and

if this is money now if this is money you're paying back it ends

more but if this is money you're putting in the bank earning

benefit so that thousand dollars five percent for three years but

per year i use the extra parentheses for any fraction so there you

it was one plus r over n i put that in parentheses also and you

to make sure that stays because that is the whole exponent nt and

know in math we might skip the multiplying symbol because we're

python have to put that asterisk there so there's the annuity the

about this instead of just amount that's annuity and principle is

out with and annuity is the word for the amount we end up with so

monthly look at that 1161 so we go back from that simple interest

annual 1157 much better to 1161 because we're compounding it

better but first i'm going to start talking about this irrational

these very practical money in the bank things that we can

about this irrational number e and yes it's e but it's a number

the next formula all right so just a quick note about e we're

import math because e is a weird number and so when i print math

there we go and as a good estimate you can just call it 2.7 but in

want to use this whole number so that's why i want to import math

you know say 2.7 and call it good another interesting thing about

we see it's 1828 and then the 1828 repeats again that's kind of

a couple times and then it just all falls apart so math dot e use

and it's e because it's euler's number his name began with an e

logarithms and other things when he came up with this number he

mathematician that you liked euler so sometimes people call it

now if we're talking about compounding n times per year we go from

about e to being very rational well what if i just keep

we get to continuous growth so in that previous formula n could be

it 100 because we're going to do some things with this uh you know

every week you know i just made it a nice round number 100 could

and bigger number then we get the annuity becomes this formula p e

talking about e so i get this irrational number e and then the

i'll just demonstrate this so let's say n is 100 and if we do that

52 so compounding every week and then here i'm going to compare

i call the variable n times so there's the there's our other

nt and then we're going to compare that with our continuous growth

and then we're going to print it this one or this one

one so n times we remember that from before 1161 that should be

and then the continuous growth 1161.83 so it's pretty close but

1161.75 1161.83 it's pretty close but the continuous growth gives

and if n gets to be a much larger number like a thousand

you'll see it gets really close 1161.829 or 1161.834

so we see if we took it to two decimal places they actually round

and that's the idea with continuous growth you see it it's not

than this it's that as n gets larger it approaches this value so

a lot or other things beyond money in the bank or things like that

growth applications you know population increase or things like

and you know i can call it annuity i can call it the you know the

this formula so it's very useful for anything continuous growth it

throughout math and calculus some version of this comes up a lot

formula so i have this continuous growth formula if i want to

the mortgage formula and the mortgage payment formula is if i have

and that's usually a mortgage you know as an example i think this

close to the average mortgage these days 240 000 so that's a large

is 30 years that's a long amount that's a long time so for such a

such a long period of time then it actually really makes a

little bit you know every month the fact that i'm paying you know

principle reduces just a little then the next month i'm earning

principle well that really makes a difference so if we use any of

mortgage payments it's going to be a lot more than you know you

how can i afford that but when we use this more complicated

that we're paying it off gradually and looking at all that then we

amount of time what would my monthly payment be so that it'll work

paying over 30 years times 12 months that's 360 payments so i can

this amount of money after exactly 360 payments my balance is zero

given that i can figure out the payment and then we'll look at you

being a complicated formula if we want to write code for this and

to you this you don't even usually enter this into your calculator

that you can plug in the numbers here so that's what we're going

to break that down i just took the numerator that r over 12 one

numerator i made it a variable and we see r over 12 times one plus

and they did the same thing with the denominator so just the

over 12 so the exponent 12 t and then minus one separate from that

two separate variables so then the payment see i didn't forget

the capital p for principle so i'll talk about the so i'm going to

going to be a weird number and the first part is p times numerator

my mortgage payment so p times numerator divided by denominator

because that's going to be a very long decimal i'm going to round

argument and then the second one how many decimal places there we

and when we run it mortgage payment 1362 and 69 cents so given

that will make this work out that after 360 payments of this so

still accruing interest and we're still making a payment and it's

to see how that plays out we call it the amortization schedule so

that's the mortgage payment schedule and yes all these have the

means death you know kind of like you know what's what's true life

and i guess mortgage but yeah it was very morbid when people first

came up with this word but it stuck we will end this after this

so you know don't worry i don't want to leave you in the more

schedule if i take this there we go given these three things i

all this how we're going to calculate the payment all right so i'm

like what i did before so that's all the same as what we just did

is i want to have a loop to show the interest amounts and we can

that's pretty common and later on in the course we'll talk about

a loop so i'm going to set this new variable called balance and

to be that original p principle okay then i'm going to print this

month i'm going to put a tab there balance put a tab and then

the number you know what month am i talking about what's the

the interest that i'm paying for that month so here's what we have

and how many months 12 t okay so the interest is round so again

because calculating these you know it's going to be a lot of weird

two decimals dollars and cents so the interest that we're going to

again because that short term this is only for one month and then

it here in this loop so i do want simple interest it still has

p for that month times r and then t is 1 12th so i'm not just

going to divide by 12 but that's what it is this is the original

the balance is the principle for that month rate and then time is

by 12 so there we go and all that i'm going to round it to two

the interest this will still calculate the interest and i just put

of the displaying is that i only want if a is divisible by 24 you

a remain a divided by 24 if the remainder is zero then or if a is

almost the last month so these are the ones so i'm rather than

would just want to skip a few normally if i know if i put this in

schedule and i'd print out all 360 months but only for these we're

so i want a and then tab so that you know a is going to be what

range and then tab balance tab and then interest so there we go

to line up month balance interest so then after i do this so i put

first time through i want it to be the current amount and then i'm

so you see what i'm doing to increase the balance here all right

balance plus interest minus payment so you see that's what's

so we're manually compounding it the balance at that month plus

month minus the payment and then round it to two decimal places so

so we have the payment is 1362 just so we figured and then here we

balance interest so month zero we start out owing 240 000 as we

why it makes a difference you're paying 1100 in interest just for

and you know that's it it's you know it's almost the whole

at first after you do this enough times you come to terms with it

paying almost all interest at the beginning you know and then 1362

notice that only 262 is going toward the payment now we could see

but at month 24 we've paid down some things but it's probably not

you'd think like oh i'm paying so much i'd expect this balance to

just chip away at it and even then you're paying less interest but

dollars and that's you know month 24 so after two years and you

little bit less interest we do see this moving in the right

and you're paying a little bit less interest but probably not

and that's why i just wanted to skip ahead you know every two

and you know how much interest we're paying you know we take a

now you know we take and we look at this so if we have just over

borrowed 240 still a 140 but at this point now that difference we

start start building equity in in you know if this is a mortgage

the house is worth is worth a lot more than what we owe and so you

you know home equity loans that's where that comes in as you start

and you know the the value of the house is worth a certain amount

amount the difference you know you can actually borrow money

for another time so we have this and finally now it's still a lot

you know to think about wow i'm still you know look at that 20

than five hundred dollars a month just in interest you know it but

dollars a month and you know we're getting the vast majority of

point to paying off the mortgage and now when we take a look you

instead of three digits thirty thousand and then the amount of

and forty one dollars so uh i wanted to skip ahead and then just

now the balance at that point is just under what the mortgage

is this tiny amount and that would be if you know month 360 when

up being like four or five dollars more than the normal mortgage

so it works out that that that monthly payment you know using that

payment 1362 and 69 cents and that monthly payment did work out

as you know we're look at the amortization table at 360 months it

and there we go for an extra five dollars the last month i'm not

so these are you know interesting things to think about seeing how

payments applied to loans how that plays out how you're paying off

of the simple interest and then paying down this balance so this

for any large amount of money over a significant amount of time we

have to be a mortgage car payments not always because a lot of

if it was a car payment for like particularly expensive car or

of time then we might use this doesn't have to go to 30 years that

this you know paying off this debt over time so let's use

it around and have something positive retirement account

money you know this p is not money that i'm borrowing it's money

into an account and instead of owing money every month i'm going

formula for this that i'm just going to estimate this oh okay

see you'll see how this plays out and i just picked these numbers

principle is a thousand and this is just estimation here well

dollars now this is where the estimation comes in most of the time

any decent fund manager is going to be able to get you point eight

so now this is your money growing in an account this isn't you

growing in an account so if you had a thousand dollars and we

percent per year that's great and i made this time 38 years as you

just thought like age 22 till age 60 38 years so you start it put

account and then you start saving that you know from that point

you put away 350 every month in this account okay and i'm going to

annuity which i'm going to start out the original is p so now

and the percent growth so again 38 years so for a in range 12 t so

we're contributing something so notice what we're doing with this

that we already have plus the monthly contribution and then that's

one plus r over 12 because that's a monthly notice that percent

so the exponent if it was nt the exponent n would be 12 and t

so i don't need to write the exponent of one but that's where this

formula r over n to the nt but nt and is n is 12 t is one over 12

so each month then if we take this and that's why i wanted to

because i want to allow for this contribution here the annuity

we want that to happen each time and then it's going to grow so in

to grow so at the end we're going to print this out and i'm going

that'll end up being a weird decimal annuity okay so now the

but this is what we want to break this down if we're if our rate

let's say it's still earning eight percent when you start taking

interest you see and so that's annuity times the rate that's it

places but that so if i have this this amount of money that i'm

then the idea is that money now is yours in the bank or in this in

that the annual income is just that percentage rate

so take a look at this annuity here so one zero six one five nine

million dollars and if it continues to earn that eight percent the

interest is eighty four thousand dollars or almost eighty five

this money in the bank or in some investment not not in the bank

bank is going to get you this eight percent interest that makes

investment if you're you know you have this and you can contribute

constantly contributing and it's growing so at some point then you

million dollars that earns this amount of interest and i just

the earlier you start the better and the later you stop start uh

can grow even more that you could take the money out at you know

estimated did age 22 to 60 you know if you if you start as early

you know till 68 if you look at that that'd be like 50 years that

because remember this snowball so that eight percent in those

of larger and larger amounts of money and then if you get it to

of a million is more than 80 000 so that's that's the idea we want

of money that just the interest on that you know if this

on that is enough to live on and you take the interest and then

which will earn interest the next year so we want to leave you

we can use all this you know that's that's what use all all these

to you know turn that into money in the bank that's that's the

hopefully then it works out and if you have these this code you

know how much do you start out with you know keep this rate maybe

you know my fund manager is doing really well change the time

tinker with these and run it and you know see the results so you

you need to do to invest for retirement so let's look at some more

mortgage formulas here now like we were doing before i'm going to

formula as its own function so there we go uh def and then payment

pr t run it through this formula and i'll just return the payment

we can use this and all of these you know we're going to apply

know you're looking for a house you're comparing different things

to run this and there we go and i don't need to run that again now

so i'm going to do a similar function and i get something

show you this one i'm just going to define it pmt and i'm only

because i'm going to just give a set rate and time so with this

taking the input as whatever the principle is i'm going to

i'm doing this and we see these imports is because i'm going to

i'm just going to go from zero to three hundred thousand and see

that principle of course the slider shows it somewhere in the

four hundred and fifty thousand the monthly payment would be eight

and we can see how this goes up and here's something you can do

comparing different mortgages or houses you can see how does this

you know and we look at this oh two hundred and twenty one

should i buy a house it's more expensive i know my payment will go

much you see and we can look at this now that's great to see these

to see the trend maybe there's i want to graph it so notice our

case i just define my x maximum is three hundred thousand and my y

going to be plenty all these standard things you would already

points you're going to use for np dot linspace and everything

something that i don't always put in there plt dot grid just to

axe dot set x label and set y label so the x will be the amount

from zero to 300 000 and the y will be the monthly payment so now

earlier i can just use that as my y value i only need the x that

here and i'm still going to give it just a consistent rate and

the input all these are going to given the same rate and time how

monthly payment and when we run this we can see that the payment

that same rate and time and this would be realistic maybe you get

you would know your interest rate and you would know the time so

it is linear how much would it increase you know for how much

for increases in the amount borrowed well all right this is a good

get down to the numbers so it's linear i'm just going to do the

and i'm just going to pick two values you know and you know x1 and

this i just put rate and time here and i define them up here just

i'm using the same numbers consistently throughout this i feel

for the consistent example and just pick two x values calculate

values and calculate the slope formula and what we'll find so

this is you know dollars and cents so really it would go up you

of you know price increase or we can you know multiply it by a

go up six dollars you know because 5.99 i'll round up it would go

six dollars for every thousand extra dollars you borrow and now

an extra thousand that's going to be an extra six dollars in the

and you know you can put things in perspective do i like it that

thousand it's only six more dollars so we see some things we can

formulas that we we've been working with and we can define them as

use them in sliders we can graph them and finding aspects of the

and finding aspects of the graph we can do other analysis this is

building upon for our data analysis so this is all good and i want

here beyond the python i know this is almost all python but i want

do this in a spreadsheet because that will give you like that full

you also how to build that because these are some good spreadsheet

the balance here is a formula but we'll get to that in a second so

i picked this two hundred and fifty thousand that's somewhere

in the united states right now i want it to be some realistic

down payment it's going to be we want 20 percent so notice if i

up here is the formula and i just did 20 times g2 0.02 times g2 so

and i make g2 you know 300,000 instead click somewhere else then

calculate the down payment i think i do want to go back to 250,000

i feel like that's a good middle of the road number for us to use

and click and it'll adjust again now then the borrowed amount i

going to double click in this and equals g2 minus g3 actually

sometimes google will prompt you will give you that and notice

can see the borrowed amount and then up here this is the formula

minus the down payment all right so let's just say the interest

now i'm going to write the word years here i'm not going to use

but here the interest rate i will all right now monthly payment we

to this and calculate the monthly payment so we see here i just

you go six percent 30 years and i already have this so what if i

which was 200,000 even and i can just go to this borrowed amount

oh 200,000 now i could actually also just make another since i

i could actually right under here write print

payment 250 uh 200 comma 0.06 comma 30

so you see i could just actually just put this in here

output the monthly payment all right there we go 11.99 and 10

so monthly payment will be 11.99 and 10 cents

so now notice here this is just a couple of things that we can do

and 10 cents so now notice here this is just a good reference for

price down payment and everything but these notice i already had

the monthly payment and if you'll notice i have these dollar signs

some things we're going to do later i mean i don't need them over

things later we're going to drag these formulas down and it's

the row and the dollar sign will keep that the same so dollar sign

if i ever adjust this that's not going to change it's going to

that cell g4 and same with the payment that's always going to

that's what i want now the interest we've been doing all these

actually this is going to be monthly each of these is going to be

going to be a month so for monthly interest we can just do simple

balance which is a2 times the rate which is 0.06 times the time

so one month is one twelfth of a year so i'm just going to say

and this works out i mean you know that's very nice but it's not

so what i want to do is i want to add another function around here

round open parentheses and i will put you know so it rounds this

you see it'll round it to the whole number but i want to put comma

to two decimal places and this is google sheets if you you do this

have that comma and then however many decimal places even if it's

that it'll just round it to the whole number so there we go so

then my new balance equals the original balance which was a2 plus

so that's b2 minus the payment which is c2 and i want to take this

a thousand dollars in interest you know just you know just

about that and all that the paid you just took a little bit off

dollars in interest and then the payment was only 11.99 so a

to the paying down the original principle so i want to take this

it accrues interest i make a payment new balance and i want to

so that was cell d2 so i'm just going to put equals d2 here

here and here now notice if i drag this down

it changed because it's calculating the interest now based on this

times 0.06 divided divided by 12 and then here when i drag that

because i put the dollar sign in there it won't adjust anything

and we see again doing that it adjusts it for this row now

two everything here is based on row three and we got that by

hand corner you get a little tiny plus sign and this is magnified

the that thin plus sign and you click and drag that but we're

so i'm holding down shift in the arrow key to highlight the entire

over that i get that thin plus sign and i'm going to click and

amateurization table now notice my mouse is down below the bottom

that's 360 months so somewhere around there is where i'll stop

and these are all negatives so i'm going to take this highlight

then just delete them so now here this is you know after this the

so we can just make this 11.99 plus that additional dollar 13 and

decimals like that and that should be yes 10 cents negative 10

three and that should be enough to make that we're just going to

we're going to call it zero and what we have the row number is

first row as headings so three given that mortgage payment it

months you have a balance of zero so we're going to take take this

amateurization table and you know you can look through it and

the payment stays the same but what happens is the amount of

gradually because for such a large amount of money for such a long

fact that you're paying interest on just that little bit less that

and then that means that look you know that same payment a little

more of it goes to the principal so you know you're you're making

paying this off but it takes a while until see look at this i mean

to go into you know 225 months you know we're coming up on 20

payment goes to interest or less than half of your payment goes to

interesting to see that whole table and then this balance here at

amount you know supposing you come into some money and you say oh

it you know that that's the balance that's your payoff amount and

showing you the spreadsheet and how you can set this up and how we

other cells and how we can add formulas and how we can drag all

and you know mortgage mort you know it does sound like death

mort so let's do something more positive with a very similar type

i set up the same thing balance interest i'll capitalize it and

contribution and then new balance so let's look at this in a more

this was this amount that you owed that you're paying off but

account that you're adding to so i don't know let's let's start by

five thousand dollars and a lot of retirement accounts you know

you know get like eight percent interest that's pretty reasonable

so we have equals so this is column i two times zero point zero

eight percent interest and then for one month so then you know one

by 12 there we go and maybe you know we could do this we could do

i'll round it that might that might work out it'll just look nicer

to two decimal places again dollars and cents okay now monthly

you know you can contribute a few hundred dollars let's start with

start with like 200 and we're going to change that later so with

and you contribute so it's all plus equals i2 plus j2 plus k2

and then we're going to carry that around so that's l2 equals l2

l2 and we're going to do the same thing drag this down one drag

drag this down one and what we have is you know given that you

given the contribution you could always add more different times

for about the same amount of time so if we drag this down

now this being a retirement account a lot of times people will

so we don't really even have to stop at 360 let's stop with like

um there we go uh 401 because the first rose so what we have is

you know this amount that you have is 469 thousand dollars and the

this 3000 so you could easily like take that interest and then

you take that 3106 out every month and live on that and then

so you know it it you know goes from this and then back to this

back to this that's one way to do it i just want to show you so

here and say then we contribute something like you know 400

that down in a little bit and this one let's say you know 0.085 i

more optimistic on the there we go drag that down a little bit and

correct and we'll drag this down and the other thing i'm going to

longer maybe even like uh 450 months or something like that 40

so we can take this even further and you'll see that just you know

some poss some nice possibilities here all right and because look

because look at this final number here so yes that one is one

we're talking about here you know one million dollars and the

eight hundred eighty five dollars so this is something you know

take this same math and turn it around and make it positive

to and then at this point you have this money you know in this

out every month or or less and you know not have to worry about it

take that out and then you're still left with this you know more

live off the interest and you still have this every month take

to generate interest the next month and these are some of the

that you know from the other side of it you know you owe all this

but then you own this house and on the same time you're

have money that you can live off of and you know these you know

want to do with our math skills look at how we can manage manage

in life then have this you know property that has equity and you

in the bank that you can live off of and there we go you know

and seeing how you can make this work for you seeing the trends

now you might be working hard and owing a lot and not having that

many years you build things and then you have things oh i have

they've built over decades and can live off of it so that's really

some of this with the other uh python you know generating just

like these you know these are some of the ways you can make this

at the trends and hopefully this helps you make some decisions

whether it be real estate or borrowing money or other retirement

just you know some bonus and there we go we're in the home stretch

this all come together you know skills that you're building

so let's talk about exponents and logarithms so the exponential

is this two to the exponent three two to the third power equals

we've been doing this so far in algebra then logarithm is the same

so this same information the log base two of eight equals three so

exponent that's what we want to know that's what we want to know

far and in most calculators you know two to the third no problem i

calculator in python they don't even have to import anything but

and i know what i often call the result if i know the base i know

is what i don't know then that sometimes becomes a problem

nicely like this works out but supposing you know it wasn't a nice

even nice exponent like three supposing you know two to what

it's going to be some weird decimal but how do i how do i get that

to appreciate that you know uh john napier you know spent like

20 years of his life figuring this out and making tables and now

button on the calculator and we're going to look at also how to

logarithms we do have to import something but you know we'll

than spending 20 years years of your life figuring this out so if

um you know the the base is always going to be whatever you know

what i call the base so if i have you know log base two of eight

now what's interesting is our count in our calculators we you know

you'll see a log button now if i have the log if i just say the

i don't see any number there if i don't see a number then it's

and we call that the common log because our number system is base

write that and so the log of a hundred is two because 10 to the

and we're going to use this a lot of times for working with

converting decimals so that you know that's that's where a lot of

the common log and then i might even if i'm doing things like

the floor function because if you know what's the log uh you know

two point something but then for converting it to scientific

the two because then that tells me to move the decimal place too

1.5 times 10 to the second you see you know different things that

converting this for scientific notation all right other

that exponents and logarithms they're inverses of each other so if

um if i have the log base two of two to the third then or actually

a definite number i'm going to call this x because then anytime

the log of the exponent anytime you have a function of its inverse

now notice then what we're saying here the log base two so two to

x oh x is the exponent you know makes sense and the same type of

to the log base two of x exponent same thing that's exponent and

look at this log base two of x so it's two to what exponent gets

want the exponent two to what exponent gets me x and whatever the

am and then in fact taking two to that exponent so i would end up

we're going to use these inverses and this is where it becomes

for whatever base i have if i'm solving something if i have three

that doesn't work out nicely if it was 27 x is three that's nice

on the calculator what am i going to put on the calculator well

x equals something well if it's three to the x then if i take the

of this three to the x and then on the other side it's going to be

so in algebra in the things you know i do the same thing to both

do well one of the things i can do to both sides is take the log

same base so if i take the log and i pick base three because of

down to x and then over here i have the log base three of 25 which

calculator and we'll see in the code you know we're going to

able to do this you know no problem on the calculator or in the

thing is if you're on a calculator or you know you want to do

i could just do the common log for this it actually works out also

divided by the log of three so the log of this divided by the log

use common log or whatever i'd want for that so you know that's

you know what another way that like that works so change of base

in a situation where you don't have the ability to do the log of

you can use whatever logarithm you have and do this so that's

we use that inverse property when we're solving and that comes up

to look at things you know things that are growing exponentially

sometimes it's you know radioactive decay for dating carbon all

in this situation exponent my unknowns in the exponent and so i

figure that out the other log is going to be the so we have all

the common log which is base 10 but then i also have the natural

and if i was writing this it's i'm going to i'm going to write it

the logarithm functions in python they actually if i just say log

not base 10 so that that's really interesting in python the

another base there but if i was writing it out it's l n for

natural log of of anything so the natural log means that it's base

e is about equal to that 2.71828 1828 and then after that it's

it just that pattern stops so oh 2.71828 so he's about 2.71828

for a lot of continuous growth formulas so if i have you know the

well then my answer is going to be you know about one it's going

it's like 0.99 or something like that you know 0.99 and i can you

for for some other things and we'll see some of the applications

code but that's where if i have base e so if if it's base e i just

i can just write log and if it's any other base and the bases we

whole numbers but if it's any other base then i'm just going to

so okay so we take a look at this we'll see how we can actually

in the code and that's going to be pretty useful for some of the

so let's take a look at the code so when using logarithms in

library and now we have math dot log there we go math dot log and

and then the base so this is math dot log of 10 000 base 10 so

four this works out nicely for things that we know are going to be

show you that python weirdness sometimes even when you think it's

math dot log of a thousand base 10 so i expect this to be exactly

it gives me this long decimal so any of these even when you think

the way that this works behind the scenes you might want to just

so i'm going to round all this math dot log of a thousand base 10

to four decimal places so you just might want to build build that

get that answer and notice i still said four decimal places but

not rounding not needing decimal places it didn't have to give me

there we go so that's just the usefulness of it that it's just

in there all right so that said i have a few of these that i don't

just to show you what the exact answer might be so if i have base

and then we're going to do math dot log in this case 16 base 2 so

four and i don't have the log in there so if i have something like

rounding so you see it'll give you all that if you need these

for something it will give you that but there you go just whatever

natural log i'm going to import math again and notice here i can

and if i don't give it that second argument of what the base is

and e is about 2.7 so if we we do this we see you know it gives it

want to show you is as we were talking about before uh so math dot

math dot e is going to be the most exact you know it'll give you

have to just you know guess at it but also e to the fourth then

so if i have e to the fourth and then i take the log which is the

four because the log of that exponent or if i have e to the third

i use that math dot e to get the most exact uh approximation of e

argument then it will cancel out and that'll that'll work out the

you know 10 to the fifth and then log you know base 10 you know

just just to show you this all right so other good uses of log or

answering the question how long will it take for an investment to

and i still wanted to put the algebra in here so if we have our

so i have p and then i have the annuity what number p what what it

what number it ends up so if it's doubling then whatever i have

whatever the number is so i can divide both sides by p and then i

so as we were looking before about getting to exponents if my

it what's my base and then i'm going to take that log of both

take the natural log so on the left i have the natural log of two

of e to the rt as we just saw is going to be that exponent so

i'll take natural log and then they will cancel out the function

want to solve for t now i have rt divide by r so i would have this

by r gives me how long it would take money to double so we can

just define r as the rate here you know two percent you know sort

natural log of two divided by the rate so i am going to round this

argument there log two and divided by r and also the rounding is

whole number there we go and we get it to be 35 years so if

35 years from now it'll be double and like i said as a low

whatever now 35 years from now it'll be about double that price so

you know some investment you know oh nine you know nine percent

to be doing okay i put money in this investment how long will it

in this case eight years so there we go so you know some people

for what they invest in and there we go answering that question

i want to show you this pattern of graphing the exponent and the

all the usual imports for any graph we're going to do all the

dimensions here and we've done this before when we use numpy i

points here because then i'm going to create my numpy array np dot

using this many points so there's my array now for this i also

array specifically because my x min i don't want it to go down to

at 0.01 and we'll see that in a second and then again set up all

i'm going to graph this first line here y1 equals math dot e to

i imported math just for this using math dot e so if i'm graphing

and then line two so then if it's e to the x i want to do the

and i didn't use the math one here i wanted to use the np dot log

graphing so numpy has log built in also so i want to use that

with with the numpy array and that's this is why i wanted x2 as my

can't be zero and they can't be negative so i didn't this actually

but it would still graph it just with that error and i didn't feel

that other array starting at 0.01 so there we go so now i have e

it's just y equals x all right so this is what this looks like and

upward that's the exponential function and going to the left it

it just gets really really close beyond what this display can show

and then this orange line that logarithm it the x value can't be

has this curve to it as x gets bigger this increases but it not so

goes on and on and then the green line is y equals x so if it

each other across that line that's true like any inverse functions

the line y equals x so kind of cool how that works out so we see

e to the x natural log of x and just as a slight comparison i

two to the x and i'm going to use this and comment out this other

have two to the x all right there you go two exponent x and then

and again i'm going to use numpy and you see numpy's log instead

have different functions so this is so log is natural log log two

there's log 10 so those are the main ones you'd you'd want to be

gonna have log two here and then the line y equals x so if i run

different because two is not as big as 2.7 but very similar and

drawing i i wouldn't be able to draw the subtle differences so i

look the same you know exponential logarithmic y equals x there we

good good estimates here all right we can also use logs for

it print out 3.2 times 10 to the fifth now i wrote that in

with with all the zeros here and here i have it as a negative

second there we go and you might be wondering about that comment

why that's an important thing and there we go uh we see the output

three gets weird very interesting because the exponent of positive

know got a little weird too and you see just the way it works

we would think that it would stop there and that'd be fine but it

and then a five at the end unnecessary but as i was mentioning

will solve a lot of these problems so you know i just use this

so i have this 10 4.5 times 10 to the negative four and i'm going

places so if i have a negative exponent let's say negative four

decimal places and then that gives all of these because it'd be

place to show that so you see rounding rounding again works out

negative five and greater it forces the scientific notation so as

going to print this out i already have it in scientific notation i

decimal places doesn't matter the output python wants to put in

this e is not e 2.7 but it's 4.5 times 10 to this exponent so

calculators use that notation also we can convert something to

if i take this so i want to get the number of decimal places so if

variable a so if i take the log of a and base 10 so this one i'm

take the log but since it's not 0.001 it's not going to work out

it's going to be like one point you know it's going to be like uh

is going to be four point something so that's why i want to do

exponent i want to chop off all the the extra decimals in my

that the floor of that is going to be the exponent so that's

four and then if i take n i'm going to round that 10 divided by

works for larger numbers too but if i take that and then round it

two decimal places that's fine that's n i could stop there but i

in case there's rounding errors because it could work out

greater than 10 you know that that happens and so in that case

i still get true scientific notation one number than the decimal

if it's greater than i'm going to increase the exponent now for

point zero zero zero five and there we go it gives you point zero

ten to the negative four and then this should work for all kinds

converting it to scientific notation if you wanted to so these are

logarithms you know we can see the graph that's pretty interesting

the applications the solving sometimes if my unknowns in the

to and from scientific notation so these are a few applications of

you will find even more so you can make use of make use of this

here we have the foundational math three certification three and

going to make a copy first of all and work through this whole

google drive and then from your copy we'll do first step like

testing library and later on in this unit or the next unit we'll

and how you can set up your own virtual your own library hosted on

into whatever notebooks you want so you know as you can see here

whatever notebooks you want so you know as we get through this

of setting it up and then you'll learn how to set up all these

comments here you know install requests here's the library and

library take it uh as as a new file locally here uh behind the

it and it's imported there we go and this last thing says that yes

on to the next step there we go and remember the runtime run uh it

consecutively at the most or about a half hour of inactivity so

you know working on part of this you leave come back then you go

so we were looking a lot of into graphing before now we're going

i have let's say y is greater than or equal to 2x well if i have

do i shade that well we have this argument here fill between and

so we're going to go here import matplot library and import numpy

and we're going to set our window like we were doing before how

linspace give our points setting up the graph all this is very

fancy put a title on there so we have the plot title y1 just like

i was graphing this just as a line we see all the similarities

and remember this we have plot these two and then we have this

we had this third argument so if you're working through this and

watching this video maybe hopefully you already did work through

you know double checking anything you did or if you want any

we have the second arguments are the fill between so this one

fill between so we still have one x value and the y values are

zero and i'm just going to run this to show you you see it fills

that doesn't matter you know that's not a consistent you know over

is up top and over on the right it's down to zero but thinking

oh what if i just change that to if i want to shade above what if

and then we're always shading above between that line and the

below you can change it to y min and there we go and when we do

so part two well we had this one is a nice solid line and remember

to but what if i wanted you know not that solid line so we have

here we know the logic of this so now if i have y min now that's

below and this is the only thing we're going to change if i have

gives me a line until i do the fill between but what if i wanted

so we see you know see if i have b dash dash r dash dash gives it

and we run this so now we see and sometimes i'll do that i'll make

stands out if i made the blue just blend in so now we know this

and there we go yes and knowing now that we can make some lines

you can define these lines we can make art for example we can do

fill between two different graphs two different functions oh that

this defaulted to notice i didn't even define the color blue here

but i can define that face color what color do i want to do i want

so if you know like a lot of your basic colors you actually can

that you know i want to be green put it in green in quotes single

color names this will work out so that'll work so the directions

and then reverse the order so we see here like before here's the

plotting and remember we defined the x value up here once so

i'm not worried about making a particular line so two arguments

between and see what i have x y1 and between that and 10 or i

max but that's fine and then i add the other argument face color

and fill between now notice between y2 and y1 so you know we can

you know between other values that's this is this will come up

you can define your lines and then fill between these lines i mean

so then change this you know we have this color and same thing all

you know this line y3 and i'm going to fill between y3 and y2 next

function fill between y4 and y3 and so when we run this see that

there up to the top and then i did not decide to have another one

just left it here so if we take this and reverse the colors so

colors and make the first one blue there we go and we'll reverse

red at the end so the next one will be green and the next one will

be red there we go so now we have this and we see when we run it

so there we go interesting things you can do and if you were

whatever formulas you want remember you know we were looking at

you know quadratic you know uh exponential functions so if you

and you can just define that you want it to be between something

where i define this linspace here once so that's the reason why

but you could always for each of them you could define lines you

within certain values and you can make all kinds of shapes and

with other graphing i kept these axis lines but you could always

you just keep your window size but then you could always eliminate

again you can get creative with this make some different art all

looking further into factoring into representing equations so

mono is one and then it actually overlaps n o m is name that exact

and then i al just makes it a noun so one thing one term so each

and variable and an exponent we don't think about that all the

definitely i see a negative sign i see the coefficients three x to

but if i have you know all of these the term x also has that so

first power just things to think about because that really comes

that we do with factoring etc even even a constant like five is

five x to the zero because x to the zero is one and this will help

each of these you can have a monomial and just make sure that you

sometimes these values might be one or zero and you might not

now this is just a very interesting within simp i we have eq which

and you see how this equation is eq and then in parentheses i'm

and then what's on the right so i define it as an equation and

by a comma so you see what we have here so just showing you that

could use this within simp i for some other things that you might

of this course you know if you wanted to make sure you have a nice

up eq you know use your variables that you already have and you

symbol here so we could do this and then just display it and then

and so we have monomials one thing binomial bi is two trinomial

is three and after that we kind of stop counting so polynomial is

three and we'll usually put them in order so we take x as the

the highest exponent and if we want to graph this then it would be

so there we go here's your example and it continued on to the next

know this i this is the normal standard standard form setup so you

coefficients and then display them and so there's what we have

so we're going to use this display and we're also going to use

to import everything from simpy so now we have these symbols and

i do want to cast it as an integer though i mean realistically the

integers so there's our coefficient and a and b continue to prompt

you can do is you can copy this and make this c input coefficient

all right so we've done that and now we want to display the full

display ax to the third bx squared and then we do the rest plus c

plus d and notice we you might have remembered this from before

i just put these right next to each other ax means i'm multiplying

notation i have to put that multiplying in there so it is ax to

go now we have this this is great we have this in python notation

we want to display this in a nice way oh well i have to convert

way we're going to you know to display math terminal math formulas

i'm going to convert that python to there you go so if we have

so you see we can take this and convert it that way you have all

thinking ahead we're building things that you can use you have all

coefficients you could do other math to this here and then maybe

this this formula in a nice way so there we go that's what we have

so this is going through so interactive polynomial so now we're

this you know here's well quadratic but there we go and given this

we're going to adjust each of these coefficients so again in doing

certification notebook you should see that this is coming up you

and hopefully you've already worked through this and you know

you were stuck on or if you forgot so remember we were doing

and we're going to bring in the interactive from the widgets there

and remember these the function of a b and so for this one we're

third slider so there we go so my function needs to include c

because i need these three inputs great and then all the rest of

ax squared plus bx i have to add this plus c

c and so that will show the plot and then we want to make all

i'm going to run this function a b i just need c

so we can just copy that comma

so now we will have three sliders a b and c and we have the

display the interactive plot so there we go so that's how we can

you know using the same thing you can modify this to have four

looks like nothing because everything is zeroed out but as soon as

then we have a quadratic you know negative all right so that's

and it's always interesting to move these and see what this does

so you can tinker with this and see remember that just moves it up

there we go so you can take these and you can tinker with you know

code test pass but these notebooks you can always tinker with

you know you've learned that step you can go back in there and

see let's see what it looks like so now we're going to do

our exponential function y equals a times b to the x and there we

zero then we'll have an upward or as long as a is greater than one

thing as long as b is greater than one we'll curve upward all

some of the exponential functions the so what we're going to do is

see the sliders and then change the slider so that a actually does

instead of positive so here we have all these importing same

minimum and y maximum a lot bigger things grow exponentially so

decent graph here we just have a and b and you see one to nine we

so when we run this we're just going to see what this looks like

either one you see as i increase this it gets to be steeper

but we see the general shape of this it just gets to be steeper

it just gets to be steeper and then if i change a

i'll even make this low because it don't it still will be

so that's that's really all these you know they kind of do the

thing maybe b has a little more of an effect but then you see it's

upward curve and then as the x values get negative this really

get it's beyond what the resolution of any computer screen could

in you know it looks like it doesn't touch and then it looks like

then it doesn't so you know that's the general trend so if we make

need to do one more thing because this interactive graph takes

net making it negative negative one is not the lowest we have to

is the low value and negative one is the high value remember

we don't switch those also then we would get an error so we get

see just making the a value negative would make the whole graph go

b to the x would be positive still so then a being negative there

and just a note making b negative i'm not going to even worry

we did that it works for integer values of the exponent but since

it would be integer it would be work it would work and it would be

an even exponent negative if it's an odd exponent and it would go

those in between values because that would be mathematically

would actually be imaginary they wouldn't they would not

sometimes if we had you know some of them would and it would just

not even going to mess with that changing b to negative it just

anything but behind the scenes that's what's happening it works

doesn't work out nicely for others and you know a graph would be a

that does work out nicely percent increase so hopefully you

working through it in the previous unit but then we have you know

and if i were to graph this so we have a equals p times one plus r

and a is annuity and that would be the y value if you were

p and you would know r so t would be time in years but that would

graphing it and so we have all this yes p is the principal the

rate converted to a decimal and t is time in years and just the

capitalized so it's just interesting in in my variables i kept

times you'll see that written you know they're lowercase but p

um all right so you have your starting amount so you can prompt

worry about changing the code well we are going to change this

amount as input and converting all these to float numbers so you

numbers percentage rate converted to a decimal and how many years

going to take this equation and put it down here in python terms

gotten from here so p times and remember i have to put the times

one plus the rate and all that to the exponent of time and there

equation to to python and then we'll print out the annuity so when

one thousand just take these nice even numbers all right and we

that'd be five percent and how many years then let's see uh let's

and i didn't worry about putting in another function around this

but there you go to two decimal places this is what the annuity

percent for seven years and then we have percent decrease very

subtracting inside the parentheses because it's one minus three

does happen we have car value values of cars or some other

certain percentage rate every year the decay of some elements so

certain elements like carbon 14 that are very predictable so that

knowing how much that we can measure how much that is we can

the age of something some sales discounts maybe there's something

at a percentage you know every week if it's not sold so you have

and notice we're going to do same thing prompt for pr and t and

very similar p times and then in parentheses one minus r close the

so there we go and we see you know the jupyter notebook the colab

us tool tips and advice on what variable and everything so there

run it all right so there you go so let's just say i had my you

that uh decreases at 0.09 which might be the you know rate of

how many years will decrease you know let's just say 10

so there you go that twenty thousand car twenty thousand dollar

depreciation is nine percent so it loses nine percent of its value

then ten years later that car is worth seventy seven hundred

that might be the case i mean you know this is not uh don't quote

for the sake of practicing it that that rate's probably realistic

all right so now we have compound interest which is very much like

but we're taking it you know we could use compound interest we

before if it's compounded once a year so now we have this formula

but it's r divided by n and then the exponent is nt so if it was

and is one so that doesn't that goes back to that formula we were

but where n is multiple times per year you know this is the

and remember that that's really and that's why we wanted for any

wanted to convert it to that formula that it's something that's a

easily adjust the rate of decrease that's why we wanted to convert

and we can then easily adjust this for how many years something

and in this case now we had the other twist also then for how many

every year i mean n being 12 that's pretty common you know a lot

you know they calculate things you know once a month so we're

amount all the rest very similar and then we're going to add this

compounded so using this formula then we'll do p times and in

r over n i'd rather use extra parentheses just to make sure that

uh r divided by n and all that to the

so you see r over n in the extra parentheses to the nt but again

in parentheses n times t in parentheses so that way it stays

python might not but i want to make sure it doesn't accidentally

you know well then multiply by t so there we go p times one plus r

and that's going to give us our annuity so

just to be interesting let's use the same one we did two steps ago

converted to a decimal and we were saying that that was at 5.05

and and 12 is pretty common but i'm going to say 52 let's say it

you know every every week all year

here so what do we get 14 18 and 82 cents and let's just quick go

increase 1407 and 10 cents so look at that just that little amount

made us another 11 because it was compounded that often it's a

you have that little bit extra advantage this is money growing and

percent for that little bit more time you know gets you a few more

well okay 52 then maybe 365 would give you a little bit more

quick do that so remember that was 1418 and if we did this so that

we're going to do the same 0.05 the same seven years but then this

every day so you see 1419 it gave you another dollar so compared

made about 11 more dollars doing this every week but from every

one more dollar so there is a point to where it doesn't really get

we start thinking about well okay can I compound it multiple times

into continuous growth and continuous growth if i use that formula

times a day if i compounded that like three times a day and who

make n like an even thousand it starts approaching this other

recap so there you go that's you know money compounding annually

increase but we take it as annually and then n times per year we

n gets to be really big like i said like a thousand or something

formula pe to the rt and this is useful because as we see later

what the rate or the time would be if i know these other values so

complicated to i actually think this is a little bit easier

e is not just a random variable e is this constant equal to about

through that twice the next time it's not exact so it's not just

that would make it even easier but it actually after that second

different okay so we give the code that we're already going to

we're going to add one more import statement we're going to import

math.e you know you don't have to worry about typing in this

to take it to even more decimal places so we have the same thing

there you go time and that can be an integer that doesn't have to

and is going to be an integer because you can you can't like maybe

do you compound it you can't say like oh one time i kind of like

okay so now we're going to use these these same formulas we did

annual so that's going to be p times times one plus r exponent t

and then this one is going to be p times one plus the extra

exponent of nt so these are those two formulas and now this one

math.e so that's going to be p times math.e to the exponent of r

so there we go we see pe to the rt

t and now we have these three so we can prompt and let's compare

compounded n times per year i'm going to put an extra space in

so let's take something here and let's just say a principle of

and time and let's just say you know 15 years n and we only need

but let's say n is a normal 12 so given this compounded annually

2632 compounded 12 times a year 26,995 so wow that made you know

just by compounding it but look at that compounded continuously

see from here from once a year to multiple times we're talking

but then from that to compounding it continuously you know

up to about 700 more and it kind of evens out that oh and this one

it gave this error because i used the parentheses so if you're

you know these errors messages are supposed to be hints just to

parentheses and that is okay because python does no word of

it should give us not the error so there we go and should give us

so 11,000 rate 0.06 time 15 years and and we said it was 12 so we

message saying you know maybe i'll go in and change that that the

right so there we go prompting once and comparing these three

interest so now some of this comes in where we have you know some

often you know something you might be contributing to for

value so we have any of these formulas but we also contribute

will say month monthly that's that's kind of a normal normal thing

about like all the different values that that could be okay so we

to increase by a percentage and we're using just that simple

principal times rate times time so that is just simple interest

and for one month t equals one over 12 because time in years but

by 12 so here if i have this we can set up a loop here so prt but

ask for the monthly contribution all right now this annuity at the

have i have this i have this amount done so we'll keep that but

that we're going to keep updating so somebody said time in years

12 times t so monthly contributions for this long for a and range

t and now let's see we're just going to say first of all the

all right so now that annuity value is greater and then now we

so that's why simple interest is just the interest equals annuity

time and then times one over 12 and i'm just going to say divide

in principle times rate times time and then 1 12 is divided by 12

and now again i'm going to update annuity annuity equals annuity

annuity plus interest and because i'm making this a loop then you

are useful for not having to do a loop but we also wanted to add

and then there we go so you know we have a few things going on in

the loop it's going to go through and each time you know add the

percentage for that little bit of time you know now update annuity

this so starting amount maybe we can say 5 000 and annual

for you know some uh you know retirement account 0.08 and then

some of the things you might contribute and then have it grow for

like long amount of time 35 years there we go monthly contribution

we go so that's that's what we have you start out and then every

and you know we hope it's growing at that rate

so given this notice how notice what this number is that's a

is and to get some of these the the rate it it changes that's

it's a reasonable very reasonable estimate so putting money away

put five thousand dollars away and some good account for as eight

then each you know each paycheck you're contributing contributing

with a million dollars there you go things to think about that you

that you can put away this little bit forget about it for 35 years

and you know have this because if it's earning if it still is

estimate eight percent of that million is eighty thousand dollars

thousand dollars interest you can live on that and it still has

keep generating more interest so there you go

okay so we go from the positive saving money and having a million

payments and yes more like the word for death somebody originally

this death payment but nonetheless mortgage for any large amount

of time those other percent increase formulas won't quite do it

paying a lot more than is reasonable so this is the much more

that's the value of you know setting these up and writing the code

remember this formula or you know plug the numbers in every time

we were doing before pr and t and figure out what your monthly

so given that some large some huge amount of money and how banks

um something else but knowing how much you're going to borrow and

it through this formula and figure out what would your monthly

useful you and again this applies to mortgage or possibly car

large enough sum and a long enough time that this is a much better

this is the formula and again p is principal r is rate t is time

but n would be 12 each time so i just put 12 in there so it kind

formula but n is 12 each time you know there's no other setup you

payments or yearly no it's it's always monthly all right there we

so now i gave you a hint to do this use other variables i like

look at this r over 12 comes up a few times so you know we can

um there you go r over 12 is uh you know multiplier or if you

other variables you can do this there we go and since i don't have

um i could have i might have like just like math n for numerator

and so the payment would be n divided by d because that just might

um and then it would be p times n divided by d so i could have

r over 12 times and then in parentheses one over r over 12

oh one oh sorry one plus r over 12

12 and all that to the exponent in parentheses 12 times t

so r over 12 times one plus r over 12 to the exponent 12 t there

not bad and then i can have the denominator equals so one plus r

12 put up with some parentheses 12 t and now when i subtract one

you see then that's definitely not a part of that

so yeah and again you could add other variables you can make r

something like that you can make one plus r over 12 its own

do that i feel i felt like this wasn't that bad so but breaking up

and then payment times numerator over denominator

so we take a look at this and the amount borrowed we could just

uh annual percentage rate 0.07 you know some of the rates have

to say that for now and number of years 30 is pretty common and so

there we go so 1330 and change and this helps because then you say

the cost of the house and i have a down payment you know you get

that's one thing okay but what's really going to make the

your monthly payment be that's the important thing and you know

whatever your income is the monthly payment should be for mortgage

fourth of your income or less so just to remember you know if

month you should be making at least four times that or if you're

house because sometimes you might say oh yeah i'm making more than

end up being house poor yes you can afford your your mortgage

else all right so let's go on to exponents and logarithms with

they're inverse functions they're the same information rearranged

here two to the fourth power equals 16 so there you go two base

and then the log base two of 16 equals four there we go both of

rearranged and that's what logarithms are you know picture

sign oh two to the fourth power i know how to do that plug that

it's the exponent you don't know the log base two of 16 the answer

exponent gets me to that so uh this guy john napier like 400 years

400 years ago he realized that that was a missing piece that that

wanted to know the exponent for different things and he spent like

developing this and it was you know very useful you know the

you could like look them up now we have this at the benefit of you

and we're going to take it one more we're going to even write the

couple numbers and it'll calculate it so if you use the math

numpy library that's another way but if you use the math library

and then the arguments are you know the log of what so 16 and then

so there we go and if you don't have that first argument or if you

if you just say the log of 16 it takes it as base e but we'll get

we go the log base two of 16 so here's the code we're going to

always call the the exponent the result just because it reminds me

what i want so there we go so or you know that's the the result of

i had somebody input the base and we're going to input cast as a

but usually it's an integer and the result and cast that as a

change the next line okay math dot log and we were just saying

and it'll import this for it and it'll do this for you so that's

and we'll run it and this should check it for you there we go base

three result 81 so it should return four there we go nice you

in your head but if you have you know base you know three to what

you can do that in your head that's four but this is useful three

or gets you 85 you know when it becomes some weird decimal all

talked about e so uh this is the this is another use of a

if you have this formula annuity equals pe to the rt there we go

how long it will take for something to double and you'll see the

you know whatever i start out with p the annuity needs to be

then whatever p is and annuity is twice then i could cancel it out

so it simplifies to this and then now one of the many situations i

exponent well i have base e so math terms i would say natural log

the natural log of two on the left and the natural log of e to the

the natural log of e to the rt because that says e the natural log

this so e to what exponent gets me e to the rt well it's obviously

we know that take the natural log of both sides now we're in this

of two good thing i have a calculator to do this and then if i

double if i know the rate then i would just divide by the rate and

that math dot log and you see no other argument necessary if i

as what base it is it takes it as base e natural log so there we

rate as a decimal and for this one i just put the end just it's

the input comes in as a new line on a new line and once the person

just math dot log of two divided by r is time and we'll print out

this is pretty useful so you can run this now this one remember

let's say i have some investment growing at eight percent and it

six six years so some of the you know you might say all right if

percent and we can even round this down to eight and a half and

okay well eight and a half years it'll double 17 years it'll

and you know you could use this to estimate some outcomes all

10 it's the common log if you were writing this in math notation

log so uh with without the little number because then you know if

then oh we'll take it as base 10 because that's our number system

we're going to be using when we want to have you know how many

things for scientific notation so in the code you see we'll just

so you see uh if i don't say the base it's e but 10 even though

base there we go so if we have this and our number with several

and this one i just separated into two lines to show you can do

you know cast it as n the other reason to put this into two lines

line over here maybe even before writing python you know you might

you only want the lines to be so long you don't have to scroll

a design decision if i continued beyond this line it would still

that code but just as a convention for writing python code we

80 characters brings it over to that line so there we go i have

want to go further on that line so i just made it a separate line

all right now we have the round the code so this one we have math

take the log of that but i'm not gonna round but instead of

of this because i don't want the decimals i don't i don't want to

and i would just you know drop the decimals math dot floor that's

and that's actually what what we really want and this is a weird

exactly three it sometimes doesn't give you the answer you want i

python quirk so just bonus insight here after doing this enough

if statement here all right and we're going to print the exponent

there we go several digits so if i have point you know zero zero

there we go and if we remove the math dot four function

then it will just give us it'll be you know negative four point

just do that there we go so you can see and i'll just to compare

zero zero five all right and that's it now notice in positive

something or three if this was positive three point something and

it'd be down to three but with negative numbers the floor from

floor of that is negative four so just interesting insight and

because if you've gone through the the foundational math three and

on something you're what we want some other insight into some of

this helps all right now all that you know we can use these

something to double you know things like this uh number of decimal

exp for exponent because well first of all that's what we're

but also that leads to scientific notation you know how can i

numbers without all those zeros so if i had like 45 million oh but

to the seventh or point zero zero zero all these could be 4.5

and there we go so it's always n times 10 to an exponent and n is

so that's what we want you know because 4.5 that's i want that one

that that's the true scientific notation now this is just changing

not actually writing code this is determine the value by just by

just to see so you know as we count so how many decimal place how

five six seven eight nine ten eleven because it has to be down to

one is 1.56 that's where the decimal would go and a2 would be 11

out a that that is a1 times 10 to a2 so then we'll do this for

so the decimal point would be there so then it's yeah so then we

five six seven eight nine ten so it would be 4.13 and we have to

so it'd be to the negative 10 and so then that would print b is

there we go and we run it there we go now

now this one works displays nicely this one how we want to display

automatically and this is fine you know you go on the next step

this one a positive 11th power fine it didn't see a reason to

the python notation 4.13 e negative 10 so and this is not the same

so times 10 to the negative 10 so there we go in this you know

watching this it's you know illustrating like why there's some of

so we can use these logs for scientific notation because if i want

this to scientific notation in an easy way so what do i do so i

together the things we did in the last two exercises so for a i'm

of the log of a base 10 there we go and now i'm going to round it

so it was a i called it x1 for exponent one and so if i have this

and it rounds it to two decimal places so if that's the if that's

multiply this times 10 to the negative x1 and it's going to print

i'll just show you the code and then we'll run it and you'll see

what i want x2 and i want to show you that this actually works for

so if we have so but this one i want this for b so take the log of

and if i have this then i'm going to round that so i'm going to

again referencing b this time

to there we go so it's what exponent got me there but then when i

then that will divide it out that's why it's that that's why it

because i want to divide it out to get that early number there all

and this just took those so you see we get a is 2.34 times 10 to

and b is 1.23 times 10 to the 13th

so pretty cool how you could just take this and whether i wanted

a decimal so a negative exponent or a huge number with the

this this works take the log base 10 and the floor of that and

that's how we get n1 you know then we're then multiplying it by

and we can display so this way now you have it you can have a code

to scientific notation pretty cool

and that's what we get into step 19 there we go now this one very

before and we're just added one more thing of entering a number to

notation so see this is where you can make it as your calculator

this and it works either way i'll copy this code that we had from

i don't need the number ones because i'm only doing one thing

right and in that case if we want to print you see i even left

so we can reuse this and this is the key with copying and reusing

be aware to notice these little things if you just copy and paste

there there might be a little subtle things that don't line up you

good you have to understand the code well enough to know these

oh okay i needed to delete the ones here all around and you know

copy paste with the follow-up editing now that works and see what

standalone uh you know input you didn't have to like define the

and there you go enter in whatever number all right so one two

three there we go there we go

two three i got to one point two three ah look at this it see as i

i didn't delete the one there and that variable was still in in

so it gave me this you know exponent that i was not expecting one

all right pretty good so anything that you have now now we have

and then later on what we're going to do is and then maybe i'll

put all of this code within one function and so then now you can

convert to scientific notation and all this code is within that

then that function we can use wherever all right so let's go on to

graphing exponents and logarithms so just connecting with the math

putting the adjective first natural log if i was just writing it

and that would be base e and so comparing y equals e to the x and

those two are the inverse functions and mirrored over the line y

we'll see these and then we'll see the line y equals x so we were

library if i just want to calculate a logarithm i'll use math.log

using numpy and we're using the linspace function to create an

graph so i have to keep with numpy which i usually import as np so

you see if i want to do one thing yeah that works if i'm using the

i'm using np.log and np has it slightly different np has np.log 10

base 2 and if i just do np.log it's going to be base e

other base because we have the change of base formula which i'll

we go so we have the positive x values now e to the if i was

all my x values whatever i want but when i do natural log then the

because remember they're inverses so in this case e to whatever

zero as the inverse here the x value will never be zero so when i

familiar from other graphing we might have done i'm going to

minimum maximum you know based on here what i did here that's fine

log i need i'm going to use a different linspace i won't do zero

tiny number 0.001 and that should be enough to illustrate this

so i'm graphing these i'll use x1 is my array of x values and

and then within that range i want a thousand points i mean you'll

smooth graph even if you don't have a thousand points all right

and i'll just write the y value here instead of separating it out

and then that to the to that exponent there we go and i'll make

the red line for y equals x then x1 and we see if i have this x1 i

it's y equals x so i don't need to define a new y value which i

and here now for my log y equals log of x i'll do this linspace

all right so we're just going to run this and then change it so we

and there we go so we have the blue line e to the x exponential

y equals x or y equals the log of x and there we go and this works

there are mirror images across the line y equals x but this there

it illustrates it nicely all right now change the blue line and

make these log base two so here instead of math dot e i'm going to

you know that's fine and this one instead of np dot log i'll do np

now we're not going to notice that much of a difference here

i went from a base of 2.7 to a base of 2 but there we go that's

one i don't want to include some applications here if you have the

as a decimal number that's really where the ph comes in so

so the way this this works it's the negative log of the hydrogen

so if we have 0.007 as scientific notation that would be seven

which would be a ph of three and then we see that as we got more

you know a really tiny decimal number then that would be ten to

know ten to the negative ten you know really tiny number and that

ten to the negative ten would be a ph of ten you know we get the

just taking this and if you knew the hydrogen concentration i was

in as like going into the chemistry and finding hydrogen

the those of you that want to uh you know pursue the chemistry

actually get you know various uh hydrogen concentrations of

work it out from there okay but nonetheless so what we want to do

concentration so it's the negative log and for this one i will use

going to make this negative math dot uh log and i'm going to log

ten there we go so that's that's really it and now that log i get

and what do i want to do i want the ceiling so i'll do math dot

perfect all right so there we go when we run this end of the

and what if i put your point two three four and there you go and

i would take this as we we look at this and yeah this is the

is and then do the math dot ceiling so that's what we'll do we'll

and these are the things that i wanted to put in here as you know

we see why this this ended up being wrong because then it gave a

putting the math dot ceiling in the wrong spot

all right so i'll do the same same one here point

i'll put a one that's okay there we go now that gives it a ph of

there we go now that gives it a ph of five you see how that makes

first with the negative log then doing the math dot ceiling there

there we go all right so now with functions here

calc these are some of the things how could we write a function

you're going to define the function and calculating mortgage

i'll call it mort pay and now notice you would think that it when

indent automatically but for other things that i do or i might use

four spaces so i backspace i delete that and then i want to make

can i do i can define this function here so and and this is i'll

you can do more here so if i have the mortgage payment now we also

um table of contents i can jump right to mortgage payments so

and then i will define these in here

here so i'm going to just copy this and actually um

there we go so i'm going to show you one way to do this and then

okay there we go and more functions here so one of the things that

is just put everything in here

you probably don't need that extra line and same here so this

the payment maybe that's all you need you want to be able to then

print out the answer so that works another way is

what i'm going to do i'm going to copy this and i'm going to show

p r t and for this one i'm going to require input input p r t

and i'm going to return payment

so this one's nice if that's the if and you know there we go so

prompt you know this could this could go a lot of different places

maybe then just put this function in and return the payment like

it's going to go so two different situations and that's why you

that's why you know i'll make it you know p r t so we know what

is different so there we go so that's that's one one way

and i have here more functions now the interactive polynomial

with that too i'm going to do one more up here because i wanted to

enter a number to convert to scientific notation so if we have

convert so if i have a as my variable so there's my scientific

and if i take that and convert this i could create a function to

a there we go and maybe it just takes that input of a

there we go so we take this and move around um

um oh but i would probably have it as a string here so i could go

uh i could instead of printing it

and so we may you know i could just make that a string and then

so i probably don't need this a equals

and it should give you n times 10 to the x so just to give you

i want you to be able to you create the functions and then the

graph with sliders we we have that so turn that into a new turn

but here's the examples how you can just take and and that's the

to you know through this course you learn and you're practicing

parts that now you know and you can turn just the important parts

into their own functions and you can you know give them other

and there you go uh create a function for you know how long it

go back to that other one again turn that into a function and all

is going to be in a financial app that you'll make so what you can

all of these and you could have this work out that it's within all

have it within uh within google co-lab where you have text that

calculate mortgage payment number two retirement account balance

the text there and then require input which option do you want and

that's that's one way to do it defining all the functions having

the menu and then when you pick your option then it runs that

want to get to is that you're putting all this together and it

you can use this for other things you know calculating your own

with what you're doing certifications one and two see now where

the end of this and you have understand the math and you have all

you know calculate things graph things solve problems and then you

the next unit which i call it week 15 is going to be the wrapping

so we reached the home stretch in our algebra journey here and now

all the different things we've been doing with math we've been

taking things that turn into x lists of x y values that we can

at how we can get the get that data from the web from the cloud

and bring that data in whether it be columns in a table or all

in those x y coordinates and then we can do math to it so really

been working with all the math content and now pretty much

beyond this we're going to bring that math content and into the

into the code and do a lot more code how can i get this data turn

then do whatever graph i can do and you know analyze the data so

get to the code so now this is some ways you can bring in data

do other things so let's talk about the other sources pandas is

with data frames tables columns rows everything like that so we're

the matplot library we've done that before and we're also going to

going to import files and that's also going to give our file

io input output there we go i separate these out because you can

takes a few seconds i already did it so run it once and that saves

because we want to go through you'll see test some things now we

ways that we can get the files all right here's one option if you

you can upload it and because we imported that google dialogue or

this is all we need files dot upload and we're going to upload

file dialogue you can search for on your computer and that's what

that as that variable uploaded and then so i have my file name

take that uploaded file and iteration so iter and the argument is

the next one and what that is going to do is just one time you

going to select that and store it as file name and here that file

know that it's a csv so pandas has this nice building pd dot read

get to and don't worry about this it's because i already stored

for you know whatever file you upload so uploaded file name and it

it will store it as this table so there we go these three lines

you know give you the file dialogue upload it from your computer

when you select that file it's stored as table one awesome so

in in a minute here another way is you can get a csv from a url so

looking at different websites and you found you know this uh url

uh yeah this this guy uh seems to put put out a few good examples

dot csv old faithful geyser you know it rups for a little you know

regular basis that's why it's called old faithful and that just

can look at so i'll just store it as this variable url which is

the csv there you go plenty you could also and i didn't include

you wanted to you know make your own like you know input statement

somebody else using you know you made this program for somebody

to just copy and paste the url and store it you know that works

so you just put the url here store it as variable url now same

dot uh because we imported pan does it as pd pd dot read csv and i

and that's enough and i'll store it as table one

okay and yes those imports and everything that you run i just even

you so that's good for maybe 20 minutes google colab will uh have

on their server and you know colab has a runtime once we started

30 minutes of inactivity or even if you're active like you know

busy day i think it's like about 12 hours of running it'll time

well wait i haven't been inactive but there's still an absolute

start it again so that gives that gives you time you can import

to go through and then run a few of these and tinker with it okay

to repeat those steps and again you know importing you know

few seconds to run it and then prompting you know if we want to do

test some things you don't have to upload it every time that saves

so there we go now we have this what do we want to do with it you

we're going to eventually determine what columns we want and

remember numpy arrays that's what we're that's what we end up

i even still commented this out but you could display the whole

especially the the more you get into big data you know that's

resource consuming and so i i include that in there you know just

but what else might we want to do i might just want to print out

as table one for this example so i might just want to print out

one dot head and if i don't put a number in parentheses it will

put a number in there and it's going to give me the headings and i

that give me the headings in two rows that might be enough you

you've seen the column you might want to see the see it here or

to take a look at it i want to see the heading i want to see the

two rows give me an idea of what kind of data because if i start

know that i'm not going to be able to convert that to a numpy

store column names as a variable because if i have table one in

one dot columns there you go so now this column names is a list of

um and again i can i only need to do this once and now that i have

them all right instead of printing out the heading i could and i

new line column names and then i could loop through them so i'll

range length column names so column names i'm going to get the

then each time i'll print that number the space and then that

way to do it or another way i could actually just loop through

and just print that each time so let's see what that looks like

benefits here so notice this is the first one we did uh there you

print statement that head statement gave me these but this table

and notice their index starts at one so that's just something to

and eruption wait time so there we go and notice these quotes are

with within the name of the call you know that's something the

more you get into uh working with data and maybe other people you

everything i'll tell you this you really want column names without

without spaces but you know there's ways to deal with that we can

usually makes it easier for this stuff and for like some of the

an idea of what i have and here again you see the column names you

and the the other way we did it index eruption length eruption

these exact column names so you know i could now this is how i did

here and i can see them so table one and then in those square

of that column i want and then also that name in single quotes to

that to a numpy array and then eruption length in minutes so again

square brackets and remember that extra space and those quotes

i include them and convert that to a numpy array so now i have my

can plot there you know going through each of them you know it's

x is 3.6 y is 779 x is 1.8 y is 54 oh no actually i had it as

1.8 i can graph the eruption length or i can do it for eruption

to the index again x is 1 y is 79 x is 2 y is 54 so i just graphed

can go back and do that for the second one also if you wanted to

numpy arrays i can call that function x.min and this you know if

like the entire table this just might be a good way i'll just take

because it's just a little bit of a buffer on the edge and store

five a little bit of a buffer y min y max so this goes back to

with graphing set my x min x max and each of these now notice i

doing in other other units we were defining the points here but i

defined as num numpy arrays so when i graph i can plot i can plot

you know the x refers to that array y refers to that array and i

so i have these here as numpy arrays and i can do them as a

all right let's see what this looks like

now i had it as both and it looks like a huge mess but one of the

i don't want the line graph is probably not useful so i'll just go

out right now and run it again and we see it seems to cluster it's

that's kind of interesting what if i change my x my y value maybe

what if i change this so what i can do is my y minimum and these

want to do now we know how to use your math graphing skills to

to get all this data into this graph and this is now where we get

do i can look at this and say all right you know they're all less

well actually they're all greater than zero so i can just make my

going to comment out the rest of this and i can make my y maximum

the rest of that and when we see that we might see it seems like

reason it's really rare for it to be around three and remember

here we are graphing the eruption length time so it really seems

minutes or about four minutes yeah interesting so we can look at

things that we can do now let's look at another twist on this you

names so remember earlier we made column names this array so i can

names one and that's my x name and my y name i'll make these

the actual names of your columns doing that once might save you

then you can reuse that variable here just yeah i don't have to

time now i have my x variable is that column to numpy my y

doing before but again reusing that variable save you some typing

why in a minute and i just want to show you that that same scatter

dimensions are there but you see using that as the column name i

and again without having to rewrite so now we have index and

there we go then if i wanted to talk about the eruption time or

the things that we were doing i can just change this i still want

uh columns one was the eruption length i'll change that to that

names everything else stays the same and again going to different

giving the dimensions of the graph based on that because these

i'll still have the same index because they did match but instead

and five i will have you know wait wait time in minutes you know

almost 100 minutes so we see that now again let's get to some

it looks like maybe i have this range of 40 minutes to you know

like let's say this is in yellowstone let's say you go to visit

i have to wait it just erupted i want to go do other things when

i come back in 40 minutes should i come back in 90 minutes you

could keep you you know waiting there almost an hour but let's

patterns here again the default i keep this index i want to zoom

so i'll go to that index that x value i'll keep the same minimum

value instead of that plus five maybe i'll make it like 50 and

that's interesting i'm just going to run this again doesn't help

see now instead of a scatter plot actually i'll keep the scatter

but i'll also make it a line graph and i'll make the dots red and

and you see that just looks nice and if i don't you know there's

want that then i can just make my x x minimum you know one that

comment that that way i don't even get distracted by that there we

now besides seeing the scatter pot that was a mess the line graph

interesting things with very few exceptions if it was a shorter

the next one's a longer weight closer to 90 and then the next

weight longer weight you see every so often you get one that it

but we get a shorter weight a longer weight a shorter weight you

that happens most of the time so what we can do is we could look

oh well how long was it last time and then therefore i can use

that from you know we get that from looking at the graph here so

you could probably then now connecting this to you know real-time

doing this then we could graph you know a certain amount of this

table that always just updates you know when it erupted and how

each time you know just you know maybe even we could get this to

erupted it stopped erupting and then do the calculations what was

eruption time add that new data and we could be looking at this

to give people you know real-time estimates of it just erupted

eruption and get it down even probably to some like you know uh

that you know the point is that you would want to be able to say

minutes later or 90 minutes later i want to be able to have i

you know if it just erupted and it was an 80 minute wait then you

you know 50 minutes so you get some range that you know when to

don't have to wait too long and that's really what we want to get

these patterns and be able to then predict the next one you know

if i go to here and i want to be able to predict and say oh all

of 60 minutes so i'm going to just predict that the next wait time

and if we really want we can just change this to like 55

so with the wait time of 50 minutes then the next one wasn't quite

of well i wouldn't go i wouldn't wait more than 80 minutes i might

up so there we go because remember the other graph the eruption

minutes you are two you know two minutes and it could be two

five minutes that's not even that long so you don't want to be you

going to miss it all so yeah just interesting we can use this to

math of it we have we you know once you have those column names

so once we have we get that table what once you have that table

we have those column names and then we can do other things with

whatever we want you know number them you know just loop through

pick certain ones and i'll use that as my x and y so that's the

minimum maximum we can try this because we want to make this

and maybe certain things you can comment out and certain things

modify just a little bit that what you can use the same code for

whatever table you have this is a good start you start getting

to compare what am i trying to graph and then as you get into it

you can always then copy all this and then you know do in here

some text to give you know yourself or whoever read this some some

this code cell and you know that that's the beauty of the colab

like you can delete things and retry things or you can keep that

this is interesting information keep that and then just do a new

you know let's do the next next analysis so here's another way to

then graph it this particular library so not a csv but the full

of weather data so yes may install need to only install once right

i just have this as its own cell there you go pip install medial

doing that i mean this actually might take almost 30 seconds

so sometimes that seems like forever so i already did that and

um successfully installed and then we go on to the next one and

else working with this you put put in some text areas here so now

columns i have in this library so time uh and all the other these

average temperature minimum temp maximum temperature the amount of

amount of snow wind direction wind speed the pressure so average

uh air pressure uh the wind gusts so uh after wind speed we have

and the last one is uh time in the sun so you know how many

would you get and these are really good i picked these because it

you know you can things like wind speed you know plan your uh wind

time in the sun plan for your solar panels and where to put them

um so i have these to remind me and then the reminder change start

and you know point we're going to use this a lot so what arguments

latitude longitude meters above the ground so now we have these

up for yourself and you don't use it for a while and then you

variables you can use so all this you know we can just keep this

because we have from numpy we're going to import mean because

and then the rest of these are from you know the media stat

daytime plot and for media stat we're going to just import these

it's useful put the documentation in you know that here's the

you know put that in there as a comment again just like the other

people good to keep track of these things so now we know what

period and what i wanted to do is i just picked you know sometime

start and end so i have eight time there we go year month day your

so the person who put out this one example online did this for

commented that out and i did one for philadelphia and i picked a

city because it you know the uh if you want to click places and

he had a lot of decimal places so i just wanted to round it to two

you know you know in philadelphia in general and then round it to

where we got so it's a some random street corner in the city and

how many meters above the ground so i put 10 because this could be

on top of a building and i wanted to compare this to somewhere in

closely packed buildings does that affect wind speed well we're

the ground so this is in you know residential neighborhood that is

buildings but i picked a point in the suburbs and i picked a point

open field as something to compare so now i have these two points

uh so daily so that's the function in in media stat and there we

you know these are arguments that point and then start time and

daily data dot fetch and i'm going to store it as my data now we

my data and which one do i want i want wind speed for this one and

dot to numpy and i'm going to store that so now num data is my

that we want to do this library requires this function here i'll

that variable uh got fetched that's also from media stat i'll

a numpy array so i want to make that a number i want to take that

that column and make that a numpy array so i'll print that you

to get the mean to compare so now i have the second one again i'll

function fetch the data convert it to a numpy array and in this

but remember num data and num data too they're my two numpy arrays

so we can plot the line plot the line there we go and

and so i have this you know these two things i have you know the x

since i know that i made my since i know that i made my start and

30 days i wanted to make the x value there you go same thing one

it includes the first one so there we go and then i can plot these

all right now i'll tell you why i have that commented out i'll

these because this works like uh i wanted to again connect it with

matplot library that we use i wanted to connect it to that same

using and show you that that's what you can do you know get those

matplot.show but i commented this out this works also just a

you can just take that my data dot plot and then in parentheses

these two work so i just wanted to show you i'll leave them in

to connect this with the same type of math and plotting we're

so i had it calculate and then just display you know the city mean

rounded it but and quite honestly this is why we do data analysis

i really thought the average wind speed in this suburban area that

field would have been significantly higher and it was definitely

um maybe you know a little bit of a tunnel effect maybe there is

you know the buildings maybe that wasn't high enough also at least

and if it's high enough that's a little bit closer to the river

gives off a little bit more wind and this was a little bit closer

effect but there we go so then we see the mean but beyond that we

to day and you know they're still these two points that i picked

so it tracks they're in the same area the wind speeds go up and

patterns but we see that not just the mean but at every single

you know we could take a look now i made this so that's good and

we can pick like any number of days so i could since all the rest

just pick this and say well what about over the course of the year

one and maybe i make that 12 31 and there we go so if it would

and 365 so i want to graph it this way so from one to 365 and 365

up perfectly then i might have to do one more or one less but we

oh value error and it was 365 357 oh that's really interesting all

and so we can see you know the scattering but also i liked this

you know not 365 days that seemed like that would be it i'm not

right now maybe there were some days that were missing but you

pattern but you know maybe that sometimes that happens we go and

know take out missing information or incomplete information but

the day that i picked happened to be slightly lower than average

bit you know 14 miles an hour 8 miles an hour and then we can see

wind speed you know the average then what you could do is you know

that average you actually don't need to plot it out because it

averages you can work on that so if your wind speed is you know in

almost it's like 14 miles an hour throughout the year then you can

wind turbine and it would be actually relatively small but what

you need for 14 mile an hour winds and you can do some

how much electricity would that generate for you over the course

and again you can you know you can do that with you know any other

you know maybe you know you're trying to plan something you know

temperatures just one other thing i did just was you know

for you know a month just to show you and and all the rest of this

and what i did is i just made it a loop so you know range 1 to 11

month and i have you know what month there we go and you know

this because i thought oh well is that does that change and these

um yeah the way that the x value displays sometimes it seems like

but we can see we have you know wind speed and we can just see you

over the course of it and you see each month i have the mean and

and and then the graph so we can just see it's just interesting

for each month the display it a little this might be a little

interest or you could have something that displays them all in

four the grid it gets to be small graphs though but nonetheless we

when it goes down so i just wanted to show you some of these

with this with this graph and you know beyond that that there's

to in you know any other data you find if you you know you're

them maybe it's a csv but maybe somebody you know put together an

not only have the data and then that's stored somewhere but you

so that you have all these already how can i get a point and how

from any of those points so things that you can make and things

somebody else made you know different ways that you can find this

to give you some more tools now you know now that you know math

you know some more tools and some more sources of data

so what i want to show you here is just a few more things that you

and we already looked at over here on the left the table of

get to appear disappear then we have the search you can even

and then we have this folder icon where you can click to mount

and then once you do that then you can also see give it permission

it's mounting the google drive once you do that then you'll see it

so you can unmount the google drive if you want if if you wanted

so this makes it easy to find things and move things back and

then the really big thing down here is the look at these tags you

that are already in the google colab you see adding form from

get the code right here you can copy it or even click it you know

right in the place where you want to put it you click insert all

and notice this even uses javascript within python interfaces this

again all these and some of these you know downloading files

from google drive from github all kinds of things importing

you know that are in co-laboratory all of these zip reader there's

can do open files from github from your google drive from your

just upload those so it even gets down in here into pandas data

we want to get our math skills too that we can take data from

pandas part and then bring it in and then do some different math

display it in different ways and that's what we want to take this

you'll see below this video there's other code that i'll include

every one of these but check these out i mean there's so many

into colab that you can you can do you know open up a sample you

colab notebook i just call it code tester and i just have

and you know let's try this let's try that and then you know you

of stuff that works but here's all these things available and then

another colab notebook of just a few other things you know to

continue you know some things that you can do with math and with

upon this leading to you know working with data and doing some

so check these out and you know tinker with them and enjoy

wow you made it so now having gone through this now you should

you should be familiar with all the key concepts and how to write

algebra you should have your at least one or however you organized

ready with all of your scripts for solving all different sorts of

through certification one two and three and you have the ability

problems and you know write the code or reference code that you've

to solve these problems so there you go you should be you should

foundational math because it is the math that we're going to use

so many other things in math including data science data science

formulas like this functions like you've been working on you know

graphing so now you have an understanding of all that and then you

for yeah statistics data science and a few other subjects too so

step we will do a pre-calc trig that type of topic so you know

other graphs we didn't get to in in algebra because they're a

other sorts of patterns dealing with locations of things and

triangles so that'll be something that we'll do next and again

you with statistics and data science because as you represent data

of mathematical analyses to see relationships among data points

get you to be able to detect more and more complex relationships

foundation i feel like you have a good foundation hopefully you're

you made it and on to on to the next adventure