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This video is about the exponent rules, rules\n
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two to the fifth is just shorthand for two\n
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five times. And similarly x to the n is just\n
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when we write these expressions, the number\n
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is called the base. And the number at the\n
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the base by itself is called the exponent.
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Sometimes the exponent is also called the\npower.
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The product rule says that a 5x to the power\n
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same thing as x to the n plus m power. In\n
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For example, if I have two cubed times two\n
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seventh. And that makes sense, because two\n
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two by itself three times. And then I multiply\n
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And in the end, I have to multiplied by itself\nseven times
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I'm just adding up the number of times as\n
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of times as multiplied total.
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The quotient rule says that if I have x to\n
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equal to x to the n minus m power. In other\n
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then I can subtract their exponents.
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For example, three to the six divided by three\n
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two, or three to the fourth. And this makes\n
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three by itself six times, and then I divide\n
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So when I cancel out threes, I have four threes\nleft
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notice that I have to subtract the number\n
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threes at the top to get my number of threes\n
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The power rule tells us if I have x to the\n
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same thing as x to the n times M power. In\n
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I get to multiply the exponents.
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For example, five to the fourth cubed is equal\n
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the 12th. And this makes sense, because five\n
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to the fourth times five to the fourth, times\n
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more, that's five times five times five times\n
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So I have three groups of four, or five, which\n
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The next rule involves what happens when I\n
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power, it turns out that anything to the zeroeth\n
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Usually, this is just taken as a definition.\n
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If you have something like two cubed divided\n
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equal one, anything divided by itself is just\n
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we know that this is the same thing as two\n
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divide two things with the same base, we get\n
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is the same thing as two to the zero. So two\n
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it work with the quotient rule.
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And the same argument shows that anything\nto the zero power
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What happens when we take something to a negative\npower
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x to the n is equal to one over x to the n.\n
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of a negative exponent. But here's why it\nmakes sense.
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If I take something like five to seven times\n
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roll the SAS to equal five to the seven plus\n
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and we just said that that is equal to one.
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Now I have the equation of five to the seventh\n
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one, if I divide both sides by five, the seventh,\n
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to equal one over five to the seventh. So\n
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comes from. That has to be true in order to\n
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Finally, let's look at a fractional exponents.\n
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over N really mean? Well, it means the nth\n
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means the cube root of 64, which happens to\n
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square root of nine, which is usually written\n
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Now, the square root of nine is just three.
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fractional exponents also makes sense. For\n
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cube that, then by the power role, that's\n
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which is just five to the one or five. So\n
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that when you cube it, you get five. And that's\n
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the cube root of five is also a number that\n
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The next rule tells us we can distribute an\n
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In other words, if we have a product, x times\n
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to x to the n times y to the N.
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For example, five times seven
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all raised to the third power is equal to\n
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sense, because five times seven, all raised\n
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times seven, times five times seven, times\n
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of multiplication, this is the same thing\n
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times seven times seven, or five cubed times\nseven cubed.
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Similarly, we could distribute an exponent\n
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over Y, all raised to the n power, that's\n
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example, two sevenths raised to the fifth\n
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over seven to the fifth. This makes sense,\n
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as two sevens multiply by itself five times,\n
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by itself five times divided by seven multiplied\n
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fifth, over seven to the fifth, as wanted.
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We've seen that we can distribute an exponent
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over multiplication, and division.
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But be careful, because we cannot distribute\nnext bowknot.
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over addition, or subtraction, for example,\n
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a to the n plus b to the n, a minus b to the\n
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b to the n. And if you're not sure, just try\n
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For example, two plus three squared is not\n
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and two minus three squared is definitely\n
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In this video, I gave eight exponent rules,\n
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the quotient rule, the power rule
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the zero exponent, the negative exponent,\n
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and the two rules involving distributing exponents.
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In another video, I'll use these exponent\n
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In this video, I'll work out some examples\n
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I'll start by reviewing the exponent rules.
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The product rule says that when you multiply\n
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the exponents. The quotient rule says that\n
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base, you subtract the exponents. The power\n
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power, you multiply the exponents. The power\n
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power is one, as long as the base is not zero.
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Since zero to the zero is undefined, it doesn't\nmake sense.
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negative exponents to evaluate x to the minus\n
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n. To evaluate a fractional exponent, like\n
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we can distribute an exponent over a product,\n
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times b to the n. And we can distribute an\n
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is a to the n over b to the n.
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In the rest of this video, we'll use these\n
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For our first example, we want to simplify\n
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x to the fourth, there's several possible\nways to proceed.
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For example, we could use the negative exponent\n
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all that gets divided by x to the fourth,\nstill
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notice that we only take the reciprocal of\n
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And that's because the exponent of negative\n
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Now if we think of three as three over one,\n
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numerator. And so we evaluate that by taking\n
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of the denominators, which is three over x\n
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I can think of x to the fourth as x to the\n
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of a fraction, which I can evaluate by multiplying\n
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times one divided by x squared times x to\n
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using the product rule. Since x squared times\n
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An alternate way of solving this problem
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is to start by using the quotient rule
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I can rewrite this as three times x to the\n
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quotient rule, that's three times x to the\n
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Now using the negative exponent roll, x to\n
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this product of fractions simplifies to three\n
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The second problem can be solved in similar\n
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One way to simplify would be to use the negative\n
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minus five as one over wide the fifth.
128
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thinking of this as a fraction, divided by\n
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and get four y cubed y to the fifth over one\n
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four y to the eight. And so my final answer\n
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Alternatively, I could decide to use the quotient\nrule first.
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As in the previous problem, I can write this\n
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quotient rule. And so that's for y to the\neighth as before.
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I'd like to show you one more method to solve\n
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To go on, that shortcut relies on the principle\n
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corresponds to a positive exponent in the\n
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two in the numerator here, after some manipulations\n
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Furthermore, a negative exponent in the denominator
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is equivalent to a positive exponent in the\nnumerator.
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That's what happened when we had the y to\n
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translated into a y to the positive five in\nthe numerator.
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Sometimes people like to talk about this principle,\n
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by switching the sine of the exponent that\n
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Let's see how this principle gives us a shortcut\n
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In the first problem, 3x to the minus two\n
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exponent in the numerator and make it a positive\n
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over x to the four plus two or x to the six.
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In the second example, for y cubed over y\n
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the minus five in the denominator into a y\n
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answer of four y to the three plus five or\neight.
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We'll use this principle again in the next\nproblems.
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In this example, notice that I have I have\n
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and also Z's in the numerator and the denominator.\n
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get all my y's either in the numerator or\n
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Since I have more y's in the denominator,\n
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and make it a y to the negative three. I'm\n
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in the numerator corresponds to a negative\n
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Now since I have a positive exponent, z in\n
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and I want to get rid of negative exponents,\n
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as e to the minus two in the denominator,\n
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Notice that my number seven doesn't move.\n
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because it doesn't have an exponent, and the\n
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applies to the Z not to the seven.
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Now that I've got all my Z's in the numerator\n
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to clean this up using the product rule.
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And I have my simplified expression.
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In this last example, we have a complicated\n
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I'm going to start by simplifying the expression\n
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I can bring all my y's downstairs and all\n
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In other words, I can rewrite this as 25x\nto the fourth
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I'll bring the Y to the minus five downstairs\n
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bring the x to the minus six upstairs and\n
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then I still have the Y cubed on the denominator,\n
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Using the product rule, I can rewrite the\n
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as 25x to the 10th over y to the eighth.
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Recall that we're allowed to distribute an\n
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When I distribute my three halves power
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I get 25 to the three halves times x to the\n
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Now the power rule tells me when I have a\n
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So I can rewrite this as 25 to the three halves\n
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y to the eight times three halves. In other\n
182
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15th over y to the 12th. Finally, I need to\n
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halves can be thought of as three times one\n
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25 to the three halves as 25 to the three\n
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Well, using the power rule in reverse, I can\n
186
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or as 25 to the one half cubed. Since when\n
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25 cubed to the one half might be hard to\n
188
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but 25 to the one half is just the square\n
189
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cubed, or five cubed, which is 125.
190
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Therefore, my original expression is going\n
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In this video, we use the exponent rules to\n
192
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This video goes through a few tricks for simplifying\n
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Recall that this notation means the nth root\n
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root of eight, the number that when you cube\n
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when we write the root sign without a little\n
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In this case, the square root of 25 is five\n
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Let's start by reviewing some rules for radical\n
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of a product, we can rewrite that as the product\n
199
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For example, the square root of nine times\n
200
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nine times the square root of 16, you can\n
201
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Similarly, it's possible to distribute a radical\n
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by b is the same thing as the radical of A\n
203
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the cube root of 64 over eight is the same\n
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root of eight, and you can check that both\nof these evaluated to
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you have to be a little bit careful though,\n
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sign across addition. In general, the nth\n
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of A plus the nth root of b. And similarly,\n
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If you're ever in doubt, you can always check\n
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root of one plus one is not the same thing\n
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the right side evaluates to one plus one or\n
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The second expression to show that that fails,\n
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root of one minus one it'll actually hold\n
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using say the square root of two minus one,\n
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You might notice that these Rules for Radicals,\n
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hold, remind you of rules for exponents. And\n
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be written in terms of exponents. For example,\n
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this, the nth root of A times B is the same\n
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exponent rules, I can distribute an exponent\n
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rule can be restated completely in terms of\n
220
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can be restated in terms of exponents as a\n
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one over n divided by b to the one over n.\n
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and the exponents to rewrite a to the m over\n
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a to the m with the N through taken. That's\nalso the same thing
224
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As the nth root of A, all taken to the nth\n
225
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exponent rules. So a to the m over N is the\n
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over nth power. That's because when we take\n
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and M times one over n is equal to M over\nn.
228
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But a one over nth power is the same thing\n
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is the same thing as this expression. And\n
230
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equivalence can we prove similarly, by writing\n
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M. Again, this is works because when I take\n
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one over n times M is the same thing as M\n
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the same, because the one over nth power is\n
234
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One mnemonic for remembering these relationships\n
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and root is like root, so that tells us we\n
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the power, and the n becomes the root in either\n
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Now let's use these rules in some examples.\n
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halves power, well, first I'll use my exponent\n
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one over 25 to the three halves power.
240
00:26:43,029 --> 00:26:52,839
Next, I'll use the power of a root mnemonic\n
241
00:26:52,839 --> 00:26:59,500
rooted, or, as 25 square rooted to the third\npower.
242
00:26:59,500 --> 00:27:04,650
I wrote the two's there for the square root\n
243
00:27:04,650 --> 00:27:09,900
will omit this and just write the square root\n
244
00:27:09,900 --> 00:27:15,750
Now, I could use either of these two equivalent\n
245
00:27:15,750 --> 00:27:20,569
this one because it's easier to compute without\n
246
00:27:20,569 --> 00:27:28,589
five, five cubed is 125. So my answer is one\nover 125.
247
00:27:28,589 --> 00:27:34,750
If I tried to compute the cube of 25, first,\n
248
00:27:34,750 --> 00:27:41,859
easier to compute the route before the power\n
249
00:27:41,859 --> 00:27:46,209
Now let's do an example simplifying a more\n
250
00:27:46,210 --> 00:27:52,450
cynet. I want to take the square root of all\n
251
00:27:52,450 --> 00:27:58,471
negative exponents, I'm first going to rewrite\n
252
00:27:58,471 --> 00:28:04,319
the sixth over z to the 11th. So I'll change\n
253
00:28:04,319 --> 00:28:07,879
by moving this, this factor to the denominator.
254
00:28:07,880 --> 00:28:13,450
Now, when you're asked to simplify radical\n
255
00:28:13,450 --> 00:28:17,950
much as possible, out of the radical side.
256
00:28:17,950 --> 00:28:23,000
To pull things out of the square root side,\n
257
00:28:23,000 --> 00:28:28,410
rewrite everything in terms of squares as\n
258
00:28:28,410 --> 00:28:34,890
a square, those two operations undo each other.\n
259
00:28:34,890 --> 00:28:43,210
60. So 60 is going to be two squared times\n
260
00:28:47,259 --> 00:28:52,690
Now I'll break things up into squares as much\n
261
00:28:52,690 --> 00:28:57,360
three times five, I've already got an x squared,\n
262
00:28:57,359 --> 00:29:03,579
y squared times y squared. And I'll write\n
263
00:29:03,579 --> 00:29:12,048
I guess five times 12345 times when extra\n
264
00:29:12,048 --> 00:29:15,410
add all those exponents together.
265
00:29:15,410 --> 00:29:20,430
Now I know that I can distribute my radical\n
266
00:29:20,430 --> 00:29:25,860
I'll write this with a zillion different radicals\nhere.
267
00:29:25,859 --> 00:29:32,479
And every time I see the square root of something\n
268
00:29:32,480 --> 00:29:37,410
and the squares out and get what's what's\n
269
00:29:37,410 --> 00:29:46,080
I get two times the square root of three times\n
270
00:29:46,079 --> 00:29:53,569
z times itself, I guess five times times the\n
271
00:29:53,569 --> 00:29:59,069
up with exponents. I'll write that as the\n
272
00:30:00,190 --> 00:30:07,950
times x times y cubed over z to the fifth\nthe square root of z.
273
00:30:07,950 --> 00:30:15,480
I'm gonna leave this example as is. But sometimes\n
274
00:30:15,480 --> 00:30:20,769
without radical signs in the denominator.\n
275
00:30:20,769 --> 00:30:24,789
I won't do it here, but I'll show you how\n
276
00:30:24,789 --> 00:30:31,859
This example asks us to rationalize the denominator,\n
277
00:30:31,859 --> 00:30:36,549
without radical signs in the denominator.
278
00:30:36,549 --> 00:30:44,069
To get rid of the radical sine and the denominator,\n
279
00:30:44,069 --> 00:30:49,389
root of x. But I can't just multiply the denominator\n
280
00:30:49,390 --> 00:30:55,240
the numerator by the same thing. So then just\n
281
00:30:55,240 --> 00:30:57,529
and I don't change the value of my expression.
282
00:30:57,529 --> 00:31:06,319
Now, if I just multiply together numerators\n
283
00:31:06,319 --> 00:31:12,019
squared of x 10 squared of x is the square\n
284
00:31:14,210 --> 00:31:19,620
Now I can cancel my access from the numerator\n
285
00:31:19,619 --> 00:31:24,750
times the square root of x. I rationalize\n
286
00:31:26,400 --> 00:31:34,110
In this video, we went over the Rules for\n
287
00:31:34,109 --> 00:31:37,889
by working with fractional exponents
288
00:31:37,890 --> 00:31:41,360
pulling things out of the radical sign
289
00:31:41,359 --> 00:31:46,159
and rationalizing the denominator.
290
00:31:46,160 --> 00:31:51,170
This video goes over some common methods of\n
291
00:31:51,170 --> 00:31:56,890
means to write it as a product. So we could\n
292
00:31:56,890 --> 00:32:04,670
times five, we could factor it more completely\n
293
00:32:04,670 --> 00:32:12,670
As another example, we could factor the expression\n
294
00:32:12,670 --> 00:32:16,500
x plus two times x plus three.
295
00:32:16,500 --> 00:32:23,700
In this video, I'll go over how I get from\n
296
00:32:23,700 --> 00:32:29,340
But for right now, I just want to review how\n
297
00:32:29,339 --> 00:32:35,259
factoring is correct. And that's just by multiplying\n
298
00:32:35,259 --> 00:32:40,799
two times x plus three, then I multiply x\n
299
00:32:40,799 --> 00:32:48,119
gives me 3x. Two times x gives me 2x. And\n
300
00:32:48,119 --> 00:32:54,569
to x squared plus 5x plus six, which checks\n
301
00:32:54,569 --> 00:33:00,240
of factoring as the opposite of distributing\n
302
00:33:00,240 --> 00:33:01,730
by distributing or multiplying out
303
00:33:01,730 --> 00:33:09,140
a bit of terminology, when I think of an expression\n
304
00:33:09,140 --> 00:33:17,610
I sum up are called the terms. But if I think\n
305
00:33:17,609 --> 00:33:22,929
then the things that I multiply together are\ncalled factors.
306
00:33:22,930 --> 00:33:28,350
Now let's get started on techniques of factoring.\n
307
00:33:28,349 --> 00:33:33,339
like to start by pulling out the greatest\n
308
00:33:33,339 --> 00:33:38,269
means the largest thing that divides each\nof the terms.
309
00:33:38,269 --> 00:33:46,069
In this first example, the largest thing that\n
310
00:33:46,069 --> 00:33:53,990
So the GCF is five. So I pull the five out,\n
311
00:33:53,990 --> 00:33:58,769
number. And so I get three plus 5x.
312
00:33:58,769 --> 00:34:04,990
Pause the video for a moment and see if you\n
313
00:34:08,690 --> 00:34:15,648
The biggest thing that divides both x squared\n
314
00:34:18,099 --> 00:34:24,030
One way to find this is to look for the power\n
315
00:34:24,030 --> 00:34:29,149
So that's x squared. And the power of y that\n
316
00:34:31,710 --> 00:34:38,599
Now if I factor out the x squared y from each\n
317
00:34:38,599 --> 00:34:44,129
by x squared y, if I divide the first term\n
318
00:34:44,128 --> 00:34:50,878
the second term by x squared y, I'm going\n
319
00:34:50,878 --> 00:34:57,199
out on the side just to make it more clear,\n
320
00:34:59,980 --> 00:35:07,838
Three x's on the top and two x's and a y on\n
321
00:35:07,838 --> 00:35:16,130
a Y. So I'll write the x, y here, and I factored\n
322
00:35:16,130 --> 00:35:23,329
by multiplying out. So if I multiply out my\n
323
00:35:23,329 --> 00:35:28,849
the first term and the second term, I get\n
324
00:35:28,849 --> 00:35:35,710
together and two y's multiplied together.\n
325
00:35:35,710 --> 00:35:40,880
The next technique of factoring, I'd like\n
326
00:35:40,880 --> 00:35:46,940
example, notice that we have four terms, factoring\n
327
00:35:46,940 --> 00:35:50,200
If you have four terms in your expression,\nyou need to factor
328
00:35:50,199 --> 00:35:56,368
in order to factor by grouping, I'm first\n
329
00:35:56,369 --> 00:36:00,820
of the first two terms, and then separately,\n
330
00:36:00,820 --> 00:36:08,318
last two terms. The greatest common factor\n
331
00:36:08,318 --> 00:36:14,518
I factor out the x squared, and I get x plus\n
332
00:36:14,518 --> 00:36:23,939
of forex and 12 is just four. So I factor\n
333
00:36:23,940 --> 00:36:31,009
Notice that the factor of x plus three now\n
334
00:36:31,009 --> 00:36:36,039
the greatest common factor of x plus three,\n
335
00:36:36,039 --> 00:36:45,549
of the right. And now I have an x squared\n
336
00:36:45,548 --> 00:36:52,170
this second piece. And that completes my factoring\n
337
00:36:52,170 --> 00:36:57,320
factor further by factoring the expression\n
338
00:36:57,320 --> 00:37:03,269
see later, this expression, which is a sum\n
339
00:37:03,268 --> 00:37:06,078
does not factor any further over the integers.
340
00:37:06,079 --> 00:37:13,220
Next, we'll do some factoring of quadratics.\n
341
00:37:15,210 --> 00:37:20,480
just a term with x in it, and a constant term\nwith no x's in it.
342
00:37:20,480 --> 00:37:29,500
I'd like to factor this expression as a product\n
343
00:37:32,289 --> 00:37:37,329
The key idea is that if I can find those two\n
344
00:37:37,329 --> 00:37:41,440
this expression, those two numbers would have\n
345
00:37:41,440 --> 00:37:48,889
eight. And these two numbers would end up\n
346
00:37:48,889 --> 00:37:53,319
because when I multiply out, this number will\n
347
00:37:53,320 --> 00:37:59,509
be also another coefficient of x, they'll\n
348
00:37:59,509 --> 00:38:04,820
So if I look at all the pairs of numbers that\n
349
00:38:04,820 --> 00:38:11,519
could be one and eight, two, and four, four\n
350
00:38:11,519 --> 00:38:16,340
as I had before. And that's sort of the same\n
351
00:38:16,340 --> 00:38:20,470
negatives, I could have negative one, negative\n
352
00:38:20,469 --> 00:38:25,399
four, those alternate multiply together to\n
353
00:38:25,400 --> 00:38:31,579
if there's a pair of these numbers that add\n
354
00:38:31,579 --> 00:38:39,190
that these ones will work. So now I can write\n
355
00:38:39,190 --> 00:38:45,720
two times x minus four. And it's always a\n
356
00:38:45,719 --> 00:38:53,558
going to get x squared minus 4x minus 2x,\n
357
00:38:53,559 --> 00:39:00,730
I want. Now this second examples a bit more\n
358
00:39:00,730 --> 00:39:04,750
my coefficient of x squared is not just one,\nit's the number 10.
359
00:39:04,750 --> 00:39:08,789
Now, there are lots of different methods for\n
360
00:39:08,789 --> 00:39:13,889
going to show you one method, my favorite\n
361
00:39:13,889 --> 00:39:18,139
but to start out, I'm going to multiply my\n
362
00:39:18,139 --> 00:39:25,548
so I'm multiplying 10 by negative six, that\n
363
00:39:25,548 --> 00:39:31,690
coefficient of x the number 11. And write\n
364
00:39:31,690 --> 00:39:38,280
two numbers that multiply to give me negative\n
365
00:39:38,280 --> 00:39:42,410
that this is exactly what we were doing in\n
366
00:39:42,409 --> 00:39:47,210
have to multiply the coefficient of x squared\n
367
00:39:47,210 --> 00:39:53,769
was just one. So to find the two numbers that\n
368
00:39:53,769 --> 00:39:58,528
might just be able to come up with them in\n
369
00:39:58,528 --> 00:40:00,550
can figure it out. Pretty simple.
370
00:40:00,550 --> 00:40:05,528
thematically by writing out all the factors,\n
371
00:40:05,528 --> 00:40:12,579
60. So I can start with negative one and 60,\n
372
00:40:12,579 --> 00:40:19,960
And keep going like this until I have found\n
373
00:40:19,960 --> 00:40:28,318
me the number 11. And, and now that I look\n
374
00:40:28,318 --> 00:40:34,159
gives me 11. So I don't have to continue with\n
375
00:40:34,159 --> 00:40:40,858
factors, I write out my expression 10x squared,\n
376
00:40:40,858 --> 00:40:49,210
4x plus 15x. Now I copy down the negative\n
377
00:40:49,210 --> 00:40:55,679
11x. That's how I chose those numbers. And\n
378
00:40:55,679 --> 00:41:01,018
as, as this expression, I haven't changed\n
379
00:41:01,018 --> 00:41:05,250
that I can apply factoring by grouping on\n
380
00:41:05,250 --> 00:41:11,139
I factor out my greatest common factor of\n
381
00:41:11,139 --> 00:41:18,009
it's 2x. So I factor out the 2x, I get 5x\n
382
00:41:18,010 --> 00:41:25,160
common factor of 15x and negative six, that\n
383
00:41:25,159 --> 00:41:31,409
again, this is working beautifully. So I have\n
384
00:41:31,409 --> 00:41:38,920
the 5x minus two on the right, and I put what's\n
385
00:41:38,920 --> 00:41:43,710
plus three. And I have factored my expression.
386
00:41:43,710 --> 00:41:50,929
There are a couple special kinds of expressions\n
387
00:41:50,929 --> 00:41:55,149
just memorize the formula for. So the first\n
388
00:41:55,150 --> 00:42:02,460
something of the form a squared minus b squared,\n
389
00:42:02,460 --> 00:42:08,230
a minus b. And let's just check that that\n
390
00:42:08,230 --> 00:42:17,358
multiply that out, I get a squared minus a\n
391
00:42:17,358 --> 00:42:21,710
two terms cancel out. So it gives me back\n
392
00:42:21,710 --> 00:42:30,590
it. So for this first example, I if I think\n
393
00:42:30,590 --> 00:42:34,170
squared, then I can see that's a difference\n
394
00:42:34,170 --> 00:42:41,329
as x plus four times x minus four. And the\n
395
00:42:41,329 --> 00:42:50,140
that's the same thing as three p squared minus\n
396
00:42:54,139 --> 00:42:57,949
Notice that if I have a sum of squares
397
00:43:01,170 --> 00:43:09,389
x squared plus four, which is x squared plus\n
398
00:43:09,389 --> 00:43:13,239
The difference of squares formula doesn't\n
399
00:43:13,239 --> 00:43:19,989
for a sum of squares. There is, however, a\n
400
00:43:19,989 --> 00:43:27,449
a sum of cubes. The difference of cubes formula,\n
401
00:43:27,449 --> 00:43:34,879
squared plus a b plus b squared. The formula\n
402
00:43:34,880 --> 00:43:41,318
you just switch the negative and positive\n
403
00:43:41,318 --> 00:43:49,018
b times a squared minus a b plus b squared.\n
404
00:43:49,018 --> 00:43:55,608
multiplying out. Let's look at one example\n
405
00:43:55,608 --> 00:44:03,298
actually a sum of two cubes because it's y\n
406
00:44:03,298 --> 00:44:10,969
using the sum of cubes formula by plugging\n
407
00:44:10,969 --> 00:44:18,028
y plus three times y squared minus y times\n
408
00:44:18,028 --> 00:44:23,849
that up a little bit to read y plus three\n
409
00:44:23,849 --> 00:44:31,190
So in this video, we went over several methods\n
410
00:44:31,190 --> 00:44:40,259
common factor. We did factoring by grouping.\n
411
00:44:40,259 --> 00:44:47,789
difference of squares. And we did a difference\n
412
00:44:47,789 --> 00:44:54,210
and more complicated problems, you may need\n
413
00:44:54,210 --> 00:44:58,470
to get through a single problem. For example,\n
414
00:45:00,329 --> 00:45:03,920
Add to a factoring of quadratics, or something\nsimilar.
415
00:45:07,699 --> 00:45:13,848
This video gives some additional examples\n
416
00:45:13,849 --> 00:45:20,009
which of these first five expressions factor\n
417
00:45:20,009 --> 00:45:27,469
The first expression can be factored by pulling\n
418
00:45:27,469 --> 00:45:34,199
So that becomes x times x plus one.
419
00:45:34,199 --> 00:45:41,358
The second example can be factors as a difference\n
420
00:45:41,358 --> 00:45:44,630
something squared, minus something else squared.\n
421
00:45:44,630 --> 00:45:55,099
like a squared minus b squared, that's a plus\n
422
00:45:55,099 --> 00:45:58,509
X plus five times x minus five.
423
00:45:58,509 --> 00:46:05,139
The third one is a sum of two squares, there's\n
424
00:46:05,139 --> 00:46:09,690
real numbers. So this is the one that does\nnot factor.
425
00:46:09,690 --> 00:46:16,608
just for completeness, let's look at the next\n
426
00:46:16,608 --> 00:46:23,259
When we factor by grouping, we pull up the\n
427
00:46:23,260 --> 00:46:29,720
terms, that would be an x squared, that becomes\n
428
00:46:29,719 --> 00:46:36,019
as much as we can add the next two terms,\n
429
00:46:36,019 --> 00:46:42,420
that the x plus two factor now occurs in both\n
430
00:46:42,420 --> 00:46:51,528
x plus two out and get x plus two times x\nsquared plus three
431
00:46:51,528 --> 00:46:57,050
we can't factor any further because x squared\n
432
00:46:57,050 --> 00:47:04,810
Finally, we have a quadratic, this also factors.\n
433
00:47:04,809 --> 00:47:12,849
by grouping trick. So first, what I do is\n
434
00:47:12,849 --> 00:47:19,019
the constant term, five times eight is 40.\n
435
00:47:19,019 --> 00:47:24,318
take the coefficient of the x term, that's\n
436
00:47:24,318 --> 00:47:32,480
part of the x. Now I'm looking for two numbers\n
437
00:47:32,480 --> 00:47:37,659
Sometimes I can just guess numbers like this,\n
438
00:47:37,659 --> 00:47:46,868
40. So factors of 40, I could do one times\n
439
00:47:46,869 --> 00:47:51,568
add to a negative number. So if I use two\n
440
00:47:51,568 --> 00:47:56,929
to add to a negative number. It's better for\n
441
00:47:56,929 --> 00:48:02,278
a negative times a negative still multiplies\n
442
00:48:02,278 --> 00:48:06,460
a negative number. So but negative one and\n
443
00:48:06,460 --> 00:48:11,220
add to negative 14, they add to negative 41.\n
444
00:48:11,219 --> 00:48:15,838
biggest number that divides 40, besides one\n
445
00:48:15,838 --> 00:48:22,518
20. Those add to negative 22. That doesn't\n
446
00:48:22,518 --> 00:48:29,818
so I'll try negative four and negative 10.\n
447
00:48:29,818 --> 00:48:35,460
negative 10 is negative 14, negative four\n
448
00:48:35,460 --> 00:48:41,869
it. Alright, so the next step is to use factoring\n
449
00:48:41,869 --> 00:48:50,809
this negative 14x as negative 4x minus 10x.\n
450
00:48:50,809 --> 00:48:57,300
Notice that this works, because I picked negative\n
451
00:48:57,300 --> 00:49:04,680
14, so so negative 4x minus 10x, will add\n
452
00:49:04,679 --> 00:49:09,528
just just expand it out a little bit. Now\n
453
00:49:09,528 --> 00:49:13,989
so I can group the first two terms and factor\n
454
00:49:13,989 --> 00:49:19,348
x times 5x minus four. And now I'll factor\n
455
00:49:19,349 --> 00:49:26,499
including the the negative. So that becomes,\n
456
00:49:26,498 --> 00:49:33,588
and that becomes 5x minus four since negative\n
457
00:49:33,588 --> 00:49:40,068
I've got the same 5x plus four in both my\n
458
00:49:40,068 --> 00:49:45,858
I can factor out the 5x minus four from both\n
459
00:49:45,858 --> 00:49:51,440
I factored this quadratic. If I want to, of\n
460
00:49:51,440 --> 00:49:59,179
out by multiplying out. So a check here would\n
461
00:49:59,179 --> 00:50:08,288
squared 5x minus 10 to minus two is minus\n
462
00:50:08,289 --> 00:50:13,460
four times minus two is plus eight. So let's\n
463
00:50:13,460 --> 00:50:22,119
should be. So that was the method of factoring\na quadratic.
464
00:50:22,119 --> 00:50:27,358
And all of these factors except for the sum\nof squares.
465
00:50:27,358 --> 00:50:35,009
So we saw that factoring by grouping is handy\n
466
00:50:35,009 --> 00:50:39,929
also handy for factoring the quadratic indirectly,\n
467
00:50:44,389 --> 00:50:52,279
into two terms. So how can you tell when a\n
468
00:50:52,280 --> 00:50:57,690
by grouping, there's, there's an easy way\n
469
00:50:57,690 --> 00:51:00,099
that's that it has four terms.
470
00:51:00,099 --> 00:51:05,539
So if you see four terms, or in the case of\n
471
00:51:05,539 --> 00:51:09,030
then that's a good candidate for factoring\n
472
00:51:09,030 --> 00:51:16,210
two terms group the second two terms, factoring\n
473
00:51:16,210 --> 00:51:21,559
with four terms. But, but that's like the\n
474
00:51:21,559 --> 00:51:25,829
what are the same main techniques of factoring.\n
475
00:51:25,829 --> 00:51:32,230
there was pull out common factors. There's\n
476
00:51:36,230 --> 00:51:41,849
There's factoring quadratics.
477
00:51:41,849 --> 00:51:49,970
And one more that I didn't mention is factoring\n
478
00:51:49,969 --> 00:51:57,328
that uses the formulas, aq minus b cubed is\n
479
00:51:57,329 --> 00:52:07,930
squared. And a cubed plus b cubed is a plus\n
480
00:52:07,929 --> 00:52:11,068
One important tip when factoring
481
00:52:11,068 --> 00:52:17,150
I always recommend doing this first, pull\n
482
00:52:17,150 --> 00:52:21,798
That'll simplify things and making the rest\n
483
00:52:21,798 --> 00:52:28,130
you might need to do several these factoring\n
484
00:52:28,130 --> 00:52:31,869
you might have to first pull out a common\n
485
00:52:31,869 --> 00:52:35,338
And then you might notice that one of your\n
486
00:52:35,338 --> 00:52:41,038
and you have to apply a difference of squares\n
487
00:52:41,039 --> 00:52:43,910
bit, keep factoring as far as you can go.
488
00:52:43,909 --> 00:52:49,411
Here are some extra examples of factoring\n
489
00:52:49,411 --> 00:52:52,619
the video and give these a try.
490
00:52:52,619 --> 00:53:01,130
For the first one, let's multiply two times\n
491
00:53:01,130 --> 00:53:06,338
then we'll bring the three down in the bottom\n
492
00:53:06,338 --> 00:53:14,269
that multiply to negative 28 and add to three.\n
493
00:53:14,269 --> 00:53:23,518
28, we'll need one of them to be negative\n
494
00:53:23,518 --> 00:53:26,808
negative 128, or one, negative 28, those don't\n
495
00:53:26,809 --> 00:53:30,309
14, those don't work. Hey, I just noticed\n
496
00:53:30,309 --> 00:53:37,400
the negative number, so they add to a positive\n
497
00:53:37,400 --> 00:53:43,460
negative four times seven, four times negative\n
498
00:53:43,460 --> 00:53:49,199
will work. So I'll write those here at negative\nfour, seven
499
00:53:49,199 --> 00:53:58,009
copy down the two z squared. And I'll split\n
500
00:53:58,009 --> 00:54:05,170
z and then minus 14. Now factoring by grouping,\n
501
00:54:05,170 --> 00:54:12,650
pull out a seven and that becomes z minus\n
502
00:54:12,650 --> 00:54:18,259
seven times z minus two as my factored expression.
503
00:54:18,259 --> 00:54:23,048
My second expression, I could work at the\n
504
00:54:23,048 --> 00:54:28,619
grouping that kind of thing. But it's actually\n
505
00:54:28,619 --> 00:54:32,170
I can pull out a common factor from all of\n
506
00:54:32,170 --> 00:54:36,130
simpler to deal with. So notice that a five\n
507
00:54:36,130 --> 00:54:39,660
I'm going to go ahead and pull out the negative\n
508
00:54:39,659 --> 00:54:44,690
in front of my squared term. So I'm going\n
509
00:54:44,690 --> 00:54:49,470
Again, it would work if I forgot to do this,\n
510
00:54:49,469 --> 00:54:56,699
negative five v squared, this becomes minus,\n
511
00:54:56,699 --> 00:54:59,868
negative five is negative 45. And this becomes\nminus
512
00:54:59,869 --> 00:55:06,650
10 cents native 10 times negative five is\n
513
00:55:06,650 --> 00:55:10,849
by grouping, or I can use kind of a shortcut\n
514
00:55:10,849 --> 00:55:15,588
I can just put these here, and then I know\n
515
00:55:15,588 --> 00:55:21,150
have to multiply to the negative 10. And they're\n
516
00:55:21,150 --> 00:55:26,289
be plus 10, and a minus one will do the trick.
517
00:55:26,289 --> 00:55:32,230
Those are all my factoring examples for today.\n
518
00:55:32,230 --> 00:55:37,929
a chance to spend some time working in ALEKS.\nBye.
519
00:55:37,929 --> 00:55:43,028
This video is about working with rational\n
520
00:55:43,028 --> 00:55:49,460
usually with variables in it, something like\n
521
00:55:49,460 --> 00:55:55,360
rational expression. In this video, we'll\n
522
00:55:55,360 --> 00:56:01,660
and dividing rational expressions and simplifying\n
523
00:56:01,659 --> 00:56:07,808
We'll start with simplifying to lowest terms.\n
524
00:56:07,809 --> 00:56:14,599
numbers in it, something like 21 over 45,\n
525
00:56:22,199 --> 00:56:26,019
and then canceling common factors.
526
00:56:26,019 --> 00:56:35,478
So in this example, the three is cancel, and\n
527
00:56:35,478 --> 00:56:40,259
If we want to reduce a rational expression\n
528
00:56:40,259 --> 00:56:47,309
we proceed the same way. First, we'll factor\n
529
00:56:47,309 --> 00:56:54,680
and then factor the denominator. In this case\n
530
00:56:54,679 --> 00:57:00,710
could also write that as x plus two squared.\n
531
00:57:00,710 --> 00:57:08,579
left with three over x plus two. Definitely\n
532
00:57:08,579 --> 00:57:15,900
Next, let's practice multiplying and dividing.\n
533
00:57:15,900 --> 00:57:20,789
just numbers in them, we simply multiply the\n
534
00:57:20,789 --> 00:57:28,599
So in this case, we would get four times two\n
535
00:57:28,599 --> 00:57:35,338
If we want to divide two fractions, like in\n
536
00:57:35,338 --> 00:57:43,170
as multiplying by the reciprocal of the fraction\n
537
00:57:43,170 --> 00:57:50,970
times three halves, and that gives us 12 tenths.\n
538
00:57:53,818 --> 00:58:00,000
we use the same rules when we compute the\n
539
00:58:00,000 --> 00:58:06,548
with the variables. And then here, we're trying\n
540
00:58:06,548 --> 00:58:12,889
we can multiply by the reciprocal. I call\n
541
00:58:12,889 --> 00:58:18,828
And now we just multiply the numerators.
542
00:58:18,829 --> 00:58:21,859
And multiply the denominators.
543
00:58:21,858 --> 00:58:28,889
It might be tempting at this point to multiply\n
544
00:58:28,889 --> 00:58:32,748
denominator. But actually, it's better to\n
545
00:58:32,748 --> 00:58:38,139
even more completely. That way, we'll be able\n
546
00:58:38,139 --> 00:58:46,048
the common factors. So let's factor even more\n
547
00:58:46,048 --> 00:58:52,440
plus one, and x squared minus 16. And that's\n
548
00:58:52,440 --> 00:58:59,510
four times x minus four, the denominator is\n
549
00:58:59,510 --> 00:59:06,569
it over. And now we can cancel common factors\n
550
00:59:06,568 --> 00:59:12,659
x minus four. This is our final answer.
551
00:59:12,659 --> 00:59:16,170
Adding and subtracting fractions is a little\n
552
00:59:16,170 --> 00:59:23,088
find a common denominator. A common denominator\n
553
00:59:23,088 --> 00:59:29,190
into, it's usually best of the long run to\n
554
00:59:29,190 --> 00:59:34,028
the smallest expression that both denominators\ndivided into.
555
00:59:34,028 --> 00:59:39,219
In this example, if we just want a common\n
556
00:59:39,219 --> 00:59:46,058
is 90 because both six and 15 divided evenly\n
557
00:59:46,059 --> 00:59:55,150
the best way to do that is to factor the two\n
558
00:59:55,150 --> 00:59:59,900
times five, and then put together only the\nfactors we need for
559
00:59:59,900 --> 01:00:07,789
Both six and 50 into divider numbers. So if\n
560
01:00:07,789 --> 01:00:14,159
is 30, we know that two times three will divide\n
561
01:00:14,159 --> 01:00:19,440
it. And we won't be able to get a denominator\n
562
01:00:19,440 --> 01:00:24,970
three, and five, in order to ensure both these\n
563
01:00:24,969 --> 01:00:31,598
denominator, we can rewrite each of our fractions\n
564
01:00:31,599 --> 01:00:38,940
I need to get a 30 in the denominator, so\n
565
01:00:38,940 --> 01:00:45,539
and multiply by the factors that are missing\n
566
01:00:45,539 --> 01:00:54,068
my least common denominator of 30. For the\n
567
01:00:54,068 --> 01:00:57,829
30. So I'm going to multiply by two over two
568
01:00:57,829 --> 01:01:08,971
I can rewrite this as 3530 s minus 8/30. And\n
569
01:01:08,971 --> 01:01:15,608
just subtract my two numerators. And I get\n27/30.
570
01:01:15,608 --> 01:01:20,119
If I factor, I can reduce this
571
01:01:20,119 --> 01:01:28,108
to three squared over two times five, which\n
572
01:01:28,108 --> 01:01:33,759
sum of two rational expressions with variables\n
573
01:01:33,759 --> 01:01:39,440
we have to find the least common denominator,\n
574
01:01:39,440 --> 01:01:45,659
So 2x plus two factors as two times x plus\n
575
01:01:45,659 --> 01:01:51,858
of two squares. So that's x plus one times\n
576
01:01:51,858 --> 01:01:57,558
I'm going to take all the factors, I need\n
577
01:01:57,559 --> 01:02:02,329
into, so I need the factor two, I need the\n
578
01:02:02,329 --> 01:02:08,010
minus one, I don't have to repeat the factor\n
579
01:02:08,010 --> 01:02:14,410
And so I will get my least common denominator\n
580
01:02:14,409 --> 01:02:17,868
not going to bother multiplying this out,\n
581
01:02:17,869 --> 01:02:24,920
form to help me simplify later. Now I can\n
582
01:02:24,920 --> 01:02:29,950
by multiplying by whatever's missing from\n
583
01:02:29,949 --> 01:02:36,098
denominator. So what I mean is, I can rewrite\n
584
01:02:36,099 --> 01:02:41,278
plus two is two times x plus one, I'll write\n
585
01:02:41,278 --> 01:02:45,778
compared to the least common denominator,\n
586
01:02:45,778 --> 01:02:52,099
I multiply the numerator and the denominator\n
587
01:02:52,099 --> 01:02:56,660
But I can't get away with just multiplying\n
588
01:02:56,659 --> 01:03:00,788
I have to multiply by it on the numerator\n
589
01:03:00,789 --> 01:03:06,690
by one and a fancy form and not changing the\n
590
01:03:06,690 --> 01:03:10,950
the second rational expression, I'll I'll\n
591
01:03:10,949 --> 01:03:16,989
make it easier to see what's missing from\n
592
01:03:16,989 --> 01:03:21,588
compared to my least common denominator is\n
593
01:03:21,588 --> 01:03:27,590
and the denominator by two. Now I can rewrite\n
594
01:03:27,590 --> 01:03:36,059
becomes three times x minus one over two times\n
595
01:03:36,059 --> 01:03:44,849
five times two over two times x plus 1x minus\n
596
01:03:44,849 --> 01:03:51,519
So I can just add together my numerators.\n
597
01:03:51,518 --> 01:03:58,228
two times x plus 1x minus one. I'd like to\n
598
01:03:58,228 --> 01:04:04,218
is to leave the denominator in factored form.\n
599
01:04:04,219 --> 01:04:12,599
so that I can add things together. So I get\n
600
01:04:12,599 --> 01:04:22,130
1x minus one, or 3x plus seven over two times\n
601
01:04:22,130 --> 01:04:29,160
factor. And there's therefore no factors that\n
602
01:04:29,159 --> 01:04:33,129
As much as it can be. This is my final answer.
603
01:04:33,130 --> 01:04:40,778
In this video, we saw how to simplify rational\n
604
01:04:42,980 --> 01:04:47,949
We also saw how to multiply rational expressions\n
605
01:04:47,949 --> 01:04:53,538
the denominator, how to divide rational expressions\n
606
01:04:53,539 --> 01:05:00,579
and subtract rational expressions by writing\n
607
01:05:00,579 --> 01:05:07,119
This video is about solving quadratic equations.\n
608
01:05:07,119 --> 01:05:11,960
the square of the variable, say x squared,\n
609
01:05:11,960 --> 01:05:20,449
The standard form for a quadratic equation\n
610
01:05:25,498 --> 01:05:31,419
represent real numbers. And a is not zero\n
611
01:05:31,420 --> 01:05:40,450
Let me give you an example. 3x squared plus\n
612
01:05:40,449 --> 01:05:49,788
in standard form, here a is three, B is seven,\n
613
01:05:49,789 --> 01:05:56,839
equals minus 7x plus two is also a quadratic\n
614
01:05:56,838 --> 01:06:02,449
The key steps to solving quadratic equations\n
615
01:06:02,449 --> 01:06:07,449
form, and then either factor it
616
01:06:07,449 --> 01:06:14,548
or use the quadratic formula, which I'll show\n
617
01:06:14,548 --> 01:06:21,119
Let's start with the example y squared equals\n
618
01:06:21,119 --> 01:06:26,650
and we need to rewrite this quadratic equation\n
619
01:06:26,650 --> 01:06:33,680
18 from both sides and adding seven y to both\n
620
01:06:33,679 --> 01:06:40,848
minus 18 plus seven y equals zero. And I can\n
621
01:06:40,849 --> 01:06:48,300
seven y minus 18 equals zero. Now I've got\n
622
01:06:48,300 --> 01:06:54,589
to try to factor it. So I need to look for\n
623
01:06:57,489 --> 01:07:02,659
two numbers that work are nine and negative\n
624
01:07:02,659 --> 01:07:12,039
left as y plus nine times y minus two equals\n
625
01:07:12,039 --> 01:07:17,389
that multiply together to give you zero, either\n
626
01:07:17,389 --> 01:07:22,478
second quantity has to be zero, or I suppose\n
627
01:07:22,478 --> 01:07:27,879
this is really handy, because that means that\n
628
01:07:27,880 --> 01:07:36,410
or y minus two equals zero. So I can as my\n
629
01:07:36,409 --> 01:07:43,278
y plus nine equals zero, or y minus two equals\n
630
01:07:46,619 --> 01:07:51,039
It's not a bad idea to check that those answers\n
631
01:07:51,039 --> 01:07:56,998
equation, negative nine squared, does that\n
632
01:07:56,998 --> 01:08:04,459
and you can work out that it does. And similarly,\n
633
01:08:04,460 --> 01:08:11,449
In the next example, let's find solutions\n
634
01:08:11,449 --> 01:08:17,729
is a quadratic equation, because it's got\n
635
01:08:17,729 --> 01:08:24,278
in standard form by subtracting 121 from both\nsides.
636
01:08:24,279 --> 01:08:31,839
Notice that A is equal to one b is equal to\n
637
01:08:31,838 --> 01:08:39,340
to negative 121 in the standard form, a W\n
638
01:08:39,340 --> 01:08:47,980
I'm going to try to factor this expression.\n
639
01:08:47,979 --> 01:08:56,129
of two squares, and it factors as w plus 11,\n
640
01:08:59,560 --> 01:09:07,560
I get w plus 11 equals zero or w minus 11\n
641
01:09:07,560 --> 01:09:14,289
11. In this example, I could have solved the\n
642
01:09:14,289 --> 01:09:20,539
said that if W squared is 121, and then w\n
643
01:09:20,539 --> 01:09:26,189
of 121. In other words, W is plus or minus\n11.
644
01:09:26,189 --> 01:09:30,939
If you saw the equation this way, it's important\n
645
01:09:30,939 --> 01:09:36,019
11 squared equals 121, just like 11 squared\ndoes.
646
01:09:36,020 --> 01:09:41,790
Now let's find the solutions for the equation.\n
647
01:09:41,789 --> 01:09:46,630
might be tempted to say that, oh, if two numbers\n
648
01:09:46,630 --> 01:09:51,529
better equal one and the other equals seven\n
649
01:09:51,529 --> 01:09:57,590
But that's faulty reasoning in this case,\n
650
01:09:57,590 --> 01:10:00,489
whole numbers. They could be crazy.
651
01:10:00,489 --> 01:10:08,729
fractions or even irrational numbers. So instead,\n
652
01:10:08,729 --> 01:10:16,879
To do that, I'm first going to multiply out.\n
653
01:10:16,880 --> 01:10:23,310
That equals seven, and I'll subtract the seven\n
654
01:10:23,310 --> 01:10:28,770
seven is zero. Now I'm looking to factor it.\n
655
01:10:28,770 --> 01:10:34,770
seven and add to two, since the only way to\n
656
01:10:34,770 --> 01:10:39,580
seven or seven times negative one, it's easy\n
657
01:10:39,579 --> 01:10:46,750
will do will work. So there's no way to factor\n
658
01:10:46,750 --> 01:10:53,140
let's use the quadratic equation. So we have\n
659
01:10:53,140 --> 01:11:00,000
So A is one, B is two, and C is minus seven.\n
660
01:11:00,000 --> 01:11:06,869
quadratic equation, which goes x equals negative\n
661
01:11:06,869 --> 01:11:10,279
minus four, I see all over two
662
01:11:10,279 --> 01:11:15,059
different people have different ways of remembering\n
663
01:11:15,060 --> 01:11:22,420
it x equals negative b plus or minus the square\n
664
01:11:22,420 --> 01:11:28,060
to a, but you can use any pneumonic you like.\n
665
01:11:28,060 --> 01:11:34,140
negative two plus or minus the square root\n
666
01:11:34,140 --> 01:11:40,730
negative seven support and remember the negative\n
667
01:11:40,729 --> 01:11:48,429
Now two squared is four, and four times one\n
668
01:11:48,430 --> 01:11:55,360
whole quantity under the square root sign\n
669
01:11:55,359 --> 01:12:00,849
I can rewrite this as x equals negative two\n
670
01:12:01,890 --> 01:12:11,079
Since 32, is 16 times two and 16 is a perfect\n
671
01:12:11,079 --> 01:12:15,819
plus or minus the square root of 16 times\n
672
01:12:15,819 --> 01:12:19,920
negative two plus or minus four times the\n
673
01:12:19,920 --> 01:12:26,159
Next, I'm going to split out my fraction
674
01:12:26,159 --> 01:12:32,159
as negative two over two plus or minus four\n
675
01:12:32,159 --> 01:12:38,729
those fractions. This becomes negative one\n
676
01:12:38,729 --> 01:12:45,039
answers are negative one plus two square root\n
677
01:12:45,039 --> 01:12:52,039
of two. And if I need a decimal answer for\n
678
01:12:52,039 --> 01:12:57,390
As our final example, let's find all real\n
679
01:12:57,390 --> 01:13:04,600
equals 1/3 y minus two. I'll start as usual\n
680
01:13:04,600 --> 01:13:12,810
me one half y squared minus 1/3 y plus two\n
681
01:13:12,810 --> 01:13:17,940
to factor or use the quadratic formula right\n
682
01:13:17,939 --> 01:13:22,649
of annoying. So I'd like to get rid of them.\n
683
01:13:22,649 --> 01:13:27,879
that means I'm going to multiply the whole\n
684
01:13:27,880 --> 01:13:33,020
In this case, the least common denominator\n
685
01:13:33,020 --> 01:13:37,300
the whole equation by six have to make sure\n
686
01:13:37,300 --> 01:13:42,220
in this case, six times zero is just zero.\n
687
01:13:42,220 --> 01:13:47,140
y squared minus two y plus 12 equals zero.
688
01:13:47,140 --> 01:13:51,760
Now, I could try to factor this, but I think\n
689
01:13:54,000 --> 01:14:01,909
So I get x equals negative B, that's negative\n
690
01:14:01,909 --> 01:14:12,539
root of b squared, minus four times a times\nc, all over to a.
691
01:14:12,539 --> 01:14:17,590
Working out the stuff in a square root sign,\n
692
01:14:21,699 --> 01:14:28,449
So this simplifies to x equals to plus or\n
693
01:14:28,449 --> 01:14:35,210
negative 140. All of our sex. Well, if you're\n
694
01:14:35,210 --> 01:14:40,399
the square root sign, you should be we can't\n
695
01:14:40,399 --> 01:14:45,469
and get an N get a real number is our answer.\n
696
01:14:45,470 --> 01:14:54,650
number. And therefore, our conclusion is we\n
697
01:14:54,649 --> 01:14:59,689
In this video, we solve some quadratic equations\n
698
01:15:00,689 --> 01:15:03,669
Either factoring or using the quadratic formula.
699
01:15:03,670 --> 01:15:08,440
In some examples, factoring doesn't work,\n
700
01:15:08,439 --> 01:15:13,969
But in fact, using the quadratic formula will\n
701
01:15:13,970 --> 01:15:19,011
solve it by factoring. So you can't really\n
702
01:15:19,011 --> 01:15:23,000
just sometimes it'll be faster to factor instead.
703
01:15:23,000 --> 01:15:29,539
This video is about solving rational equations.\n
704
01:15:29,539 --> 01:15:34,189
that has rational expressions in that, in\n
705
01:15:35,359 --> 01:15:41,069
There are several different approaches for\n
706
01:15:41,069 --> 01:15:46,719
start by finding the least common denominator.\n
707
01:15:46,720 --> 01:15:52,350
three and x, we can think of one as just having\n
708
01:15:52,350 --> 01:15:57,250
Since the denominators don't have any factors\n
709
01:15:57,250 --> 01:16:00,689
just by multiplying them together.
710
01:16:00,689 --> 01:16:04,359
My next step is going to be clearing the denominator.
711
01:16:04,359 --> 01:16:12,460
By this, I mean that I multiply both sides\n
712
01:16:12,460 --> 01:16:19,270
x plus three times x, I multiply on the left\n
713
01:16:19,270 --> 01:16:22,310
same thing on the right side of the equation.
714
01:16:22,310 --> 01:16:28,250
Since I'm doing the same thing to both sides\n
715
01:16:28,250 --> 01:16:33,020
of the equation. Multiplying the least common\n
716
01:16:33,020 --> 01:16:38,650
is equivalent to multiplying it by all three\n
717
01:16:41,189 --> 01:16:47,909
I'll rewrite the left side the same as before,\n
718
01:16:47,909 --> 01:16:56,029
right side to get x plus three times x times\n
719
01:16:56,029 --> 01:17:01,920
x. So I've actually multiplied the least common\n
720
01:17:01,920 --> 01:17:07,920
Now I can have a blast canceling things. The\n
721
01:17:10,119 --> 01:17:15,579
The here are nothing cancels out because there's\n
722
01:17:15,579 --> 01:17:18,470
numerator cancels with the x in the denominator.
723
01:17:18,470 --> 01:17:26,310
So I can rewrite my expression as x squared\n
724
01:17:26,310 --> 01:17:29,550
x plus three. Now I'm going to simplify.
725
01:17:29,550 --> 01:17:36,690
So I'll leave the x squared alone on this\n
726
01:17:36,689 --> 01:17:44,039
plus x plus three, hey, look, the x squared\n
727
01:17:44,039 --> 01:17:51,199
equals 4x plus three, so 4x is negative three,\n
728
01:17:51,199 --> 01:17:57,409
I'm going to plug in my answer to check. This\n
729
01:17:57,409 --> 01:18:01,609
it's especially important for a rational equation\n
730
01:18:01,609 --> 01:18:05,699
you'll get what's called extraneous solution\n
731
01:18:05,699 --> 01:18:09,949
original equation because they make the denominator\n
732
01:18:09,949 --> 01:18:16,029
we're going to get any extraneous equations\n
733
01:18:16,029 --> 01:18:20,670
to make any of these denominators zero, so\n
734
01:18:29,489 --> 01:18:34,119
the denominator here, negative three fourths\n
735
01:18:34,119 --> 01:18:42,680
nine fourths. And this is one I'll flip and\n
736
01:18:42,680 --> 01:18:49,570
I can simplify my complex fraction, it ends\n
737
01:18:49,569 --> 01:18:55,960
four thirds is negative 1/3. So that all seems\nto check out.
738
01:18:55,960 --> 01:19:00,890
And so my final answer is x equals negative\nthree fourths.
739
01:19:00,890 --> 01:19:05,310
This next example looks a little trickier.\n
740
01:19:05,310 --> 01:19:13,410
First off, find the least common denominator.\n
741
01:19:13,409 --> 01:19:22,029
c plus one, and C squared minus four c minus\n
742
01:19:22,029 --> 01:19:29,189
five times c plus one. Now, my least common\n
743
01:19:29,189 --> 01:19:35,269
to that each of these denominators divided\n
744
01:19:35,270 --> 01:19:39,960
I need the factor c plus one. And now I've\n
745
01:19:39,960 --> 01:19:47,480
denominator. So here is my least common denominator.\n
746
01:19:47,479 --> 01:19:53,589
So I do this by multiplying both sides of\n
747
01:19:53,590 --> 01:20:01,560
In fact, I can just multiply each of the three\n
748
01:20:01,560 --> 01:20:06,789
I went ahead and wrote my third denominator\n
749
01:20:06,789 --> 01:20:16,619
what cancels. Now canceling time dies, this\n
750
01:20:16,619 --> 01:20:21,229
out the denominator is the whole point of\n
751
01:20:21,229 --> 01:20:25,189
you're multiplying by something that's big\n
752
01:20:25,189 --> 01:20:28,649
you don't have to deal with denominators anymore.
753
01:20:28,649 --> 01:20:32,069
Now I'm going to simplify by multiplying out.
754
01:20:32,069 --> 01:20:41,569
So I get, let's see, c plus one times four\n
755
01:20:41,569 --> 01:20:51,130
I get minus just c minus five, and then over\n
756
01:20:51,130 --> 01:21:00,440
I can rewrite the minus quantity c minus five\n
757
01:21:00,439 --> 01:21:06,829
And now I can subtract the three c squared\n
758
01:21:06,829 --> 01:21:15,710
here, and the four c minus c that becomes\na three C.
759
01:21:15,710 --> 01:21:21,829
And finally, I can subtract the three from\n
760
01:21:21,829 --> 01:21:28,979
two equals zero. got myself a quadratic equation\n
761
01:21:28,979 --> 01:21:36,229
this factors to C plus one times c plus two\n
762
01:21:36,229 --> 01:21:44,279
or C plus two is zero. So C equals negative\n
763
01:21:44,279 --> 01:21:49,689
Now let's see, we need to still check our\nanswers.
764
01:21:49,689 --> 01:21:54,519
Without even going to the trouble of calculating\n
765
01:21:54,520 --> 01:22:01,160
one is not going to work, because if I plug\n
766
01:22:01,159 --> 01:22:09,250
zero, which doesn't make sense. So C equals\n
767
01:22:09,250 --> 01:22:16,789
actually satisfy my original equation. And\n
768
01:22:19,869 --> 01:22:23,809
I can go if I go ahead, and that doesn't make\n
769
01:22:23,810 --> 01:22:30,780
made any mistakes, it should satisfy my original\n
770
01:22:39,850 --> 01:22:46,690
So my final answer is C equals negative two.\n
771
01:22:46,689 --> 01:22:53,349
equations using the method of finding the\n
772
01:22:55,600 --> 01:22:59,490
we cleared the denominator by multiplying\n
773
01:22:59,489 --> 01:23:04,389
denominator or equivalently. multiplying each\n
774
01:23:04,390 --> 01:23:10,250
There's another equivalent method that some\n
775
01:23:10,250 --> 01:23:16,390
we find the least common denominator, but\n
776
01:23:16,390 --> 01:23:22,100
over that least common denominator. So in\n
777
01:23:22,100 --> 01:23:27,829
denominator of x plus three times x. But our\n
778
01:23:27,829 --> 01:23:32,350
rational expressions over that common denominator\n
779
01:23:32,350 --> 01:23:38,880
appropriate things. So one, in order to get\n
780
01:23:38,880 --> 01:23:44,310
to multiply the top and the bottom by x plus\n
781
01:23:44,310 --> 01:23:49,640
the top and the bottom just by x plus three\n
782
01:23:49,640 --> 01:23:53,380
x. Now, if I simplify a little bit
783
01:23:53,380 --> 01:24:00,090
let's say this is x squared over that common\ndenominator, and
784
01:24:00,090 --> 01:24:09,150
here I have just x plus three times x over\n
785
01:24:09,149 --> 01:24:17,139
over that common denominator. Now add together\n
786
01:24:18,949 --> 01:24:27,189
So this is x plus three times x plus x plus\n
787
01:24:27,189 --> 01:24:32,939
that are equal that have the same denominator,\n
788
01:24:32,939 --> 01:24:37,759
also. So the next step is to set the numerators\nequal.
789
01:24:37,760 --> 01:24:46,860
So I get x squared is x plus three times x\n
790
01:24:46,859 --> 01:24:52,079
the previous way, we solve this equation,\n
791
01:24:52,079 --> 01:24:57,809
here on we just continue as before.
792
01:24:57,810 --> 01:25:00,020
When choosing between these two methods, I\npersonally tend
793
01:25:00,020 --> 01:25:03,600
prefer the clear the denominators method,\n
794
01:25:03,600 --> 01:25:07,000
don't have to get rid of those denominators\n
795
01:25:07,000 --> 01:25:12,569
times. But some people find this one a little\n
796
01:25:12,569 --> 01:25:15,359
understand either of these methods is fine.
797
01:25:15,359 --> 01:25:21,529
One last caution, don't forget at the end,\n
798
01:25:23,460 --> 01:25:30,670
These will be solutions that make the denominators\n
799
01:25:30,670 --> 01:25:36,250
This video is about solving radical equations,\n
800
01:25:36,250 --> 01:25:40,789
square root signs in them, or cube roots or\n
801
01:25:40,789 --> 01:25:45,621
When I see an equation with a square root\n
802
01:25:45,621 --> 01:25:51,079
root. But it'll be easiest to get rid of the\n
803
01:25:51,079 --> 01:25:56,750
root. In other words, I want to get the term\n
804
01:25:56,750 --> 01:26:01,739
of the equation by itself, and everything\n
805
01:26:01,739 --> 01:26:07,489
I start with my original equation, x plus\n
806
01:26:07,489 --> 01:26:12,929
x from both sides, then that does isolate\n
807
01:26:12,930 --> 01:26:19,850
everything else on the right. Once I've isolated\n
808
01:26:19,850 --> 01:26:27,079
rid of the square root. And I'll do that by\nsquaring both sides
809
01:26:27,079 --> 01:26:36,479
of my equation. So I'll take the square root\n
810
01:26:36,479 --> 01:26:42,309
Now the square root of x squared is just x,\n
811
01:26:45,369 --> 01:26:54,170
to work out 12 minus x squared, write it out\n
812
01:26:54,170 --> 01:27:02,539
minus x is minus 12x. I get another minus\n12x from here.
813
01:27:02,539 --> 01:27:09,869
And finally minus x times minus x is positive\n
814
01:27:09,869 --> 01:27:18,640
that's minus 24x. And now I can subtract x\n
815
01:27:18,640 --> 01:27:26,550
25x plus x squared. That's a quadratic equation,\n
816
01:27:27,890 --> 01:27:34,170
So now I've got a familiar quadratic equation\n
817
01:27:34,170 --> 01:27:38,800
I'll just proceed to solve it like I usually\n
818
01:27:38,800 --> 01:27:46,079
So I'm going to look for two numbers that\n
819
01:27:46,079 --> 01:27:53,079
I'm going to need negative numbers to get\n
820
01:27:53,079 --> 01:27:57,800
negative numbers. So they still multiply to\n
821
01:27:57,800 --> 01:28:04,730
factors of 144, I could have negative one\n
822
01:28:04,729 --> 01:28:12,079
72, negative four, negative 36, and so on.\n
823
01:28:12,079 --> 01:28:17,780
it's not hard to find the two that add to\n
824
01:28:17,780 --> 01:28:28,029
16. So now I can factor in my quadratic equation\n
825
01:28:28,029 --> 01:28:34,269
that means that x minus nine is zero or x\n
826
01:28:34,270 --> 01:28:41,000
16. I'm almost done. But there's one last\n
827
01:28:41,000 --> 01:28:49,170
Solutions so that we can eliminate any extraneous\n
828
01:28:49,170 --> 01:28:54,149
that we get that does not actually satisfy\n
829
01:28:54,149 --> 01:29:00,269
can happen when you're solving equations with\n
830
01:29:00,270 --> 01:29:06,000
nine. If we plug in to our original equation,\n
831
01:29:06,000 --> 01:29:11,539
want that to equal 12. Well, the square root\n
832
01:29:11,539 --> 01:29:18,390
indeed equal 12. So that solution checks out.\n
833
01:29:18,390 --> 01:29:25,020
get 16 plus a squared of 16. And that's supposed\n
834
01:29:25,020 --> 01:29:31,880
is supposed to equal 12. But that most definitely\n
835
01:29:31,880 --> 01:29:47,840
to be extraneous solution, and our only solution\n
836
01:29:47,840 --> 01:30:26,100
This next equation might not look like an\n
837
01:30:26,100 --> 01:30:32,210
we can think of a fractional exponent as being\n
838
01:30:32,210 --> 01:30:38,829
the same thing we did on the previous problem\n
839
01:30:41,590 --> 01:30:46,730
that involves the fractional exponent. So\n
840
01:30:46,729 --> 01:30:53,299
times P to the four fifths equals 1/8. And\n
841
01:30:53,300 --> 01:31:00,000
I can multiply both sides by one half, that\n
842
01:31:00,000 --> 01:31:05,739
And I've effectively isolated the part of\n
843
01:31:05,739 --> 01:31:12,920
as much as possible. Now, in the previous\n
844
01:31:12,920 --> 01:31:18,980
radical. In this example, we're going to get\n
845
01:31:18,979 --> 01:31:24,799
to actually do this in two stages. First,\n
846
01:31:26,689 --> 01:31:34,509
That's because when I take an exponent to\n
847
01:31:34,510 --> 01:31:41,840
so that becomes just p to the fourth equals\n
848
01:31:41,840 --> 01:31:48,360
get rid of the fourth power by raising both\n
849
01:31:48,359 --> 01:31:53,299
root, there's something that you need to be\n
850
01:31:53,300 --> 01:32:01,880
an even root, or the one over an even number\n
851
01:32:04,630 --> 01:32:10,869
It's kind of like when you write x squared\n
852
01:32:10,869 --> 01:32:16,449
of both sides, x could equal plus or minus\n
853
01:32:16,449 --> 01:32:23,659
plus or minus two, since minus two squared\n
854
01:32:23,659 --> 01:32:32,800
why when you take an even root, or a one over\n
855
01:32:32,800 --> 01:32:39,440
to include the plus or minus sign, when it's\n
856
01:32:39,439 --> 01:32:46,159
don't need to do that. If you had something\n
857
01:32:46,159 --> 01:32:51,180
equals the cube root of negative eight, which\n
858
01:32:51,180 --> 01:32:55,560
you don't need to do the plus or minus because\n
859
01:32:55,560 --> 01:33:02,390
So that aside, explains why we need this plus\n
860
01:33:02,390 --> 01:33:09,590
1/4. When I raise a power to a power, I multiply\n
861
01:33:09,590 --> 01:33:17,630
which is equal to plus or minus 1/16 to the\n
862
01:33:17,630 --> 01:33:23,010
Now I just need to simplify this expression,\n
863
01:33:23,010 --> 01:33:28,760
power, because 16 to the fifth power is like\n
864
01:33:32,420 --> 01:33:42,649
as P equals plus or minus 1/16. I'll write\n
865
01:33:42,649 --> 01:33:49,239
And as I continue to solve using my exponent\nrules
866
01:33:49,239 --> 01:33:58,939
I'm going to prefer to write this as 1/16\n
867
01:33:58,939 --> 01:34:03,649
it's going to be easier to take the fourth\n
868
01:34:03,649 --> 01:34:08,149
the same thing as the fourth root of one over\n
869
01:34:08,149 --> 01:34:14,359
fifth power. fourth root of one is just one\n
870
01:34:14,359 --> 01:34:19,279
to the fifth power, that's just going to be\n
871
01:34:24,819 --> 01:34:28,949
The last step is to check answers.
872
01:34:28,949 --> 01:34:35,880
So I have the two answers p equals 130 seconds,\n
873
01:34:36,880 --> 01:34:43,199
1/32 to the fourth fifth power.
874
01:34:43,199 --> 01:34:51,109
That gives me two times one to the fourth\n
875
01:34:51,109 --> 01:34:59,659
two times one over 32/5 routed to the fourth\n
876
01:34:59,659 --> 01:35:07,519
Raisa to the fourth power, I get 16. So this\n
877
01:35:07,520 --> 01:35:16,910
wanted in the original equation up here. Similarly,\n
878
01:35:16,909 --> 01:35:24,039
actually does satisfy the equation, I'll leave\n
879
01:35:24,039 --> 01:35:30,710
So our two solutions are p equals one over\n
880
01:35:30,710 --> 01:35:35,310
to point out an alternate approach to getting\n
881
01:35:35,310 --> 01:35:41,270
gotten rid of it all in one fell swoop by\n
882
01:35:45,340 --> 01:35:51,199
five fourths is the reciprocal of four fifths.\n
883
01:35:51,199 --> 01:35:56,859
when I raise the power to the power, I multiply\n
884
01:35:56,859 --> 01:36:03,670
fifths times five fourths is plus or minus\n
885
01:36:03,670 --> 01:36:10,569
P to the One Power, which is just P is plus\n
886
01:36:10,569 --> 01:36:16,759
an alternate and possibly faster way to get\n
887
01:36:16,760 --> 01:36:22,440
comes from the fact that when we take the\n
888
01:36:22,439 --> 01:36:27,859
a fourth root, and so we need to consider\n
889
01:36:27,859 --> 01:36:33,449
This video is about solving radical equations,\n
890
01:36:33,449 --> 01:36:39,059
square root signs in them, or cube roots or\n
891
01:36:39,060 --> 01:36:45,680
In this video, we solved radical equations\n
892
01:36:48,270 --> 01:36:54,670
and then removing the radical sine or the\nfractional exponent
893
01:36:54,670 --> 01:37:01,630
by either squaring both sides or taking the\n
894
01:37:01,630 --> 01:37:06,550
This video is about solving equations with\n
895
01:37:06,550 --> 01:37:12,130
Recall that the absolute value of a positive\n
896
01:37:12,130 --> 01:37:17,761
value of a negative number is its opposite.\n
897
01:37:17,761 --> 01:37:23,170
of a number as representing its distance from\n
898
01:37:25,840 --> 01:37:32,360
and the number of negative four are both at\n
899
01:37:32,359 --> 01:37:35,939
value of both of them is four.
900
01:37:35,939 --> 01:37:44,339
Similarly, if I write the equation, the absolute\n
901
01:37:44,340 --> 01:37:50,319
to be three units away from zero on the number\nline.
902
01:37:50,319 --> 01:37:56,319
And so X would have to be either negative\n
903
01:37:56,319 --> 01:38:00,439
three times the absolute value of x plus two\nequals four.
904
01:38:00,439 --> 01:38:08,639
I'd like to isolate the absolute value part\n
905
01:38:14,090 --> 01:38:17,670
subtracting two from both sides
906
01:38:17,670 --> 01:38:23,920
and dividing both sides by three.
907
01:38:23,920 --> 01:38:31,100
Now I'll think in terms of distance on a number\n
908
01:38:31,100 --> 01:38:40,671
means that x is two thirds away from zero.\n
909
01:38:43,289 --> 01:38:50,420
And the answer to my equation is x is negative\n
910
01:38:50,420 --> 01:38:52,711
I can check my answers by plugging in
911
01:38:52,711 --> 01:39:00,090
three times the added value of negative two\n
912
01:39:00,090 --> 01:39:04,739
four. Well, the absolute value of negative\n
913
01:39:04,739 --> 01:39:10,969
three times two thirds plus two, which works\nout to four.
914
01:39:10,970 --> 01:39:20,050
Similarly, if I plug in positive two thirds,\n
915
01:39:20,050 --> 01:39:24,239
The second example is a little different,\n
916
01:39:24,239 --> 01:39:29,210
a more complicated expression, not just around\nthe X.
917
01:39:29,210 --> 01:39:35,930
I would start by isolating the absolute value\npart.
918
01:39:35,930 --> 01:39:41,730
But it's already isolated. So I'll just go\n
919
01:39:41,729 --> 01:39:49,899
on the number line. So on my number line,\n
920
01:39:49,899 --> 01:39:53,979
to be at a distance of four from zero.
921
01:39:53,979 --> 01:39:59,939
So that means that 3x plus two is here at\nfour or 3x plus two
922
01:39:59,939 --> 01:40:03,839
though is it negative four, all right, those\nas equations
923
01:40:03,840 --> 01:40:12,090
3x plus two equals four, or 3x plus two is\n
924
01:40:12,090 --> 01:40:20,369
So this becomes 3x equals two, or x equals\n
925
01:40:20,369 --> 01:40:29,489
minus six, or x equals minus two. Finally,\n
926
01:40:29,489 --> 01:40:32,800
I'll leave it to you to verify that they both\nwork.
927
01:40:32,800 --> 01:40:38,590
A common mistake on absolute value equations\n
928
01:40:38,590 --> 01:40:44,569
like we did here, and then just solve for\n
929
01:40:44,569 --> 01:40:48,759
Another mistake sometimes people make is,\n
930
01:40:48,760 --> 01:40:52,640
assume that the negative of that works also.
931
01:40:52,640 --> 01:40:57,900
But that doesn't always work. In the first\n
932
01:40:57,899 --> 01:41:02,710
of each other. But in our second examples,\n
933
01:41:02,710 --> 01:41:06,619
of each other one was two thirds and the other\nwas negative two.
934
01:41:06,619 --> 01:41:12,659
In this third example, let's again, isolate\n
935
01:41:12,659 --> 01:41:16,760
So starting with our original equation
936
01:41:16,760 --> 01:41:22,550
we can subtract 16 from both sides
937
01:41:22,550 --> 01:41:31,320
and divide both sides by five or equivalently,\nmultiply by 1/5.
938
01:41:31,319 --> 01:41:35,049
Now let's think about distance on the number\nline
939
01:41:35,050 --> 01:41:41,770
we have an absolute value needs to equal negative\n
940
01:41:41,770 --> 01:41:48,400
absolute value sign needs to be at distance\n
941
01:41:48,399 --> 01:41:52,849
be at distance negative three from zero. Another\n
942
01:41:52,850 --> 01:41:57,220
the absolute value of something and end up\n
943
01:41:57,220 --> 01:42:05,530
positive, or zero. So this equation doesn't\n
944
01:42:07,510 --> 01:42:15,010
In this video, we solved absolute value equations.\n
945
01:42:15,010 --> 01:42:20,949
will have two solutions. But in some cases,\n
946
01:42:25,079 --> 01:42:30,979
This video is about interval notation, and\n
947
01:42:30,979 --> 01:42:39,159
Before dealing with interval notation, it\n
948
01:42:39,159 --> 01:42:45,420
Our first example of an inequality is written\n
949
01:42:46,750 --> 01:42:53,260
First, we used to write down our variable\n
950
01:42:53,260 --> 01:43:01,320
values for this problem that is one and three.\n
951
01:43:01,319 --> 01:43:07,529
numbers, meaning we will have one inequality\n
952
01:43:07,529 --> 01:43:12,779
it says not including one and three, which\n
953
01:43:12,779 --> 01:43:18,670
beneath each inequality. Here we will put\n
954
01:43:18,670 --> 01:43:24,489
put three the highest key value. Next, we're\n
955
01:43:27,069 --> 01:43:34,029
Here we write our key values one, and three.\n
956
01:43:34,029 --> 01:43:40,189
we have an empty circle around each number.\n
957
01:43:41,289 --> 01:43:50,069
The last step of this problem is writing this\n
958
01:43:50,069 --> 01:43:54,949
writing things in interval notation is kind\n
959
01:43:54,949 --> 01:43:57,579
put one and then here we will put three.
960
01:43:57,579 --> 01:44:03,189
Next we need to put brackets around these\nnumbers.
961
01:44:03,189 --> 01:44:09,429
For this problem, it is not including one\n
962
01:44:09,430 --> 01:44:15,970
However, if it were including one and three,\n
963
01:44:15,970 --> 01:44:22,960
It is important to note for interval notation,\n
964
01:44:22,960 --> 01:44:29,039
left and the biggest value always goes on\nthe right.
965
01:44:29,039 --> 01:44:32,680
You also include a comma between your two\nkey values.
966
01:44:32,680 --> 01:44:41,770
Now let's work on problem B. Once again we\n
967
01:44:41,770 --> 01:44:47,170
negative four and two but this time it is\n
968
01:44:47,170 --> 01:44:53,859
us to have the or equal to assign below the\n
969
01:44:53,859 --> 01:44:59,739
here and their highest here. The number line\n
970
01:45:00,739 --> 01:45:07,399
have our key values negative four and two.\n
971
01:45:07,399 --> 01:45:15,759
right here, we instead use a closed circle,\n
972
01:45:15,760 --> 01:45:21,320
four and two, you complete this with a line\n
973
01:45:21,319 --> 01:45:27,170
notation. In the last problem, I said that\n
974
01:45:27,170 --> 01:45:35,470
we would use these hard brackets. Now we are\n
975
01:45:35,470 --> 01:45:40,650
on each side because isn't including foreign\n
976
01:45:40,649 --> 01:45:46,799
left, and our highest value on the right,\n
977
01:45:46,800 --> 01:45:51,029
You now know how to correctly write these\n
978
01:45:51,029 --> 01:45:58,689
Now we are going to practice transforming\n
979
01:45:58,689 --> 01:46:05,569
notation, and vice versa. This is slightly\n
980
01:46:05,569 --> 01:46:11,939
of having two soft brackets or two hard brackets,\n
981
01:46:11,939 --> 01:46:18,559
left side will have a hard bracket because\n
982
01:46:18,560 --> 01:46:24,039
right side, it is a soft bracket because it\n
983
01:46:24,039 --> 01:46:31,880
put a comma in the middle, a lower key value\n
984
01:46:31,880 --> 01:46:37,140
we're taking an equation already written an\n
985
01:46:38,430 --> 01:46:47,600
For the second problem, we can see are key\n
986
01:46:47,600 --> 01:46:54,340
Now, this sounds a little bit weird, but let's\n
987
01:46:54,340 --> 01:47:02,470
x, which is less than because of the soft\n
988
01:47:02,470 --> 01:47:09,409
or equal to because it has a soft bracket\n
989
01:47:09,409 --> 01:47:16,319
greater than negative infinity, we can take\n
990
01:47:17,739 --> 01:47:25,269
It is important to note that a soft bracket\n
991
01:47:25,270 --> 01:47:32,010
infinity is not a real number, so we cannot\n
992
01:47:32,010 --> 01:47:38,110
The next problem is a little tricky, because\n
993
01:47:38,109 --> 01:47:43,829
But let's just start off the equation as a\n
994
01:47:43,829 --> 01:47:51,510
of an order equal to sign. And negative 15\n
995
01:47:51,510 --> 01:47:57,190
On the other side, we put the highest possible\n
996
01:47:57,189 --> 01:48:05,289
a soft bracket. We can see the relation of\n
997
01:48:05,289 --> 01:48:12,119
into this problem when x is greater than a\n
998
01:48:12,119 --> 01:48:19,189
Once again, we have our sauce bracket for\n
999
01:48:19,189 --> 01:48:26,279
Part D brings up an important point about\n
1000
01:48:26,279 --> 01:48:30,130
our other problems, we have had the inequalities\npointing left
1001
01:48:30,130 --> 01:48:41,609
instead of rights. When seeing Part D, you\n
1002
01:48:41,609 --> 01:48:47,799
would go here, and zero is on the right who\n
1003
01:48:47,800 --> 01:48:53,940
When writing inequalities, you must always\n
1004
01:48:53,939 --> 01:49:01,550
this problem is zero. To fix this, we can\n
1005
01:49:01,551 --> 01:49:06,800
this, it is still identical just written in\n
1006
01:49:06,800 --> 01:49:13,070
notation, then we can see that zero has a\n
1007
01:49:13,069 --> 01:49:19,899
zero, but r comma four or other key value\n
1008
01:49:21,859 --> 01:49:29,420
For interval notation, you must always have\n
1009
01:49:29,420 --> 01:49:36,190
This was our video on interval notation, an\n
1010
01:49:36,189 --> 01:49:41,739
This video is about solving inequalities that\n
1011
01:49:41,739 --> 01:49:45,979
Let's look at the inequality absolute value\n
1012
01:49:45,979 --> 01:49:52,529
Thinking of absolute value as distance. This\n
1013
01:49:52,529 --> 01:50:01,380
is less than five units. So x has to live\n
1014
01:50:01,380 --> 01:50:06,560
We can express this as an inequality without\n
1015
01:50:06,560 --> 01:50:12,850
is less than x, which is less than five. Or\n
1016
01:50:12,850 --> 01:50:16,890
negative five, five, soft bracket.
1017
01:50:16,890 --> 01:50:20,760
Both of these formulations are equivalent\n
1018
01:50:23,649 --> 01:50:28,670
In the second example, we're looking for the\n
1019
01:50:28,670 --> 01:50:32,649
x is greater than or equal to five.
1020
01:50:32,649 --> 01:50:39,299
on the number line, this means that the distance\n
1021
01:50:42,069 --> 01:50:48,130
A distance bigger than five units means that\n
1022
01:50:48,130 --> 01:50:54,730
over here, where it's farther than five units\n
1023
01:50:54,729 --> 01:51:00,320
distance equal to five units. So I'll fill\n
1024
01:51:00,320 --> 01:51:04,569
the number line that satisfy my inequality.
1025
01:51:04,569 --> 01:51:10,130
Now I can rewrite the inequality without the\n
1026
01:51:10,130 --> 01:51:17,090
less than or equal to negative five, or x\n
1027
01:51:17,090 --> 01:51:20,239
also write this in interval notation
1028
01:51:20,239 --> 01:51:26,689
soft bracket, negative infinity, negative\n
1029
01:51:26,689 --> 01:51:33,829
hard bracket five infinity soft bracket, I\n
1030
01:51:33,829 --> 01:51:38,269
I'm trying to describe all these points on\n
1031
01:51:39,920 --> 01:51:44,520
Let's take this analysis a step further with\n
1032
01:51:44,520 --> 01:51:49,010
want the absolute value of three minus two\n
1033
01:51:49,010 --> 01:51:56,020
And an absolute value less than four means\n
1034
01:51:56,020 --> 01:52:00,260
But it's not the variable t that lives in\n
1035
01:52:00,260 --> 01:52:04,960
zero, it's the whole expression, three minus\ntwo t.
1036
01:52:04,960 --> 01:52:13,980
So three minus two t, live somewhere in here.\n
1037
01:52:13,979 --> 01:52:21,209
absolute value signs by saying negative four\n
1038
01:52:21,210 --> 01:52:27,730
four. Now I have a compound inequality that\n
1039
01:52:27,729 --> 01:52:33,739
three from all three sides to get negative\n
1040
01:52:33,739 --> 01:52:40,719
than one. And now I'll divide all three sides\n
1041
01:52:40,720 --> 01:52:46,140
number, this reverses the directions of the\ninequalities.
1042
01:52:46,140 --> 01:52:52,050
Simplifying, I get seven halves is greater\n
1043
01:52:52,050 --> 01:52:58,720
So my final answer on the number line looks\n
1044
01:53:01,880 --> 01:53:07,340
But not including the endpoints, and an interval\n
1045
01:53:07,340 --> 01:53:09,920
a half, seven, half soft bracket
1046
01:53:09,920 --> 01:53:14,539
please pause the video and try the next problem\non your own.
1047
01:53:14,539 --> 01:53:19,909
thinking in terms of distance, this inequality\n
1048
01:53:19,909 --> 01:53:25,880
three minus two t and zero is always bigger\n
1049
01:53:27,609 --> 01:53:33,779
If three minus two t has a distance bigger\n
1050
01:53:33,779 --> 01:53:39,869
region that's near zero, it has to be on the\n
1051
01:53:39,869 --> 01:53:48,569
That is three minus two t is either less than\n
1052
01:53:48,569 --> 01:53:54,369
than four. I solve these two inequalities\n
1053
01:53:54,369 --> 01:54:01,390
from both sides, I get negative two t is less\n
1054
01:54:01,390 --> 01:54:08,350
I get T is bigger than seven halves. And then\n
1055
01:54:08,350 --> 01:54:15,820
greater than one. So t is less than negative\n
1056
01:54:15,819 --> 01:54:21,259
last step due to dividing by a negative number.\n
1057
01:54:21,260 --> 01:54:28,750
again, the first piece says that t is greater\n
1058
01:54:28,750 --> 01:54:41,029
And the second piece says that t is less than\n
1059
01:54:41,029 --> 01:54:46,380
Because these two statements are joined with\n
1060
01:54:46,380 --> 01:54:52,690
in this one, or in this one. That is I want\n
1061
01:54:52,689 --> 01:55:00,029
interval notation, this reads negative infinity\n
1062
01:55:03,130 --> 01:55:09,289
This last example looks more complicated.\n
1063
01:55:09,289 --> 01:55:14,500
value part, it looks pretty much like the\n
1064
01:55:14,500 --> 01:55:20,310
seven from both sides. And then I'll divide\n
1065
01:55:21,310 --> 01:55:28,060
And I'm looking for this expression for x\n
1066
01:55:28,060 --> 01:55:33,471
than or equal to negative three from zero,\n
1067
01:55:33,470 --> 01:55:36,759
to negative three, well, distance is always\n
1068
01:55:36,760 --> 01:55:43,340
always greater than equal to zero. So this,\n
1069
01:55:43,340 --> 01:55:50,119
And so the answer to my inequality is all\n
1070
01:55:50,119 --> 01:55:55,930
In other words, all real numbers.
1071
01:55:55,930 --> 01:56:01,400
Once solving absolute value inequalities,\n
1072
01:56:01,399 --> 01:56:08,500
value of something that's less than a number\n
1073
01:56:12,869 --> 01:56:18,579
On the other hand, an absolute value is something\n
1074
01:56:18,579 --> 01:56:25,829
whatever's inside the absolute value sign\n
1075
01:56:25,829 --> 01:56:29,960
from zero is bigger than that certain number.
1076
01:56:29,960 --> 01:56:34,739
drawing these pictures on the number line\n
1077
01:56:34,739 --> 01:56:40,899
and equality as an inequality that doesn't\n
1078
01:56:40,899 --> 01:56:47,179
it would be negative three is less than x\n
1079
01:56:47,180 --> 01:56:53,630
case, it would be either x plus two is less\n
1080
01:56:55,880 --> 01:57:01,670
This video is about solving linear inequalities.\n
1081
01:57:01,670 --> 01:57:09,170
involve a variable here x, but don't involve\n
1082
01:57:09,170 --> 01:57:15,739
The good news is, we can solve linear inequalities,\n
1083
01:57:15,739 --> 01:57:19,779
adding and subtracting terms to both sides\n
1084
01:57:19,779 --> 01:57:26,719
both sides. The only thing that's different\n
1085
01:57:26,720 --> 01:57:32,840
number, then you need to reverse the direction\n
1086
01:57:32,840 --> 01:57:39,319
the inequality, negative x is less than negative\n
1087
01:57:39,319 --> 01:57:45,710
by negative one to get rid of the negative\n
1088
01:57:45,710 --> 01:57:50,560
the inequality. With this caution in mind,\n
1089
01:57:50,560 --> 01:57:56,170
Since our variable x is trapped in parentheses,\n
1090
01:57:58,569 --> 01:58:04,849
That gives me negative 5x minus 10 plus three\n
1091
01:58:04,850 --> 01:58:10,020
Negative 10 plus three is negative seven,\n
1092
01:58:10,020 --> 01:58:16,890
seven is greater than eight. Now I'll add\n
1093
01:58:16,890 --> 01:58:23,100
greater than 15. Now I'd like to divide both\n
1094
01:58:23,100 --> 01:58:31,260
is a negative number that reverses the inequality.\n
1095
01:58:31,260 --> 01:58:36,130
five. In other words, x is less than negative\nthree.
1096
01:58:36,130 --> 01:58:40,390
If I wanted to graph this on a number line,\n
1097
01:58:40,390 --> 01:58:44,400
an open circle around it, and shade in to\nthe left
1098
01:58:44,399 --> 01:58:51,049
I use an open circle, because x is strictly\n
1099
01:58:51,050 --> 01:58:56,070
three. If I wanted to write this in interval\n
1100
01:58:56,069 --> 01:59:01,989
infinity, negative three soft bracket. Again,\n
1101
01:59:01,989 --> 01:59:10,090
is not included. This next example is an example\n
1102
01:59:10,090 --> 01:59:15,900
Either this statement is true, or this statement\n
1103
01:59:15,899 --> 01:59:21,529
satisfy either one. I'll solve this by working\n
1104
01:59:21,529 --> 01:59:28,079
them at the end. For the inequality on the\n
1105
01:59:32,149 --> 01:59:36,189
then subtract x from both sides
1106
01:59:36,189 --> 01:59:40,149
and then divide both sides by two.
1107
01:59:40,149 --> 01:59:45,170
I didn't have to reverse the inequality sign\n
1108
01:59:45,170 --> 01:59:51,640
on the right side, I'll copy the equation\n
1109
01:59:51,640 --> 01:59:59,329
both sides by 696 is the same as three halves.\n
1110
02:00:00,329 --> 02:00:07,300
row four, make this statement true. Let me\n
1111
02:00:07,300 --> 02:00:14,329
x is less than or equal to negative two, means\n
1112
02:00:14,329 --> 02:00:22,670
to the left. X is greater than three halves\n
1113
02:00:25,239 --> 02:00:33,050
My final answer includes both of these pieces,\n
1114
02:00:33,050 --> 02:00:38,880
allowed to be an either one or the other.\n
1115
02:00:38,880 --> 02:00:44,340
The first piece on the number line can be\n
1116
02:00:44,340 --> 02:00:50,900
negative two hard bracket. And the second\n
1117
02:00:50,899 --> 02:00:57,379
halves, infinity soft bracket to indicate\n
1118
02:00:57,380 --> 02:01:01,590
I use the union side, which is a U.
1119
02:01:01,590 --> 02:01:09,340
That means that my answer includes all x values\n
1120
02:01:09,340 --> 02:01:15,079
This next example is also a compound inequality.\n
1121
02:01:15,079 --> 02:01:22,029
I'm looking for all y values that satisfy\n
1122
02:01:22,029 --> 02:01:25,340
I can solve each piece separately
1123
02:01:25,340 --> 02:01:34,090
on the left, to isolate the Y, I need to multiply\n
1124
02:01:34,090 --> 02:01:40,460
that gives me Why is less than negative 12\n
1125
02:01:40,460 --> 02:01:46,649
flip to a less than because I was multiplying\n
1126
02:01:47,729 --> 02:01:54,469
By clean up the right side, I get why is less\nthan 18.
1127
02:01:54,470 --> 02:01:59,590
On the right side, I'll start by subtracting\n
1128
02:01:59,590 --> 02:02:07,300
And now I'll divide by negative four, again,\n
1129
02:02:07,300 --> 02:02:13,270
So that's why is less than three over negative\n
1130
02:02:13,270 --> 02:02:19,710
three fourths. Again, I'm looking for the\n
1131
02:02:22,229 --> 02:02:25,819
Let me graph this on the number line
1132
02:02:25,819 --> 02:02:34,819
the Y is less than 18. I can graph that by\n
1133
02:02:34,819 --> 02:02:41,519
I don't want to include it. So I use an empty\n
1134
02:02:41,520 --> 02:02:47,490
the statement y is less than negative three\n
1135
02:02:47,489 --> 02:02:54,529
And again, I don't include it, but I do include\n
1136
02:02:54,529 --> 02:02:59,380
values for which both of these statements\n
1137
02:02:59,380 --> 02:03:05,900
both colored blue and colored red. And so\n
1138
02:03:05,899 --> 02:03:10,819
draw it above so you can see it easily. So\n
1139
02:03:10,819 --> 02:03:15,869
three fourths and lower, not including negative\n
1140
02:03:15,869 --> 02:03:21,380
the number line that have both red and blue\n
1141
02:03:21,380 --> 02:03:26,920
answer will be soft bracket negative infinity\n
1142
02:03:26,920 --> 02:03:34,720
As my final example, I have an inequality\n
1143
02:03:34,720 --> 02:03:40,380
three is less than or equal to 6x minus two\n
1144
02:03:40,380 --> 02:03:47,750
a compound inequality with two parts, negative\n
1145
02:03:47,750 --> 02:03:54,289
And at the same time 6x minus two is less\n
1146
02:03:54,289 --> 02:04:00,460
before. But instead, it's a little more efficient\n
1147
02:04:00,460 --> 02:04:08,010
same thing to all three sides. So as a first\n
1148
02:04:08,010 --> 02:04:14,400
gives me negative one is less than or equal\n
1149
02:04:14,399 --> 02:04:20,149
divide all three sides by six to isolate the\n
1150
02:04:20,149 --> 02:04:24,189
than or equal to x is less than two.
1151
02:04:24,189 --> 02:04:29,129
If we solved it, instead, in two pieces above,\n
1152
02:04:29,130 --> 02:04:35,150
get negative one six is less than or equal\n
1153
02:04:35,149 --> 02:04:40,299
than two on this piece. And because of the\n
1154
02:04:40,300 --> 02:04:45,000
negative one, six is less than or equal to\n
1155
02:04:45,000 --> 02:04:50,050
Either way we do it. Let's see if what it\n
1156
02:04:50,050 --> 02:04:58,829
line, we're looking for things that are between\n
1157
02:04:59,970 --> 02:05:05,470
But not including the to interval notation,\n
1158
02:05:08,510 --> 02:05:13,600
In this video, we solve linear inequalities,\n
1159
02:05:13,600 --> 02:05:18,670
by the conjunctions and, and or
1160
02:05:18,670 --> 02:05:22,789
remember when we're working with and we're\n
1161
02:05:24,899 --> 02:05:30,769
That is, we're looking for the overlap on\nthe number line.
1162
02:05:30,770 --> 02:05:35,510
In this case, the points on the number line\n
1163
02:05:36,649 --> 02:05:43,089
When we're working with oral statements, we're\n
1164
02:05:43,090 --> 02:05:46,380
other statement is true or both
1165
02:05:49,010 --> 02:05:54,380
this corresponds to points that are colored\n
1166
02:05:54,380 --> 02:05:57,949
that will actually correspond to the entire\nnumber line.
1167
02:05:57,949 --> 02:06:04,630
In this video, we'll solve inequalities involving\n
1168
02:06:04,630 --> 02:06:09,340
involving rational expressions like this one.
1169
02:06:09,340 --> 02:06:14,500
Let's start with a simple example. Maybe a\n
1170
02:06:14,500 --> 02:06:19,550
inequality, x squared is less than four, you\n
1171
02:06:19,550 --> 02:06:26,220
of both sides and get something like x is\n
1172
02:06:27,789 --> 02:06:35,850
To see why it's not correct, consider the\n
1173
02:06:35,850 --> 02:06:43,500
Negative 10 satisfies the inequality, x is\n
1174
02:06:43,500 --> 02:06:51,130
two. But it doesn't satisfy the inequality\n
1175
02:06:51,130 --> 02:06:58,420
10 squared is 100, which is not less than\n
1176
02:06:58,420 --> 02:07:04,230
same. And it doesn't work to solve a quadratic\n
1177
02:07:04,229 --> 02:07:09,349
both sides, you might be thinking part of\n
1178
02:07:09,350 --> 02:07:15,420
the negative two option, right? If we had\n
1179
02:07:15,420 --> 02:07:21,430
x equals two would just be one option, x equals\n
1180
02:07:21,430 --> 02:07:27,869
somehow, our solution to this inequality should\n
1181
02:07:27,869 --> 02:07:33,960
to solve an inequality involving x squares\n
1182
02:07:33,960 --> 02:07:40,399
equation first. But before we even do that,\n
1183
02:07:40,399 --> 02:07:44,449
so that my inequality has zero on the other\nside.
1184
02:07:44,449 --> 02:07:49,569
So for our equation, I'll subtract four from\n
1185
02:07:50,590 --> 02:07:58,039
Now, I'm going to actually solve the associated\n
1186
02:07:58,039 --> 02:08:06,420
zero, I can do this by factoring to x minus\n
1187
02:08:06,420 --> 02:08:12,579
I'll set my factors equal to zero, and I get\n
1188
02:08:12,579 --> 02:08:17,970
Now, I'm going to plot the solutions to my\n
1189
02:08:17,970 --> 02:08:23,240
negative two and two, those are the places\n
1190
02:08:26,170 --> 02:08:30,550
Since I want to find where x squared minus\n
1191
02:08:30,550 --> 02:08:36,029
this expression x squared minus four is positive\n
1192
02:08:36,029 --> 02:08:43,489
to plug in test values. So first, a plug in\n
1193
02:08:43,489 --> 02:08:47,369
something less than negative to say x equals\nnegative three.
1194
02:08:47,369 --> 02:08:53,720
If I plug in negative three into x squared\n
1195
02:08:53,720 --> 02:09:02,090
four, which is nine minus four, which is five,\n
1196
02:09:02,090 --> 02:09:08,029
the expression x squared minus four is positive.\n
1197
02:09:08,029 --> 02:09:12,420
the number line, my expression is going to\n
1198
02:09:12,420 --> 02:09:17,529
to negative without going through a place\n
1199
02:09:17,529 --> 02:09:21,859
x squared minus four is positive or negative\n
1200
02:09:21,859 --> 02:09:25,109
number line by plugging in test value similar\nway
1201
02:09:25,109 --> 02:09:31,849
evaluate the plug in between negative two\n
1202
02:09:31,850 --> 02:09:36,700
minus four, that's negative four and negative\n
1203
02:09:36,699 --> 02:09:43,109
minus four is negative on this whole interval.\n
1204
02:09:43,109 --> 02:09:49,349
10, something bigger than two, and I get 10\n
1205
02:09:49,350 --> 02:09:53,890
that I can tell that that's going to be a\n
1206
02:09:53,890 --> 02:09:57,980
Again, since I want x squared minus four to\n
1207
02:09:57,979 --> 02:09:59,719
on this number line where I'm getting
1208
02:09:59,720 --> 02:10:06,230
negatives. So I will share that in on my number\n
1209
02:10:06,229 --> 02:10:10,609
Because the endpoints are where my expression\n
1210
02:10:10,609 --> 02:10:12,599
I want it strictly less than zero
1211
02:10:12,600 --> 02:10:19,900
I can write my answer as an inequality, negative\n
1212
02:10:19,899 --> 02:10:25,059
interval notation as soft bracket negative\ntwo to soft bracket.
1213
02:10:25,060 --> 02:10:32,780
Our next example, we can solve similarly,\n
1214
02:10:32,779 --> 02:10:39,769
so that our inequality is x cubed minus 5x\n
1215
02:10:39,770 --> 02:10:47,420
to zero. Next, we'll solve the associated\n
1216
02:10:47,420 --> 02:10:55,449
down the equation. Now I'll factor out an\n
1217
02:10:55,449 --> 02:11:01,239
solutions to my equation are x equals 0x equals\n
1218
02:11:01,239 --> 02:11:05,489
I'll write the solutions to the equation on\nthe number line.
1219
02:11:05,489 --> 02:11:17,359
So that's negative one, zero, and six. That's\n
1220
02:11:21,050 --> 02:11:28,110
But I want to find where it's greater than\n
1221
02:11:28,109 --> 02:11:35,630
values, I can plug in, for example, x equals\n
1222
02:11:35,630 --> 02:11:40,760
or to this factored version. Since I only\n
1223
02:11:40,760 --> 02:11:45,630
it's sometimes easier to use the factored\n
1224
02:11:48,069 --> 02:11:55,539
But this factor, x minus six is also negative\n
1225
02:11:55,539 --> 02:12:03,269
Finally, x plus one, when I plug in negative\n
1226
02:12:03,270 --> 02:12:08,980
negative. And a negative times a negative\n
1227
02:12:08,979 --> 02:12:16,750
If I plug in something, between negative one\n
1228
02:12:16,750 --> 02:12:21,600
then I'm going to get a negative for this\n
1229
02:12:21,600 --> 02:12:25,020
positive for this third factor.
1230
02:12:25,020 --> 02:12:28,860
Negative times negative times positive gives\nme a positive
1231
02:12:28,859 --> 02:12:34,349
for a test value between zero and six, let's\ntry x equals one.
1232
02:12:34,350 --> 02:12:41,130
Now I'll get a positive for this factor a\n
1233
02:12:43,880 --> 02:12:50,920
positive times a negative times a positive\n
1234
02:12:50,920 --> 02:12:56,649
bigger than six, we could use say x equals\n
1235
02:12:59,220 --> 02:13:03,630
So my product will be positive.
1236
02:13:03,630 --> 02:13:08,060
Since I want values where my expression is\n
1237
02:13:12,970 --> 02:13:16,980
And the places where it's positive.
1238
02:13:16,979 --> 02:13:24,709
So my final answer will be close bracket negative\n
1239
02:13:25,909 --> 02:13:34,489
As our final example, let's consider the rational\n
1240
02:13:34,489 --> 02:13:38,229
by x minus one is less than or equal to zero.
1241
02:13:38,229 --> 02:13:43,449
Although it might be tempting to clear the\n
1242
02:13:43,449 --> 02:13:49,050
one, it's dangerous to do that, because x\n
1243
02:13:49,050 --> 02:13:53,699
it could also be a negative number. And when\n
1244
02:13:53,699 --> 02:13:59,260
you have to reverse the inequality. Although\n
1245
02:13:59,260 --> 02:14:04,869
way, by thinking of cases where x minus one\n
1246
02:14:04,869 --> 02:14:10,369
it's much easier just to solve the same way\n
1247
02:14:10,369 --> 02:14:15,340
so that we move all terms to the left and\n
1248
02:14:15,340 --> 02:14:20,750
true. So the next step would be to solve the\n
1249
02:14:20,750 --> 02:14:29,470
That is x squared plus 6x plus nine over x\n
1250
02:14:29,470 --> 02:14:35,240
That would be where the numerator is 0x squared\n
1251
02:14:35,239 --> 02:14:42,609
x plus three squared is zero, or x equals\n
1252
02:14:42,609 --> 02:14:48,069
have to do for rational expressions. And that's\n
1253
02:14:48,069 --> 02:14:55,559
not exist. That is, let's find where the denominator\n
1254
02:14:55,560 --> 02:14:59,780
I'll put all those numbers on the number line.\n
1255
02:14:59,779 --> 02:15:04,859
rational expression is equal to zero, and\n
1256
02:15:04,859 --> 02:15:13,630
exist, then I can start in with test values.\n
1257
02:15:13,630 --> 02:15:20,859
to work. If I plug those values into this\n
1258
02:15:20,859 --> 02:15:26,109
negative answer and a positive answer. The\n
1259
02:15:26,109 --> 02:15:32,009
number line where my denominators zero is\n
1260
02:15:32,010 --> 02:15:36,630
negative to positive by passing through a\n
1261
02:15:36,630 --> 02:15:41,859
exist, as well as passing by passing flew\n
1262
02:15:43,220 --> 02:15:49,570
Now I'm looking for where my original expression\n
1263
02:15:49,569 --> 02:15:56,109
I want the places on the number line where\n
1264
02:15:59,590 --> 02:16:06,090
So my final answer is x is less than one,\n
1265
02:16:08,550 --> 02:16:14,630
In this video, we solved polynomial and rational\n
1266
02:16:14,630 --> 02:16:19,869
think test values to make a sine chart.
1267
02:16:19,869 --> 02:16:26,109
The distance formula can be used to find the\n
1268
02:16:27,109 --> 02:16:34,039
if this first point has coordinates, x one,\n
1269
02:16:34,040 --> 02:16:42,010
x two, y two, then the distance between them\n
1270
02:16:42,010 --> 02:16:49,200
x two minus x one squared plus y two minus\ny one squared.
1271
02:16:49,200 --> 02:16:55,739
This formula actually comes from the Pythagorean\n
1272
02:16:55,738 --> 02:17:01,399
these two points as two of its vertices
1273
02:17:01,399 --> 02:17:09,148
then the length of this side is the difference\n
1274
02:17:09,148 --> 02:17:14,629
length of this vertical side is the difference\n
1275
02:17:14,629 --> 02:17:20,170
Now that Pythagoras theorem says that for\n
1276
02:17:20,170 --> 02:17:27,019
B, and hypotony is labeled C, we have that\n
1277
02:17:27,019 --> 02:17:31,728
Well, if we apply that to this triangle here
1278
02:17:31,728 --> 02:17:38,010
this hypotony News is the distance between\n
1279
02:17:38,010 --> 02:17:45,818
So but Tyrion theorem says, the square of\n
1280
02:17:45,818 --> 02:17:55,149
squared, plus the square of this side length,\n
1281
02:17:55,149 --> 02:18:01,658
equal the square of the hypothesis, that is\nd squared.
1282
02:18:01,658 --> 02:18:07,148
taking the square root of both sides of this\n
1283
02:18:07,148 --> 02:18:13,769
minus x one squared plus y two minus y one\n
1284
02:18:13,769 --> 02:18:18,439
we don't have to worry about using plus or\n
1285
02:18:18,439 --> 02:18:21,260
distance is always positive.
1286
02:18:21,260 --> 02:18:26,639
So we've derived our distance formula. Now\n
1287
02:18:26,638 --> 02:18:34,589
Let's find the distance between the two points,\n
1288
02:18:39,079 --> 02:18:48,359
this notation just means that P is the point\n
1289
02:18:48,359 --> 02:18:52,699
is the point with coordinates for two.
1290
02:18:52,699 --> 02:18:54,590
We have the distance formula
1291
02:18:54,590 --> 02:19:02,299
let's think of P being the point with coordinates\n
1292
02:19:04,279 --> 02:19:10,179
But as we'll see, it really doesn't matter\n
1293
02:19:10,179 --> 02:19:23,609
we get d is the square root of n x two minus\n
1294
02:19:26,769 --> 02:19:37,349
y two minus y one, so that's two minus five\nsquared.
1295
02:19:37,349 --> 02:19:41,710
Working out the arithmetic a little bit for\n
1296
02:19:41,709 --> 02:19:49,139
five squared, plus negative three squared.\n
1297
02:19:49,139 --> 02:19:55,659
are the square root of 34. Let's see what\n
1298
02:19:55,659 --> 02:20:00,200
point, x two y two instead, and a second.
1299
02:20:00,200 --> 02:20:06,390
Why'd x one y one, then we would have gotten\n
1300
02:20:12,180 --> 02:20:22,130
and added the difference of y's squared. So\n
1301
02:20:22,129 --> 02:20:27,139
That gives us the square root of negative\n
1302
02:20:31,180 --> 02:20:36,771
In this video, we use the distance formula\n
1303
02:20:36,771 --> 02:20:40,579
writing down the distance formula, sometimes\n
1304
02:20:40,578 --> 02:20:45,488
a minus and whether these are pluses or minuses.\n
1305
02:20:45,488 --> 02:20:51,629
formula comes from the Pythagorean Theorem.\n
1306
02:20:51,629 --> 02:20:56,239
And then a minus in here, because we're finding\n
1307
02:20:56,239 --> 02:20:59,421
find the lengths of the sides.
1308
02:20:59,421 --> 02:21:05,699
The midpoint formula helps us find the coordinates\n
1309
02:21:05,699 --> 02:21:09,819
as we know the coordinates of the endpoints.
1310
02:21:09,818 --> 02:21:15,328
So let's call the coordinates of this endpoint\n
1311
02:21:15,328 --> 02:21:17,600
other endpoint x two, y two
1312
02:21:17,600 --> 02:21:25,750
the x coordinate of the midpoint is going\n
1313
02:21:25,750 --> 02:21:31,068
of the endpoints. To get a number halfway\n
1314
02:21:34,340 --> 02:21:42,238
Similarly, the y coordinate of this midpoint\n
1315
02:21:42,238 --> 02:21:47,430
the y coordinates of the endpoints. So the\n
1316
02:21:47,430 --> 02:21:53,050
the average of those y coordinates.
1317
02:21:53,049 --> 02:22:00,938
So we see that the coordinates of the midpoint\n
1318
02:22:04,059 --> 02:22:07,028
Let's use this midpoint formula in an example
1319
02:22:07,029 --> 02:22:17,069
we want to find the midpoint of the segment\n
1320
02:22:24,090 --> 02:22:29,510
me draw the line segment between them. So\n
1321
02:22:29,510 --> 02:22:36,248
around here. But to find its exact coordinates,\n
1322
02:22:36,248 --> 02:22:44,209
and y one plus y two over two, where this\n
1323
02:22:44,209 --> 02:22:49,788
and the second point has coordinates x two\n
1324
02:22:49,789 --> 02:22:54,600
you decide is x one y one, which one is x\n
1325
02:22:54,600 --> 02:22:56,460
the same answer for the midpoint.
1326
02:22:56,459 --> 02:23:06,349
So let's see, I take my average of my x coordinates.\n
1327
02:23:06,350 --> 02:23:15,369
and the average of my Y coordinates, so that's\n
1328
02:23:15,369 --> 02:23:19,069
seven halves, as the coordinates of my midpoint.
1329
02:23:19,068 --> 02:23:25,449
In this video, we use the midpoint formula\n
1330
02:23:25,450 --> 02:23:30,351
by taking the average of the x coordinates\n
1331
02:23:30,351 --> 02:23:36,149
video is about graphs and equations of circles.\n
1332
02:23:36,148 --> 02:23:42,439
circle of radius five, centered at the point\nthree, two
1333
02:23:42,439 --> 02:23:47,510
will look something like this.
1334
02:23:47,510 --> 02:23:53,420
For any point, x, y on the circle, we know\nthat
1335
02:23:53,420 --> 02:24:00,930
the distance of that point xy from the center\n
1336
02:24:00,930 --> 02:24:07,398
distance formula, that distance of five is\n
1337
02:24:07,398 --> 02:24:11,498
of the x coordinates. That's x minus three
1338
02:24:11,498 --> 02:24:20,199
squared plus the difference of the y coordinates,\n
1339
02:24:20,199 --> 02:24:31,239
And if I square both sides of that equation,\n
1340
02:24:31,239 --> 02:24:40,238
In other words, 25 is equal to x minus three\n
1341
02:24:40,238 --> 02:24:43,478
square root and the squared undo each other.
1342
02:24:43,478 --> 02:24:49,519
A lot of times people will write the X minus\n
1343
02:24:49,520 --> 02:24:53,899
the left side of the equation and the 25 on\n
1344
02:24:53,898 --> 02:24:58,099
standard form for the equation of the circle.
1345
02:24:58,100 --> 02:24:59,998
The same reasoning can be generalized
1346
02:24:59,998 --> 02:25:07,119
Find the general equation for a circle with\n
1347
02:25:07,120 --> 02:25:14,609
For any point with coordinates, x, y on the\n
1348
02:25:14,609 --> 02:25:22,130
coordinates x, y and the point with coordinates\n
1349
02:25:22,129 --> 02:25:28,709
the distance is equal to r, but by the distance\n
1350
02:25:28,709 --> 02:25:36,648
between the x coordinates x minus h squared\n
1351
02:25:36,648 --> 02:25:48,559
minus k squared. squaring both sides as before,\n
1352
02:25:48,559 --> 02:25:53,078
canceling the square root and the squared,\n
1353
02:25:53,078 --> 02:26:01,090
x minus h squared plus y minus k squared equals\n
1354
02:26:01,090 --> 02:26:06,969
a circle with radius r, and center HK.
1355
02:26:06,969 --> 02:26:13,359
Notice that the coordinates h and k are subtracted\n
1356
02:26:13,360 --> 02:26:18,760
are in the distance formula, and the radius\n
1357
02:26:18,760 --> 02:26:23,960
this general formula, that makes it easy to\n
1358
02:26:23,959 --> 02:26:29,238
example, if we want the equation for a circle
1359
02:26:29,238 --> 02:26:35,618
of radius six, and center at zero, negative\nthree
1360
02:26:35,619 --> 02:26:45,670
then we have our equal six, and h k is our\n
1361
02:26:47,040 --> 02:26:57,130
we get x minus zero squared, plus y minus\n
1362
02:26:57,129 --> 02:27:04,299
or simplified this is x squared plus y plus\n
1363
02:27:04,299 --> 02:27:09,748
Suppose we're given an equation like this\n
1364
02:27:09,748 --> 02:27:13,760
of a circle, and if so what's the center was\nthe radius.
1365
02:27:13,760 --> 02:27:21,728
Well, this equation matches the form for a\n
1366
02:27:21,728 --> 02:27:30,129
equals r squared. If we let h, b five, Why\n
1367
02:27:30,129 --> 02:27:34,349
a negative six is like adding a six
1368
02:27:34,350 --> 02:27:41,140
five must be our r squared. So that means\n
1369
02:27:41,139 --> 02:27:49,500
is our radius. And our center is the point\n
1370
02:27:52,109 --> 02:27:57,930
which we could then graph by putting down\n
1371
02:27:57,930 --> 02:28:03,000
is a little bit more than two.
1372
02:28:03,000 --> 02:28:08,859
This equation might not look like the equation\n
1373
02:28:08,859 --> 02:28:13,498
To look like the equation of a circle, we're\n
1374
02:28:13,498 --> 02:28:19,270
form x minus h squared plus y minus k squared\nequals r squared.
1375
02:28:19,270 --> 02:28:24,020
First, I'd like to get rid of the coefficients\n
1376
02:28:24,020 --> 02:28:30,488
so I'm going to divide everything by nine.\n
1377
02:28:30,488 --> 02:28:38,609
8x minus two y plus four equals zero. Next,\n
1378
02:28:38,609 --> 02:28:45,238
x squared and the 8x. So write x squared plus\n
1379
02:28:49,120 --> 02:28:54,750
And I'll subtract the four over to the other\nside.
1380
02:28:54,750 --> 02:28:59,068
This still doesn't look much like the equation\n
1381
02:28:59,068 --> 02:29:05,288
to do something called completing the square,\n
1382
02:29:05,289 --> 02:29:12,210
divided by two and then square it. In other\n
1383
02:29:12,209 --> 02:29:18,608
squared, which is 16. I'm going to add 16\n
1384
02:29:24,439 --> 02:29:31,488
now I'm going to do the same thing to the\n
1385
02:29:31,488 --> 02:29:38,260
two is negative one, square that and I get\n
1386
02:29:38,260 --> 02:29:44,068
sides, but I'll put it near the y terms. So\n
1387
02:29:44,068 --> 02:29:50,639
I have to add it to the other side. On the\n
1388
02:29:50,639 --> 02:29:59,439
side, I can wrap up this expression x squared\n
1389
02:29:59,439 --> 02:30:07,460
To convince you, that's correct. If we multiply\n
1390
02:30:07,459 --> 02:30:16,919
x squared plus 4x plus 4x plus 16, or x squared\n
1391
02:30:16,920 --> 02:30:25,260
here. Similarly, we can wrap up y squared\n
1392
02:30:25,260 --> 02:30:32,340
y minus one squared. Again, I'll work out\n
1393
02:30:32,340 --> 02:30:40,030
minus y minus y plus one, or y squared minus\n
1394
02:30:40,030 --> 02:30:46,989
If you're wondering how I knew to use four\n
1395
02:30:46,988 --> 02:30:53,988
of eight, and the minus one came from taking\n
1396
02:30:53,988 --> 02:31:01,629
for a circle and standard form. And we can\n
1397
02:31:01,629 --> 02:31:10,429
negative one, and the radius, which is the\nsquare root of 13.
1398
02:31:10,430 --> 02:31:14,979
It might seem like magic that this trick of\n
1399
02:31:14,978 --> 02:31:20,568
it and adding it to both sides lets us wrap\n
1400
02:31:20,568 --> 02:31:26,988
perfect square. But to see why that works,\n
1401
02:31:26,988 --> 02:31:33,760
out x minus h squared, the thing that we're\n
1402
02:31:33,760 --> 02:31:43,609
x minus h squared, which is x minus h times\n
1403
02:31:43,609 --> 02:31:53,430
minus HX minus h x plus h squared, or x squared\n
1404
02:31:53,430 --> 02:32:01,090
out with this part, with some x squared term\n
1405
02:32:01,090 --> 02:32:06,409
decide what to add on in order to wrap it\n
1406
02:32:06,409 --> 02:32:13,430
on is h squared here, which comes from half\n
1407
02:32:13,430 --> 02:32:18,689
half of negative two H is a negative H, square\n
1408
02:32:18,689 --> 02:32:26,738
we do wrap it up, it's half of this coefficient\n
1409
02:32:26,738 --> 02:32:30,988
This trick of completing the square is really\n
1410
02:32:30,988 --> 02:32:37,260
in disguise, into the standard equation for\n
1411
02:32:37,260 --> 02:32:44,728
equation for a circle x minus h squared plus\n
1412
02:32:44,728 --> 02:32:52,159
r is the radius, and h k is the center.
1413
02:32:52,159 --> 02:32:56,648
We also showed a method of completing the\nsquare.
1414
02:32:56,648 --> 02:33:01,840
When you have an equation for a circle in\n
1415
02:33:01,840 --> 02:33:06,279
you rewrite it into the standard form.
1416
02:33:06,279 --> 02:33:10,979
This video is about graphs and equations of\nlines.
1417
02:33:10,978 --> 02:33:16,799
Here we're given the graph of a line, we want\n
1418
02:33:16,799 --> 02:33:25,059
for the equation of a line is y equals mx\n
1419
02:33:25,059 --> 02:33:34,549
B represents the y intercept, the y value\n
1420
02:33:34,549 --> 02:33:41,228
is equal to the rise over the run. Or sometimes\n
1421
02:33:41,228 --> 02:33:49,549
over the change in x values. Or in other words,\n
1422
02:33:49,549 --> 02:33:56,509
where x one y one and x two y two are points\non the line.
1423
02:33:56,510 --> 02:34:01,600
While we could use any two points on the line,\n
1424
02:34:01,600 --> 02:34:08,590
points where the x and y coordinates are integers,\n
1425
02:34:08,590 --> 02:34:13,630
grid points. So here would be one convenient\n
1426
02:34:15,639 --> 02:34:22,068
The coordinates of the first point are one,\n
1427
02:34:22,068 --> 02:34:29,319
five, negative one. Now I can find the slope\n
1428
02:34:29,319 --> 02:34:35,629
I go through a run of this distance, I go\n
1429
02:34:35,629 --> 02:34:41,078
gonna be a negative rise or a fall because\n
1430
02:34:41,078 --> 02:34:48,219
off squares. This is a run of 1234 squares\n
1431
02:34:56,270 --> 02:35:00,510
I got that answer by counting squares, but\n
1432
02:35:00,510 --> 02:35:05,630
Looking at the difference in my y values over\n
1433
02:35:08,180 --> 02:35:15,898
negative one minus two, that's from my difference\n
1434
02:35:18,459 --> 02:35:26,188
which is five minus one, that gives me negative\n
1435
02:35:26,189 --> 02:35:30,750
So my M is negative three fourths.
1436
02:35:30,750 --> 02:35:36,219
Now I need to figure out the value of b, my\n
1437
02:35:36,219 --> 02:35:41,868
the graph, it looks like approximately 2.75.\n
1438
02:35:41,869 --> 02:35:47,539
use a point that has integer coordinates that\n
1439
02:35:47,539 --> 02:35:53,689
point or that point, let's try this point.\n
1440
02:35:53,689 --> 02:36:02,389
mx plus b, that is y equals negative three\n
1441
02:36:02,389 --> 02:36:12,118
one, two, for my x and y. So that gives me\n
1442
02:36:12,119 --> 02:36:20,078
plus b, solving for B. Let's see that's two\n
1443
02:36:20,078 --> 02:36:28,090
three fourths to both sides, that's two plus\n
1444
02:36:28,090 --> 02:36:33,500
plus three fourths, which is 11. Four switches\n
1445
02:36:33,500 --> 02:36:40,559
So now I can write out my final equation for\n
1446
02:36:40,559 --> 02:36:48,289
plus 11 fourths by plugging in for m and b.\n
1447
02:36:49,420 --> 02:36:57,158
a horizontal line has slope zero. So if we\n
1448
02:36:57,158 --> 02:37:03,898
to be zero. In other words, it's just y equals\n
1449
02:37:03,898 --> 02:37:08,139
out what that that constant y value is, it\nlooks like it's
1450
02:37:08,139 --> 02:37:14,818
two, let's see this three, three and a half,\n
1451
02:37:18,998 --> 02:37:24,898
For a vertical line, like this one, it doesn't\n
1452
02:37:24,898 --> 02:37:26,449
to do the rise over the run
1453
02:37:26,450 --> 02:37:32,340
there's no run. So you'd I guess you'd be\n
1454
02:37:32,340 --> 02:37:39,859
But But instead, we just think of it as an\n
1455
02:37:39,859 --> 02:37:45,199
in this case, x equals negative two, notice\n
1456
02:37:45,199 --> 02:37:50,380
same x coordinate of negative two and the\n
1457
02:37:50,379 --> 02:37:54,879
So this is how we write the equation for a\nvertical line.
1458
02:37:54,879 --> 02:37:59,299
In this example, we're not shown a graph of\n
1459
02:37:59,299 --> 02:38:04,629
through two points. But knowing that I go\n
1460
02:38:04,629 --> 02:38:12,608
for the line. First, we can find the slope\n
1461
02:38:12,609 --> 02:38:20,090
the difference in x values. So that's negative\n
1462
02:38:26,329 --> 02:38:34,648
standard equation for the line, this is called\n
1463
02:38:34,648 --> 02:38:42,719
And we can plug in negative five thirds. And\n
1464
02:38:42,719 --> 02:38:48,488
still get the same final answer. So let's\n
1465
02:38:48,488 --> 02:38:57,779
negative five thirds times one plus b. And\n
1466
02:38:57,779 --> 02:39:03,550
thirds plus five thirds, which is 11 thirds.\n
1467
02:39:11,728 --> 02:39:17,119
method two uses a slightly different form\n
1468
02:39:17,120 --> 02:39:25,680
form and it goes y minus y naught is equal\n
1469
02:39:25,680 --> 02:39:33,380
y naught is a point on the line and again\n
1470
02:39:33,379 --> 02:39:40,309
same way by taking a difference in Y values\n
1471
02:39:40,309 --> 02:39:43,698
can simply plug in any point.
1472
02:39:43,699 --> 02:39:53,039
For example, the point one two will work we\n
1473
02:39:53,039 --> 02:39:59,420
Y not in this point slope form. That gives\nus y
1474
02:40:00,420 --> 02:40:06,719
Two is equal to minus five thirds x minus\none.
1475
02:40:06,719 --> 02:40:11,090
Notice that these two equations, while they\n
1476
02:40:11,090 --> 02:40:25,380
because if I distribute the negative five\n
1477
02:40:25,379 --> 02:40:28,529
I get the same equation as above.
1478
02:40:28,530 --> 02:40:37,100
So we've seen two ways of finding the equation\n
1479
02:40:37,100 --> 02:40:40,199
And using the point slope form.
1480
02:40:40,199 --> 02:40:48,920
In this video, we saw that you can find the\n
1481
02:40:52,158 --> 02:40:57,760
you can also find the equation for the line\n
1482
02:40:57,760 --> 02:41:03,529
the two points to get the slope and then plug\n
1483
02:41:06,430 --> 02:41:08,898
We saw two standard forms for the equation\nof a line
1484
02:41:08,898 --> 02:41:19,738
the slope intercept form y equals mx plus\n
1485
02:41:19,738 --> 02:41:27,398
And the point slope form y minus y naught\n
1486
02:41:27,398 --> 02:41:35,328
is the slope and x naught y naught is a point\non the line.
1487
02:41:35,328 --> 02:41:43,219
This video is about finding parallel and perpendicular\n
1488
02:41:43,219 --> 02:41:47,469
fourths, in other words, the rise
1489
02:41:52,529 --> 02:42:01,119
than any other line that's parallel to this\n
1490
02:42:03,939 --> 02:42:09,909
So that's our first fact to keep in mind,\n
1491
02:42:09,909 --> 02:42:15,000
on the other hand, we want to find a line\n
1492
02:42:15,000 --> 02:42:18,129
original slope of three fourths.
1493
02:42:18,129 --> 02:42:25,358
A perpendicular line, in other words, align\n
1494
02:42:25,359 --> 02:42:32,039
will have a slope, that's the negative reciprocal\n
1495
02:42:32,039 --> 02:42:37,579
slope. So we take the reciprocal of three\n
1496
02:42:37,578 --> 02:42:41,359
its opposite by changing it from positive\nto negative.
1497
02:42:41,359 --> 02:42:49,738
So I'll write this as a principle that perpendicular\n
1498
02:42:49,738 --> 02:42:54,719
get the hang of what it means to be an opposite\n
1499
02:42:54,719 --> 02:43:01,288
So here's the original slope, and this will\n
1500
02:43:01,289 --> 02:43:07,610
the slope of our perpendicular line. So for\n
1501
02:43:07,610 --> 02:43:14,319
reciprocal of two is one half opposite means\n
1502
02:43:14,318 --> 02:43:23,260
out to be, say, negative 1/3, the reciprocal\n
1503
02:43:23,260 --> 02:43:29,359
the opposite means I change it from a negative\n
1504
02:43:32,189 --> 02:43:38,609
One more example, if I started off with a\n
1505
02:43:38,609 --> 02:43:43,890
of that would be two sevenths. And I change\n
1506
02:43:47,590 --> 02:43:52,469
Let's use these two principles in some examples.
1507
02:43:52,469 --> 02:43:56,488
In our first example, we need to find the\n
1508
02:43:56,488 --> 02:44:02,908
slump, and go through the point negative three\n
1509
02:44:02,908 --> 02:44:07,538
to figure out the slope of this line. So let\n
1510
02:44:07,539 --> 02:44:14,459
the the slope intercept form. So I'll start\n
1511
02:44:14,459 --> 02:44:19,829
I'm going to solve for y to put it in this\n
1512
02:44:19,829 --> 02:44:29,439
y equals 4x minus six and then divide by three\n
1513
02:44:29,439 --> 02:44:34,850
is divided all my turns by three, I can simplify\n
1514
02:44:34,850 --> 02:44:43,439
two. Now I can read off the slope of my original\n
1515
02:44:43,439 --> 02:44:51,279
slope of my new line, my parallel line will\n
1516
02:44:51,279 --> 02:44:58,060
the equation y equals four thirds ax plus\n
1517
02:44:58,059 --> 02:45:00,029
be negative two like it was for the
1518
02:45:00,030 --> 02:45:04,189
first line, it'll be something else determined\n
1519
02:45:04,189 --> 02:45:10,359
negative three to, to figure out what b is,\n
1520
02:45:10,359 --> 02:45:16,408
negative three for x, and two for y. So that\n
1521
02:45:16,408 --> 02:45:25,449
three plus b, and I'll solve for b. So let's\n
1522
02:45:25,449 --> 02:45:32,440
this is negative 12 thirds plus b, or in other\n
1523
02:45:32,440 --> 02:45:38,479
means that B is going to be six, and by my\n
1524
02:45:42,340 --> 02:45:49,398
Next, let's find the equation of a line that's\n
1525
02:45:49,398 --> 02:45:54,019
go through a given point. Again, in order\n
1526
02:45:54,020 --> 02:45:59,869
of this given line is. So I'll rewrite it,\n
1527
02:45:59,869 --> 02:46:06,670
three, y equals four. And then I can put it\n
1528
02:46:06,670 --> 02:46:14,879
for y. So three, y is negative 6x plus four,\n
1529
02:46:14,879 --> 02:46:22,248
negative six thirds x plus four thirds. So\n
1530
02:46:22,248 --> 02:46:28,068
original slope of my original line is negative\n
1531
02:46:28,068 --> 02:46:34,028
the opposite reciprocal, so I take the reciprocal\n
1532
02:46:34,029 --> 02:46:39,380
and I change the sign so that gives me one\n
1533
02:46:39,379 --> 02:46:47,209
Now, my new line, I know is going to be y\n
1534
02:46:47,209 --> 02:46:57,148
can plug in my point on my new line, so one\n
1535
02:46:57,148 --> 02:47:04,778
I get one equals two plus b, So b is negative\n
1536
02:47:07,520 --> 02:47:14,439
These next two examples are a little bit different,\n
1537
02:47:14,439 --> 02:47:23,359
to a completely horizontal line, let me draw\n
1538
02:47:23,359 --> 02:47:29,270
is always three, which means that my line\n
1539
02:47:29,270 --> 02:47:36,329
at height y equals three, if I want something\n
1540
02:47:36,329 --> 02:47:42,799
line. Since it goes through the point negative\n
1541
02:47:45,590 --> 02:47:51,309
it's gonna have always have a y coordinate\n
1542
02:47:51,309 --> 02:47:55,510
point it goes through, so my answer will just\nbe y equals one.
1543
02:47:55,510 --> 02:48:06,260
In the next example, we want a line that's\n
1544
02:48:06,260 --> 02:48:12,328
horizontal line y equals four, but perpendicular\n
1545
02:48:12,328 --> 02:48:16,818
I need a vertical line that goes through the\npoint three, four.
1546
02:48:16,818 --> 02:48:27,748
Okay, and so I'm going to draw a vertical\n
1547
02:48:27,748 --> 02:48:32,949
x equals something for the equation. And to\n
1548
02:48:32,950 --> 02:48:38,359
the x coordinate of the point I'm going through\n
1549
02:48:38,359 --> 02:48:45,059
is three and all the points on this, this\n
1550
02:48:45,059 --> 02:48:53,748
three so my answer is x equals three.
1551
02:48:53,748 --> 02:48:58,148
In this video will use the fact that parallel\n
1552
02:48:58,148 --> 02:49:03,250
lines have opposite reciprocal slopes, to\n
1553
02:49:05,510 --> 02:49:09,988
This video introduces functions and their\ndomains and ranges.
1554
02:49:09,988 --> 02:49:17,930
A function is a correspondence between input\n
1555
02:49:17,930 --> 02:49:25,079
usually the y values, that sends each input\n
1556
02:49:25,079 --> 02:49:28,260
a function is thought of as a rule or machine
1557
02:49:28,260 --> 02:49:38,880
in which you can feed in x values as input\n
1558
02:49:38,879 --> 02:49:44,658
example of a function might be the biological\n
1559
02:49:44,658 --> 02:49:49,959
person and it gives us output their biological\nmother.
1560
02:49:49,959 --> 02:49:57,349
This function satisfies the condition that\n
1561
02:49:57,350 --> 02:50:01,068
get sent to exactly one output per
1562
02:50:01,068 --> 02:50:07,059
Because if you take any person, they just\n
1563
02:50:07,059 --> 02:50:14,049
give you a function. But if I change things\n
1564
02:50:14,049 --> 02:50:19,139
which sends to each person, their mother,\n
1565
02:50:19,139 --> 02:50:23,250
there are some people who have more than one\n
1566
02:50:23,250 --> 02:50:28,318
mother and adopted mother, or a mother and\n
1567
02:50:28,318 --> 02:50:33,590
So since there's, there's at least some people\n
1568
02:50:33,590 --> 02:50:39,029
get like more than one possible output that\n
1569
02:50:39,029 --> 02:50:43,810
would not be a function. Now, most of the\n
1570
02:50:43,809 --> 02:50:48,948
with equations, not in terms of mothers. So\n
1571
02:50:51,329 --> 02:50:57,409
This can also be written as f of x equals\n
1572
02:50:57,409 --> 02:51:02,510
notation that stands for the output value\nof y.
1573
02:51:02,510 --> 02:51:07,760
Notice that this notation is not representing\n
1574
02:51:07,760 --> 02:51:14,309
x, instead, we're going to be putting in a\n
1575
02:51:14,309 --> 02:51:21,670
of f of x or y. For example, if we want to\n
1576
02:51:21,670 --> 02:51:30,248
input for x either in this equation, or in\n
1577
02:51:30,248 --> 02:51:38,629
plus one, f of two is going to equal five.\n
1578
02:51:38,629 --> 02:51:46,719
for x, so that's gonna be five squared plus\n
1579
02:51:46,719 --> 02:51:52,420
a function on a more complicated expression\n
1580
02:51:52,420 --> 02:51:57,748
the functions value on any expression, it's\n
1581
02:51:57,748 --> 02:52:00,680
that whole expression for x.
1582
02:52:00,680 --> 02:52:09,389
So f of a plus three is going to be the quantity\n
1583
02:52:09,389 --> 02:52:17,029
we could rewrite that as a squared plus six\n
1584
02:52:19,648 --> 02:52:25,090
When evaluating a function on a complex expression,\n
1585
02:52:25,090 --> 02:52:32,119
you plug in for x. That way, you evaluate\n
1586
02:52:35,209 --> 02:52:42,590
to write f of a plus three equals a plus three\n
1587
02:52:42,590 --> 02:52:47,978
because that would imply that we were just\n
1588
02:52:52,379 --> 02:52:58,009
Sometimes a function is described with a graph\n
1589
02:52:58,010 --> 02:53:04,939
graph is supposed to represent the function\n
1590
02:53:04,939 --> 02:53:11,050
functions. For example, the graph of a circle\n
1591
02:53:11,049 --> 02:53:15,948
the graph of a circle violates the vertical\n
1592
02:53:15,949 --> 02:53:18,289
intersect the graph in more than one point.
1593
02:53:18,289 --> 02:53:24,130
But our graph, it left satisfies the vertical\n
1594
02:53:24,129 --> 02:53:29,028
graph, and at most one point, that means is\n
1595
02:53:29,029 --> 02:53:37,439
at most one y value that corresponds to it.\n
1596
02:53:37,439 --> 02:53:44,380
an x value. And we'll use the graph to find\n
1597
02:53:44,379 --> 02:53:51,278
on the x axis, and find the point on the graph\n
1598
02:53:51,279 --> 02:53:58,630
I can look at the y value of that point looks\n
1599
02:53:58,629 --> 02:54:05,559
to three. If I try to do the same thing to\n
1600
02:54:05,559 --> 02:54:11,648
is an x value, I look for it on the x axis,\n
1601
02:54:13,779 --> 02:54:21,050
Therefore, g of five is undefined, or we can\n
1602
02:54:21,049 --> 02:54:27,670
x values and y values make sense for a function\n
1603
02:54:27,670 --> 02:54:32,939
The domain of a function is all possible x\n
1604
02:54:32,939 --> 02:54:36,898
The range is the y values that make sense\nfor the function.
1605
02:54:36,898 --> 02:54:42,180
In this example, we saw that the x value of\n
1606
02:54:42,180 --> 02:54:49,090
this function. So the x value of five is not\n
1607
02:54:49,090 --> 02:54:54,389
x values in the domain, we have to look at\n
1608
02:54:54,389 --> 02:55:00,228
the graph. One way to do that is to take the\n
1609
02:55:00,228 --> 02:55:07,969
onto the x axis and see what x values are\n
1610
02:55:07,969 --> 02:55:15,538
starting at negative eight, and continuing\n
1611
02:55:15,539 --> 02:55:21,289
is the x's between negative eight, and four,\n
1612
02:55:21,289 --> 02:55:27,248
this in interval notation as negative eight\n
1613
02:55:27,248 --> 02:55:31,559
To find the range of the function, we look\n
1614
02:55:35,279 --> 02:55:42,979
By taking the shadow or projection of the\ngraph onto the y axis
1615
02:55:42,978 --> 02:55:51,398
we seem to be hitting our Y values from negative\n
1616
02:55:51,398 --> 02:55:59,340
So our range is wise between negative five\n
1617
02:55:59,340 --> 02:56:05,369
five, three with square brackets. If we meet\n
1618
02:56:05,369 --> 02:56:11,498
instead of a graph, one way to find the domain\n
1619
02:56:11,498 --> 02:56:17,478
often possible to find the domain at least\n
1620
02:56:17,478 --> 02:56:22,328
We think about what x values that makes sense\n
1621
02:56:22,328 --> 02:56:27,049
need to be excluded, because they make the\n
1622
02:56:27,049 --> 02:56:35,629
Specifically, to find the domain of a function,\n
1623
02:56:35,629 --> 02:56:38,698
denominator zero. Since we can't divide by\nzero
1624
02:56:38,699 --> 02:56:47,470
we also need to exclude x values that make\n
1625
02:56:47,469 --> 02:56:52,108
Since we can't take the square root of a negative\n
1626
02:56:52,109 --> 02:56:58,199
that make the expression inside any even root\n
1627
02:56:58,199 --> 02:57:02,520
of a negative number, even though we can take\n
1628
02:57:02,520 --> 02:57:07,488
number. Later, when we look at logarithmic\n
1629
02:57:07,488 --> 02:57:12,228
that we have to make. But for now, these two\n
1630
02:57:12,228 --> 02:57:16,639
see. So let's apply them to a couple examples.
1631
02:57:16,639 --> 02:57:23,170
For the function in part A, we don't have\n
1632
02:57:23,170 --> 02:57:28,389
So we need to exclude x values that make the\n
1633
02:57:28,389 --> 02:57:34,458
x squared minus 4x plus three to not be equal\nto zero.
1634
02:57:34,459 --> 02:57:42,239
If we solve x squared minus 4x plus three\n
1635
02:57:42,238 --> 02:57:48,609
And that gives us x equals three or x equals\n
1636
02:57:48,609 --> 02:57:54,359
All other x values should be fine. So if I\n
1637
02:57:54,359 --> 02:58:00,689
three and just dig out a hole at both of those.\n
1638
02:58:00,689 --> 02:58:06,828
the number line. In interval notation, this\n
1639
02:58:06,828 --> 02:58:12,969
infinity to one, together with everything\n
1640
02:58:15,809 --> 02:58:21,219
In the second example, we don't have any denominator\n
1641
02:58:21,219 --> 02:58:29,688
sign. So we need to exclude any x values that\n
1642
02:58:29,689 --> 02:58:35,760
words, we can include all x values for which\n
1643
02:58:35,760 --> 02:58:43,279
zero. Solving that inequality gives us three\n
1644
02:58:44,789 --> 02:58:48,209
is less than or equal to three halves
1645
02:58:48,209 --> 02:58:51,010
I can draw this on the number line
1646
02:58:51,010 --> 02:58:54,898
or write it in interval notation.
1647
02:58:54,898 --> 02:59:00,719
Notice that three halves is included, and\n
1648
02:59:00,719 --> 02:59:05,629
be zero, I can take the square root of zero,\n
1649
02:59:05,629 --> 02:59:10,238
Finally, let's look at a more complicated\n
1650
02:59:10,238 --> 02:59:14,969
and denominator. Now there are two things\nI need to worry about.
1651
02:59:18,100 --> 02:59:24,300
to not be equal to zero, and I need the stuff\n
1652
02:59:24,299 --> 02:59:30,309
than or equal to zero. from our earlier work,\n
1653
02:59:30,309 --> 02:59:36,488
not equal to three, and x is not equal to\n
1654
02:59:36,488 --> 02:59:41,879
is less than or equal to three halves. Let's\n
1655
02:59:44,180 --> 02:59:51,039
x is not equal to three and x is not equal\n
1656
02:59:51,039 --> 02:59:56,680
two dug out points. And the other condition\n
1657
02:59:56,680 --> 03:00:00,209
we can have three halves and everything
1658
03:00:00,209 --> 03:00:05,060
To the left of it. Now to be in our domain\n
1659
03:00:05,059 --> 03:00:09,219
both of these conditions to be true. So I'm\n
1660
03:00:09,219 --> 03:00:13,708
N, that means we're looking for a numbers\n
1661
03:00:13,709 --> 03:00:19,939
and blue. So I'll draw that above in purple.\n
1662
03:00:19,939 --> 03:00:24,709
one, I have to dig out one because one was\n
1663
03:00:24,709 --> 03:00:30,300
can continue for all the things that are colored\n
1664
03:00:30,299 --> 03:00:37,679
domain is going to be, let's see negative\n
1665
03:00:37,680 --> 03:00:44,540
with one, but not including it to three halves,\n
1666
03:00:46,199 --> 03:00:51,340
in this video, we talked about functions,\n
1667
03:00:51,340 --> 03:00:57,130
and ranges. This video gives the graphs of\n
1668
03:00:57,129 --> 03:01:03,849
toolkit functions. The first function is the\n
1669
03:01:03,850 --> 03:01:13,248
on the graph of this function. If x is zero,\n
1670
03:01:13,248 --> 03:01:19,059
is always equal to x doesn't have to just\n
1671
03:01:19,059 --> 03:01:24,398
and we'll connecting the dots, we get a straight\n
1672
03:01:24,398 --> 03:01:30,090
Let's look at the graph of y equals x squared.\n
1673
03:01:30,090 --> 03:01:40,398
the origin again, if x is one, y is one, and\n
1674
03:01:40,398 --> 03:01:45,248
the x value of two gives a y value of four\n
1675
03:01:45,248 --> 03:01:51,059
value of four also connecting the dots, we\nget a parabola.
1676
03:01:51,059 --> 03:01:56,828
That is this, this function is an even function.
1677
03:01:56,828 --> 03:02:02,840
That means it has mirror symmetry across the\n
1678
03:02:02,840 --> 03:02:08,139
like the mirror image of the right side. That\n
1679
03:02:08,139 --> 03:02:14,760
number, like two, you get the exact same y\n
1680
03:02:17,699 --> 03:02:24,119
The next function y equals x cubed. I'll call\n
1681
03:02:24,119 --> 03:02:31,370
x is zero, y is zero. When x is one, y is\n
1682
03:02:31,370 --> 03:02:37,319
one, two goes with the point eight way up\n
1683
03:02:37,318 --> 03:02:42,799
to give us negative eight. If I connect the\n
1684
03:02:45,049 --> 03:02:53,728
This function is what's called an odd function,\n
1685
03:02:53,728 --> 03:02:58,688
occur around the origin. If I rotate this\n
1686
03:02:58,689 --> 03:03:04,908
turn the paper upside down, I'll get exactly\n
1687
03:03:04,908 --> 03:03:12,199
this odd symmetry is because when I cube a\n
1688
03:03:12,199 --> 03:03:17,670
n cube the corresponding negative number to\n
1689
03:03:17,670 --> 03:03:25,109
gives us exactly the negative of the the y\n
1690
03:03:25,109 --> 03:03:29,828
Let's look at the next example. Y equals the\nsquare root of x.
1691
03:03:29,828 --> 03:03:37,010
Notice that the domain of this function is\n
1692
03:03:37,010 --> 03:03:42,478
because we can't take the square root of a\n
1693
03:03:42,478 --> 03:03:49,559
X is zero gives y is 0x is one square root\n
1694
03:03:49,559 --> 03:03:55,448
and connecting the dots, I get a function\nthat looks like this.
1695
03:03:55,449 --> 03:04:01,810
The absolute value function is next. Again,\n
1696
03:04:01,809 --> 03:04:11,079
with y equals 0x is one gives us one, the\n
1697
03:04:11,079 --> 03:04:17,930
the graph, and the absolute value of negative\n
1698
03:04:17,930 --> 03:04:24,470
graph. It also has even or a mirror symmetry.
1699
03:04:24,469 --> 03:04:30,379
Y equals two to the x is what's known as an\n
1700
03:04:30,379 --> 03:04:35,648
x is in the exponent. If I plot a few points
1701
03:04:41,930 --> 03:04:49,709
two to the one is two, two squared is four,\n
1702
03:04:49,709 --> 03:04:55,850
these on my graph, you know, let me fill in\n
1703
03:04:55,850 --> 03:04:59,908
is eight. That's way up here and negative\ntwo gives
1704
03:04:59,908 --> 03:05:07,318
Maybe 1/4 1/8 connecting the dots, I get something\n
1705
03:05:07,318 --> 03:05:11,988
You might have heard the expression exponential\n
1706
03:05:11,988 --> 03:05:19,359
growth, this is function is represents exponential\n
1707
03:05:19,359 --> 03:05:26,689
time we increase the x coordinate by one,\n
1708
03:05:26,689 --> 03:05:32,970
We could also look at a function like y equals\n
1709
03:05:32,969 --> 03:05:40,129
where E is just a number about 2.7. These\n
1710
03:05:40,129 --> 03:05:45,948
bigger bass makes us rise a little more steeply.
1711
03:05:45,949 --> 03:05:52,211
Now let's look at the function y equals one\n
1712
03:05:52,210 --> 03:05:58,228
I can plug in x equals one half, one over\none half is to
1713
03:05:58,228 --> 03:06:06,528
whenever one is one, and one or two is one\n
1714
03:06:06,529 --> 03:06:11,090
in the first quadrant, but I haven't looked\n
1715
03:06:11,090 --> 03:06:18,639
whenever negative one is negative one, whenever\n
1716
03:06:18,639 --> 03:06:22,318
similar looking piece in the third quadrant.
1717
03:06:22,318 --> 03:06:27,639
This is an example of a hyperbola.
1718
03:06:27,639 --> 03:06:34,788
And it's also an odd function, because it\n
1719
03:06:34,789 --> 03:06:40,369
I turn the page upside down, it'll look exactly\n
1720
03:06:42,079 --> 03:06:48,689
Again, it's not defined when x is zero, but\n
1721
03:06:48,689 --> 03:06:55,498
see one over one half squared is one over\n1/4, which is four.
1722
03:06:55,498 --> 03:07:03,369
And one over one squared, one over two squared\n
1723
03:07:03,369 --> 03:07:08,399
previous function is just a little bit more\n
1724
03:07:08,398 --> 03:07:12,959
a little more dramatically. But for negative\n
1725
03:07:12,959 --> 03:07:19,039
goes on. For example, one over negative two\n
1726
03:07:19,040 --> 03:07:27,060
fourth. So I can plot that point there, and\n
1727
03:07:27,059 --> 03:07:32,260
one. So my curve for negative values of x\n
1728
03:07:32,260 --> 03:07:38,270
third quadrant. This is an example of an even\nfunction
1729
03:07:38,270 --> 03:07:42,359
because it has perfect mirror symmetry across\nthe y axis.
1730
03:07:42,359 --> 03:07:49,809
These are the toolkit functions, and I recommend\n
1731
03:07:49,809 --> 03:07:56,180
That way, you can draw at least a rough sketch\n
1732
03:07:56,180 --> 03:07:58,850
That's all for the graphs of the toolkit functions.
1733
03:07:58,850 --> 03:08:07,029
If we change the equation of a function, then\n
1734
03:08:09,459 --> 03:08:14,460
This video gives some rules and examples for\n
1735
03:08:14,459 --> 03:08:19,108
out of this video, it's helpful if you're\n
1736
03:08:19,109 --> 03:08:25,439
functions, I call them toolkit functions like\n
1737
03:08:25,439 --> 03:08:30,350
y equals the absolute value of x and so on.\n
1738
03:08:30,350 --> 03:08:36,930
I encourage you to watch my video called toolkit\n
1739
03:08:36,930 --> 03:08:39,520
I want to start by reviewing function notation.
1740
03:08:39,520 --> 03:08:46,340
If g of x represents the function, the square\n
1741
03:08:46,340 --> 03:08:53,199
in terms of square roots. For example, g of\n
1742
03:08:55,568 --> 03:09:05,889
g of quantity x minus two means we plug in\n
1743
03:09:05,889 --> 03:09:11,978
would be the same thing as the square root\n
1744
03:09:11,978 --> 03:09:16,840
In this second example, I say that we're subtracting\n
1745
03:09:16,840 --> 03:09:21,930
we're subtracting two before we apply the\n
1746
03:09:21,930 --> 03:09:26,990
example, I say that the minus two is on the\n
1747
03:09:26,990 --> 03:09:33,359
root function first and then subtracting two.\n
1748
03:09:33,359 --> 03:09:38,590
by three on the inside of the function. To\n
1749
03:09:38,590 --> 03:09:44,549
plug in the entire 3x for x and the square\n
1750
03:09:45,568 --> 03:09:51,609
In the next example, we're multiplying by\n
1751
03:09:51,609 --> 03:09:58,720
is just three times the square root of x.\n
1752
03:10:00,059 --> 03:10:03,769
Now, this might look a little odd because\n
1753
03:10:03,770 --> 03:10:09,760
a negative number. But remember that if x\n
1754
03:10:09,760 --> 03:10:13,850
x will be negative negative two or positive\n
1755
03:10:13,850 --> 03:10:18,899
root of a positive number in that case, let\n
1756
03:10:18,898 --> 03:10:21,748
of these are outside of my function.
1757
03:10:21,748 --> 03:10:26,408
In this next set of examples, we're using\n
1758
03:10:26,408 --> 03:10:29,958
this time, we're starting with an expression\n
1759
03:10:29,959 --> 03:10:37,699
it in terms of g of x. So the first example,\n
1760
03:10:37,699 --> 03:10:43,248
because I'm taking the square root of x first,\n
1761
03:10:46,418 --> 03:10:53,520
In the second example, I'm taking x and adding\n
1762
03:10:53,520 --> 03:10:58,550
whole thing. Since I'm adding the 12 to x\n
1763
03:10:58,549 --> 03:11:04,049
So I write that as g of the quantity x plus\n12.
1764
03:11:04,049 --> 03:11:09,559
Remember that this notation means I plug in\n
1765
03:11:09,559 --> 03:11:14,738
sign, which gives me exactly square root of\n
1766
03:11:14,738 --> 03:11:20,539
the square root first and then multiplying\n
1767
03:11:20,540 --> 03:11:27,760
outside my function, I can rewrite this as\n
1768
03:11:27,760 --> 03:11:33,130
example, I take x multiplied by a fourth and\n
1769
03:11:33,129 --> 03:11:40,938
the same thing as g of 1/4 X, my 1/4 X is\n
1770
03:11:40,939 --> 03:11:44,880
it's inside the parentheses when I use function\nnotation.
1771
03:11:44,879 --> 03:11:50,809
Let's graph the square root of x and two transformations\n
1772
03:11:50,809 --> 03:11:56,019
y equals the square root of x goes to the\n
1773
03:11:56,020 --> 03:12:01,270
root of four is two, it looks something like\nthis.
1774
03:12:01,270 --> 03:12:06,248
In order to graph y equals the square root\n
1775
03:12:06,248 --> 03:12:10,629
is on the outside of the function, that means\n
1776
03:12:10,629 --> 03:12:17,299
and then subtract two. So for example, if\n
1777
03:12:17,299 --> 03:12:21,938
the square root of zero, that's zero, then\n
1778
03:12:23,680 --> 03:12:29,510
an x value of one, which under the square\n
1779
03:12:29,510 --> 03:12:36,898
a y value that's decreased by two, one minus\n
1780
03:12:36,898 --> 03:12:42,469
of four, which under the square root function\n
1781
03:12:42,469 --> 03:12:49,840
two minus two or zero, its y value is also\n
1782
03:12:49,840 --> 03:12:56,469
zero goes with negative two, one goes with\n
1783
03:13:00,078 --> 03:13:04,978
Because I subtracted two on the outside of\n
1784
03:13:04,978 --> 03:13:12,118
two, which brought my graph down by two units.\n
1785
03:13:12,119 --> 03:13:17,359
of quantity x minus two. Now we're subtracting\n
1786
03:13:17,359 --> 03:13:21,899
we subtract two from x first and then take\n
1787
03:13:21,898 --> 03:13:28,279
y value of zero as we had in our blue graph,\n
1788
03:13:28,280 --> 03:13:30,430
need our original x to be two.
1789
03:13:30,430 --> 03:13:36,689
In order to get the y value of one that we\n
1790
03:13:36,689 --> 03:13:41,659
the square root of one, so we need x minus\n
1791
03:13:41,659 --> 03:13:43,459
start with an x value of three.
1792
03:13:43,459 --> 03:13:50,819
And in order to reproduce our y value of two\n
1793
03:13:50,819 --> 03:13:55,408
root of x minus two to be two, which means\n
1794
03:13:55,408 --> 03:14:04,168
root of four, which means our x minus two\n
1795
03:14:04,168 --> 03:14:10,510
If I plot my x values, with my corresponding\n
1796
03:14:10,510 --> 03:14:18,068
get the following graph. Notice that the graph\n
1797
03:14:19,068 --> 03:14:25,629
To me, moving down by two units, makes sense\n
1798
03:14:25,629 --> 03:14:30,519
y's by two units, but the minus two on the\n
1799
03:14:30,520 --> 03:14:35,439
I expect, I might expected to to move the\n
1800
03:14:35,439 --> 03:14:41,389
be going down by two units, but instead, it\n
1801
03:14:41,389 --> 03:14:46,869
because the x units have to go up by two units\n
1802
03:14:46,870 --> 03:14:52,919
I then subtract two units again, the observations\n
1803
03:14:52,918 --> 03:14:58,590
hold in general, according to the following\n
1804
03:14:59,850 --> 03:15:06,399
In our example, y equals a squared of x minus\n
1805
03:15:06,398 --> 03:15:12,148
a result in vertical motions, like we saw,\n
1806
03:15:12,148 --> 03:15:18,119
So subtracting two was just down by two. If\n
1807
03:15:19,879 --> 03:15:24,829
numbers on the inside of the function. That's\n
1808
03:15:24,829 --> 03:15:31,379
of quantity x minus two, those affect the\n
1809
03:15:31,379 --> 03:15:35,858
these motions go in the opposite direction\n
1810
03:15:35,859 --> 03:15:40,658
two on the inside actually shifted our graph\n
1811
03:15:40,658 --> 03:15:45,918
the inside, that would actually shift our\ngraph to the left.
1812
03:15:45,918 --> 03:15:51,738
Adding results in a shift those are called\n
1813
03:15:51,738 --> 03:15:59,568
like y equals three times the square root\n
1814
03:15:59,568 --> 03:16:03,748
In other words, if I start with the square\nroot of x
1815
03:16:03,748 --> 03:16:08,539
and then when I graph y equals three times\n
1816
03:16:08,539 --> 03:16:12,590
vertically by a factor of three.
1817
03:16:13,590 --> 03:16:20,789
if I want to graph y equals 1/3, times the\n
1818
03:16:22,908 --> 03:16:29,469
Finally, a negative sign results in reflection.\n
1819
03:16:29,469 --> 03:16:34,750
y equals the square root of x, and then when\n
1820
03:16:34,750 --> 03:16:39,859
x, that's going to do a reflection in the\n
1821
03:16:39,859 --> 03:16:42,130
is on the inside of the square root sign.
1822
03:16:42,129 --> 03:16:49,318
A reflection in the horizontal direction means\n
1823
03:16:49,318 --> 03:16:53,939
If instead, I want to graph y equals negative\n
1824
03:16:53,939 --> 03:17:00,039
outside means a vertical reflection, a reflection\n
1825
03:17:00,039 --> 03:17:06,529
Pause the video for a moment and see if you\n
1826
03:17:07,779 --> 03:17:12,859
In the first example, we're subtracting four\n
1827
03:17:12,859 --> 03:17:18,380
subtracting means a translation or shift.\n
1828
03:17:18,379 --> 03:17:24,049
affects the y value, so that's moving us vertically.\n
1829
03:17:24,049 --> 03:17:31,358
root of graph and move it down by four units,\n
1830
03:17:31,359 --> 03:17:39,090
In the next example, we're adding 12 on the\n
1831
03:17:39,090 --> 03:17:44,719
we're moving horizontally. And so since we\n
1832
03:17:44,719 --> 03:17:51,148
going to go to the left by 12 units, that's\n
1833
03:17:51,148 --> 03:17:55,278
And the next example, we're multiplying by\n
1834
03:17:55,279 --> 03:18:00,689
on the outside of our function outside our\n
1835
03:18:00,689 --> 03:18:06,120
So in multiplication means we're stretching\n
1836
03:18:06,120 --> 03:18:10,310
we reflect in the vertical direction
1837
03:18:10,309 --> 03:18:15,168
here's stretching by a factor of three vertically,\n
1838
03:18:15,168 --> 03:18:19,020
minus sign reflects in the vertical direction.
1839
03:18:19,020 --> 03:18:26,029
Finally, in this last example, we're multiplying\n
1840
03:18:26,029 --> 03:18:31,040
we know that multiplication means stretch\n
1841
03:18:31,040 --> 03:18:36,220
it's a horizontal motion, and it does the\n
1842
03:18:36,219 --> 03:18:41,648
shrinking by a factor of 1/4, horizontally,\n
1843
03:18:41,648 --> 03:18:44,139
a factor of four horizontally.
1844
03:18:44,139 --> 03:18:50,998
that'll look something like this. Notice that\n
1845
03:18:50,998 --> 03:18:55,689
looks kind of like shrinking vertically by\na factor of one half.
1846
03:18:55,689 --> 03:19:01,000
And that's actually borne out by the algebra,\n
1847
03:19:01,000 --> 03:19:05,510
thing as the square root of 1/4 times the\n
1848
03:19:05,510 --> 03:19:12,119
as one half times the square root of x. And\n
1849
03:19:12,119 --> 03:19:18,520
shrink by a factor of one half is the same\n
1850
03:19:18,520 --> 03:19:22,310
at least for this function, the square root\nfunction.
1851
03:19:22,309 --> 03:19:27,988
This video gives some rules for transformations\n
1852
03:19:27,988 --> 03:19:37,039
on the outside correspond to changes in the\n
1853
03:19:37,040 --> 03:19:44,439
numbers on the inside of the function, affect\n
1854
03:19:47,760 --> 03:19:52,020
corresponds to translations or shifts.
1855
03:19:52,020 --> 03:19:57,988
multiplying and dividing by numbers corresponds\n
1856
03:19:57,988 --> 03:20:01,389
and putting in a negative sign.
1857
03:20:04,270 --> 03:20:10,100
horizontal reflection, if the negative sign\n
1858
03:20:10,100 --> 03:20:13,010
if the negative sign is on the outside
1859
03:20:13,010 --> 03:20:17,939
knowing these basic rules about transformations\n
1860
03:20:17,939 --> 03:20:24,040
much more complicated functions, like y equals\n
1861
03:20:24,040 --> 03:20:29,521
by simply considering the transformations,\none at a time.
1862
03:20:29,521 --> 03:20:35,770
a quadratic function is a function that can\n
1863
03:20:35,770 --> 03:20:46,918
plus bx plus c, where a, b and c are real\n
1864
03:20:46,918 --> 03:20:53,449
we require that A is not equal to zero is\n
1865
03:20:53,449 --> 03:21:01,449
f of x is equal to b x plus c, which is called\n
1866
03:21:01,449 --> 03:21:06,459
a is not zero, we make sure there's really\n
1867
03:21:08,959 --> 03:21:13,939
Please pause the video for a moment and decide\n
1868
03:21:15,869 --> 03:21:23,110
The first function can definitely be written\n
1869
03:21:23,110 --> 03:21:30,840
bx plus c. fact it's already written in that\n
1870
03:21:32,199 --> 03:21:39,859
The second equation is also a quadratic function,\n
1871
03:21:39,859 --> 03:21:46,559
one times x squared plus zero times x plus\nzero.
1872
03:21:46,559 --> 03:21:53,379
So it is in the right form, where A is one,\n
1873
03:21:53,379 --> 03:21:59,108
It's perfectly fine for the coefficient of\n
1874
03:21:59,109 --> 03:22:03,590
function, we just need the coefficient of\n
1875
03:22:06,379 --> 03:22:13,269
The third equation is not a quadratic function.\n
1876
03:22:17,068 --> 03:22:21,998
The fourth function might not look like a\n
1877
03:22:21,998 --> 03:22:28,708
expanding out the X minus three squared, let's\n
1878
03:22:28,709 --> 03:22:36,760
x minus three times x minus three plus four.\n
1879
03:22:36,760 --> 03:22:45,228
3x, plus nine plus four, continuing, I get\n2x squared
1880
03:22:45,228 --> 03:22:53,260
minus 12 access, plus 18 plus four, in other\n
1881
03:22:53,260 --> 03:23:02,488
22. So in fact, our function can be written\n
1882
03:23:02,488 --> 03:23:08,488
A function that is already written in the\n
1883
03:23:08,488 --> 03:23:12,520
said to be in standard form.
1884
03:23:12,520 --> 03:23:17,289
So our first example g of x is in standard\nform
1885
03:23:17,289 --> 03:23:23,270
a function that's written in the format of\n
1886
03:23:23,270 --> 03:23:33,529
equals a times x minus h squared plus k for\n
1887
03:23:35,760 --> 03:23:40,498
I'll talk more about standard form and vertex\n
1888
03:23:40,498 --> 03:23:45,869
In this video, we identified some quadratic\n
1889
03:23:54,180 --> 03:23:59,529
This video is about graphing quadratic functions.\n
1890
03:23:59,529 --> 03:24:08,069
in standard form, like this, or sometimes\n
1891
03:24:08,068 --> 03:24:11,350
graph that looks like a parabola.
1892
03:24:11,350 --> 03:24:17,229
This video will show how to tell whether the\n
1893
03:24:17,228 --> 03:24:21,788
its x intercepts, and how to find its vertex.
1894
03:24:21,789 --> 03:24:30,890
The bare bones basic quadratic function is\n
1895
03:24:30,889 --> 03:24:38,568
since f of zero is zero squared, which is\n
1896
03:24:40,738 --> 03:24:46,270
The vertex of a parabola is its lowest point\n
1897
03:24:46,270 --> 03:24:53,908
point if it's pointing downwards. So in this\n
1898
03:24:53,908 --> 03:24:58,949
The x intercepts are where the graph crosses\n
1899
03:24:58,949 --> 03:25:05,529
In this function, y equals zero means that\n
1900
03:25:05,529 --> 03:25:11,350
x is zero. So the x intercept, there's only\n
1901
03:25:11,350 --> 03:25:17,869
The second function, y equals negative 3x\n
1902
03:25:17,869 --> 03:25:25,140
functions value when x is zero, is y equals\n
1903
03:25:27,818 --> 03:25:32,100
That's because thinking about transformations\n
1904
03:25:32,100 --> 03:25:38,600
reflects the function vertically over the\n
1905
03:25:38,600 --> 03:25:43,819
upwards, reflecting the point downwards.
1906
03:25:43,819 --> 03:25:49,010
The number three on the outside stretches\n
1907
03:25:49,010 --> 03:25:55,309
So it makes it kind of long and skinny like\nthis.
1908
03:25:55,309 --> 03:26:00,299
In general, a negative coefficient to the\n
1909
03:26:00,299 --> 03:26:05,248
down. Whereas a positive coefficient, like\n
1910
03:26:06,738 --> 03:26:09,648
Alright, that roll over here.
1911
03:26:09,648 --> 03:26:16,129
So if a is bigger than zero, the parabola\nopens up.
1912
03:26:16,129 --> 03:26:22,159
And if the value of the coefficient a is less\n
1913
03:26:22,159 --> 03:26:27,728
In this second example, we can see again that\n
1914
03:26:30,408 --> 03:26:33,538
Let's look at this third example.
1915
03:26:33,539 --> 03:26:38,168
If we multiplied our expression out, we'd\n
1916
03:26:38,168 --> 03:26:43,760
be to a positive number. So that means our\n
1917
03:26:43,760 --> 03:26:50,020
But the vertex of this parabola will no longer\n
1918
03:26:50,020 --> 03:26:55,319
parabolas vertex by thinking about transformations\nof functions.
1919
03:26:55,318 --> 03:27:02,318
Our function is related to the function y\n
1920
03:27:02,318 --> 03:27:13,889
by three and up by four. Since y equals 2x\n
1921
03:27:13,889 --> 03:27:21,549
whole parabola including the vertex, right\n
1922
03:27:21,549 --> 03:27:26,099
end up at the point three, four.
1923
03:27:26,100 --> 03:27:30,170
So a parabola will look something like this.
1924
03:27:30,170 --> 03:27:36,680
Notice how easy it was to just read off the\n
1925
03:27:36,680 --> 03:27:44,050
in this form. In fact, any parabola any quadratic\n
1926
03:27:44,049 --> 03:27:53,269
h squared plus k has a vertex at h k. By the\n
1927
03:27:53,270 --> 03:27:59,779
with a vertex at the origin to the right by\nH, and by K.
1928
03:27:59,779 --> 03:28:06,300
That's why this form of a quadratic function\n
1929
03:28:06,299 --> 03:28:11,879
Notice that this parabola has no x intercepts\n
1930
03:28:11,879 --> 03:28:19,118
For our final function, we have g of x equals\n
1931
03:28:19,119 --> 03:28:23,590
we know the graph of this function will be\n
1932
03:28:26,139 --> 03:28:32,519
To find the x intercepts, we can set y equals\n
1933
03:28:32,520 --> 03:28:37,229
graph crosses the x axis, and that's where\nthe y value is zero.
1934
03:28:37,228 --> 03:28:43,858
So zero equals 5x squared plus 10x plus three,\n
1935
03:28:43,859 --> 03:28:52,029
solve that. So x is negative 10 plus or minus\n
1936
03:28:52,029 --> 03:28:59,748
five times three, all over two times five.\n
1937
03:28:59,748 --> 03:29:03,590
or minus the square root of 40 over 10
1938
03:29:03,590 --> 03:29:08,139
which simplifies further to x equals negative\n
1939
03:29:08,139 --> 03:29:16,478
of 40 over 10, which is negative one plus\n
1940
03:29:16,478 --> 03:29:21,288
negative one plus or minus square root of\n10 over five.
1941
03:29:21,289 --> 03:29:27,409
Since the square root of 10 is just a little\n
1942
03:29:27,408 --> 03:29:34,359
about negative two fifths and negative eight\n
1943
03:29:34,359 --> 03:29:40,600
our parabola is going to look something like\n
1944
03:29:40,600 --> 03:29:48,168
the x axis at y equals three, that's because\n
1945
03:29:48,168 --> 03:29:56,969
we get y equal three, so the y intercept is\n
1946
03:29:56,969 --> 03:30:00,299
Since this function is written in standard\nform
1947
03:30:00,299 --> 03:30:10,208
On the Y equals a x plus a squared plus bx\n
1948
03:30:10,209 --> 03:30:15,120
just read off the vertex like we couldn't\n
1949
03:30:15,120 --> 03:30:20,279
formula, which says that whenever you have\n
1950
03:30:20,279 --> 03:30:24,040
form, the vertex has an x coordinate
1951
03:30:24,040 --> 03:30:34,090
of negative B over two A. So in this case,\n
1952
03:30:34,090 --> 03:30:39,988
two times five or negative one, which is kind\n
1953
03:30:39,988 --> 03:30:47,578
the y coordinate of that vertex, I can just\n
1954
03:30:47,578 --> 03:30:54,799
x, which gives me at y equals five times negative\n
1955
03:30:59,420 --> 03:31:04,398
So I think I better redraw my graph a little\n
1956
03:31:04,398 --> 03:31:10,939
of negative two where it's supposed to be.
1957
03:31:10,939 --> 03:31:15,828
Let's summarize the steps we use to graph\n
1958
03:31:15,828 --> 03:31:19,680
graph of a quadratic function has the shape\nof a parabola.
1959
03:31:19,680 --> 03:31:27,318
The parabola opens up, if the coefficient\n
1960
03:31:27,318 --> 03:31:34,658
than zero and down if a is less than zero.\n
1961
03:31:34,658 --> 03:31:44,199
zero, or in other words, f of x equal to zero\n
1962
03:31:44,199 --> 03:31:53,310
either read it off as h k, if our function\nis in vertex form
1963
03:31:53,309 --> 03:31:57,189
or we can use the vertex formula
1964
03:32:01,260 --> 03:32:10,930
of the vertex to be negative B over to a if\n
1965
03:32:10,930 --> 03:32:19,520
To find the y coordinate of the vertex in\n
1966
03:32:21,789 --> 03:32:29,590
Finally, we can always find additional points\n
1967
03:32:29,590 --> 03:32:34,870
In this video, we learned some tricks for\n
1968
03:32:34,870 --> 03:32:43,220
we saw that the vertex can be read off as\n
1969
03:32:43,219 --> 03:32:50,478
form, and the x coordinate of the vertex can\nbe calculated
1970
03:32:50,478 --> 03:32:57,198
as negative B over two A, if our function\n
1971
03:32:57,199 --> 03:33:02,959
why this vertex formula works, please see\nthe my other video.
1972
03:33:02,959 --> 03:33:09,920
a quadratic and standard form looks like y\n
1973
03:33:09,920 --> 03:33:16,568
b and c are real numbers, and a is not zero.\n
1974
03:33:16,568 --> 03:33:24,028
like y equals a times x minus h squared plus\n
1975
03:33:24,029 --> 03:33:30,328
is not zero. When a functions in vertex form,\n
1976
03:33:34,488 --> 03:33:40,568
This video explains how to get from vertex\n
1977
03:33:40,568 --> 03:33:46,549
Let's start by converting this quadratic function\n
1978
03:33:46,549 --> 03:33:50,568
pretty straightforward, we just have to distribute\nout.
1979
03:33:50,568 --> 03:33:58,668
So if I multiply out the X minus three squared
1980
03:33:58,668 --> 03:34:07,289
I get minus four times x squared minus 6x\n
1981
03:34:07,289 --> 03:34:16,918
four, I get negative 4x squared plus 24x minus\n
1982
03:34:16,918 --> 03:34:26,709
squared plus 24x minus 35. And I have my quadratic\n
1983
03:34:26,709 --> 03:34:31,618
Now let's go the other direction and convert\n
1984
03:34:31,619 --> 03:34:40,510
form into vertex form. That is, we want to\n
1985
03:34:40,510 --> 03:34:51,090
x minus h squared plus k, where the vertex\n
1986
03:34:53,870 --> 03:35:00,380
The vertex formula says that the x coordinate\n
1987
03:35:00,379 --> 03:35:08,368
over to a, where A is the coefficient of x\n
1988
03:35:08,369 --> 03:35:16,649
in this case, we get an x coordinate of negative\n
1989
03:35:16,648 --> 03:35:24,118
To find the y coordinate of the vertex, we\n
1990
03:35:24,119 --> 03:35:30,918
for a g of x. So that's g of negative two,\n
1991
03:35:30,918 --> 03:35:37,360
eight times negative two plus six. And that\n
1992
03:35:37,360 --> 03:35:44,461
So the vertex for our quadratic function has\n
1993
03:35:44,460 --> 03:35:50,429
if I want to write g of x in vertex form,\n
1994
03:35:50,430 --> 03:36:00,789
two squared plus minus two. That's because\n
1995
03:36:00,789 --> 03:36:08,699
simplifies to g of x equals a times x plus\n
1996
03:36:08,699 --> 03:36:14,359
need to figure out what this leading coefficient\n
1997
03:36:14,359 --> 03:36:23,029
this out, then the coefficient of x squared\n
1998
03:36:23,029 --> 03:36:27,520
of x squared here, which is a has to be the\n
1999
03:36:27,520 --> 03:36:33,149
which we conveniently also called a, in other\n
2000
03:36:33,148 --> 03:36:39,778
I'm going to write that as g of x equals two\n
2001
03:36:39,779 --> 03:36:46,289
twos in this problem. And that's our quadratic\n
2002
03:36:46,289 --> 03:36:53,569
my answer, of course, I could just distribute\n
2003
03:36:53,568 --> 03:37:03,000
4x, plus two, minus two, in other words, 2x\n
2004
03:37:03,000 --> 03:37:09,010
to exactly what I started with. This video\n
2005
03:37:09,010 --> 03:37:14,408
form by distributing out and how to get from\n
2006
03:37:14,408 --> 03:37:17,639
vertex using the vertex formula.
2007
03:37:17,639 --> 03:37:26,269
Suppose you have a quadratic function in the\n
2008
03:37:26,270 --> 03:37:31,760
you want to find where the vertex is, when\nyou graph it.
2009
03:37:31,760 --> 03:37:40,969
The vertex formula says that the x coordinate\n
2010
03:37:42,879 --> 03:37:47,608
this video gives a justification for where\n
2011
03:37:47,609 --> 03:37:53,390
Let's start with a specific example. Suppose\n
2012
03:37:53,389 --> 03:38:00,840
vertex for this quadratic function. To find\n
2013
03:38:00,840 --> 03:38:07,510
and solve for x. So that's zero equals 3x\n
2014
03:38:07,510 --> 03:38:13,418
x, I use the quadratic formula. So x is going\n
2015
03:38:13,418 --> 03:38:22,328
root of seven squared minus four times three\n
2016
03:38:22,328 --> 03:38:29,228
That simplifies to negative seven plus or\n
2017
03:38:29,228 --> 03:38:35,039
I could also write this as negative seven,\n
2018
03:38:35,040 --> 03:38:41,029
or x equals negative seven, six minus the\n
2019
03:38:41,029 --> 03:38:47,488
root of 109 is just a little bit bigger than\n
2020
03:38:47,488 --> 03:38:54,340
six plus 10, six, and negative seven, six\nminus 10, six.
2021
03:38:54,340 --> 03:39:00,709
So pretty close to, I guess about what half\n
2022
03:39:00,709 --> 03:39:06,609
three over here, I'm just going to estimate\n
2023
03:39:06,609 --> 03:39:11,629
Since the leading coefficient three is positive,\n
2024
03:39:11,629 --> 03:39:20,270
up and the intercepts are somewhere around\n
2025
03:39:25,488 --> 03:39:30,439
Now the vertex is going to be somewhere in\n
2026
03:39:30,439 --> 03:39:37,010
to be by symmetry, it'll be exactly halfway\n
2027
03:39:37,010 --> 03:39:43,260
intercepts are negative seven, six plus and\n
2028
03:39:43,260 --> 03:39:47,978
halfway in between those is going to be exactly\n
2029
03:39:47,978 --> 03:39:56,049
right, because on the one hand, I have negative\n
2030
03:39:56,049 --> 03:39:59,858
hand, I have negative seven six minus that\nsame thing.
2031
03:39:59,859 --> 03:40:08,998
So negative seven, six will be exactly in\n
2032
03:40:08,998 --> 03:40:16,670
will be at negative seven sex. Notice that\n
2033
03:40:16,670 --> 03:40:24,818
More generally, if I want to find the x intercepts\n
2034
03:40:27,469 --> 03:40:33,198
and solve for x using the quadratic formula,\n
2035
03:40:33,199 --> 03:40:40,890
b squared minus four AC Oliver to a, b, x\n
2036
03:40:40,889 --> 03:40:47,189
the x coordinate of the vertex, which is exactly\n
2037
03:40:47,189 --> 03:40:53,710
be at negative B over two A. That's where\n
2038
03:40:53,709 --> 03:40:58,628
And it turns out that this formula works even\n
2039
03:40:58,629 --> 03:41:04,010
the quadratic formula gives us no solutions.\n
2040
03:41:06,658 --> 03:41:11,498
And that's the justification of the vertex\nformula.
2041
03:41:11,498 --> 03:41:14,898
This video is about polynomials and their\ngraphs.
2042
03:41:14,898 --> 03:41:20,379
We call that a polynomial is a function like\n
2043
03:41:20,379 --> 03:41:28,708
His terms are numbers times powers of x. I'll\n
2044
03:41:28,709 --> 03:41:35,899
a polynomial is the largest exponent. For\n
2045
03:41:37,850 --> 03:41:44,129
the leading term is the term with the largest\n
2046
03:41:44,129 --> 03:41:51,350
term is 5x to the fourth, it's conventional\n
2047
03:41:51,350 --> 03:41:56,418
of powers of x. So the leading term is first.\n
2048
03:41:56,418 --> 03:42:04,498
first term. If I wrote the same polynomial\n
2049
03:42:04,498 --> 03:42:11,658
cubed plus 5x. Fourth, the leading term would\n
2050
03:42:13,260 --> 03:42:20,029
The leading coefficient is the number in the\n
2051
03:42:21,459 --> 03:42:28,140
Finally, the constant term is the term with\n
2052
03:42:29,139 --> 03:42:34,760
please pause the video for a moment and take\n
2053
03:42:34,760 --> 03:42:40,639
figure out what's the degree the leading term,\n
2054
03:42:40,639 --> 03:42:48,760
The degree is again, four, since that's the\n
2055
03:42:48,760 --> 03:42:56,340
is negative 7x. to the fourth, the leading\n
2056
03:42:58,148 --> 03:43:04,510
In the graph of the polynomial Shown here\n
2057
03:43:04,510 --> 03:43:12,469
points, because the polynomial turns around\n
2058
03:43:12,469 --> 03:43:18,868
those same points can also be called local\n
2059
03:43:18,869 --> 03:43:25,710
maximum and minimum points. For this polynomial,\n
2060
03:43:28,068 --> 03:43:33,129
Let's compare the degree and the number of\n
2061
03:43:33,129 --> 03:43:39,698
For the first one, the degree is to and there's\n
2062
03:43:39,699 --> 03:43:50,090
For the second example, that agree, is three,\n
2063
03:43:50,090 --> 03:43:56,920
And for this last example, the degree is four,\n
2064
03:43:56,920 --> 03:44:01,699
For this first example, and the next two,\n
2065
03:44:01,699 --> 03:44:07,869
less than the degree. So you might conjecture\n
2066
03:44:07,869 --> 03:44:13,689
is not always true. In this last example,\n
2067
03:44:16,909 --> 03:44:21,510
In fact, it turns out that while the number\n
2068
03:44:21,510 --> 03:44:28,629
minus one, it is always less than or equal\n
2069
03:44:28,629 --> 03:44:33,680
Remember, when you're sketching graphs are\n
2070
03:44:33,680 --> 03:44:39,659
The end behavior of a function is how the\n
2071
03:44:39,659 --> 03:44:45,248
and bigger heads towards infinity, or x gets\n
2072
03:44:47,859 --> 03:44:54,818
In this first example, the graph of the function\n
2073
03:44:54,818 --> 03:44:59,959
goes towards negative infinity. I can draw\n
2074
03:44:59,959 --> 03:45:05,958
Down on either side, or I can say in words,\n
2075
03:45:05,959 --> 03:45:09,459
and falling also as we had right.
2076
03:45:09,459 --> 03:45:16,600
In the second example, the graph rises to\n
2077
03:45:16,600 --> 03:45:23,379
example, the graph falls to the left, but\n
2078
03:45:23,379 --> 03:45:29,489
it rises to the left and falls to the right.\n
2079
03:45:29,488 --> 03:45:36,189
might notice there's a relationship between\n
2080
03:45:36,189 --> 03:45:44,090
of the polynomials and the end behavior. Specifically,\n
2081
03:45:44,090 --> 03:45:51,728
by whether the degree is even or odd. And\n
2082
03:45:54,879 --> 03:46:00,729
When the degree is even, and the leading coefficient\n
2083
03:46:00,728 --> 03:46:08,458
leading coefficient is one, we have this sorts\n
2084
03:46:08,459 --> 03:46:14,069
When the degree is even at the leading coefficient\n
2085
03:46:14,068 --> 03:46:18,378
the end behavior that's falling on both sides
2086
03:46:18,379 --> 03:46:23,810
when the degree is odd, and the leading coefficient\n
2087
03:46:23,809 --> 03:46:30,699
the degree three and the leading coefficient\n
2088
03:46:30,700 --> 03:46:35,979
And finally, when the degree is odd, and the\n
2089
03:46:35,978 --> 03:46:40,628
example, we have this sort of NBA havior.
2090
03:46:40,629 --> 03:46:48,409
I like to remember this chart just by thinking\n
2091
03:46:48,408 --> 03:46:55,079
y equals negative x squared, y equals x cubed,\n
2092
03:46:55,079 --> 03:47:00,209
by heart what those four examples look like
2093
03:47:00,209 --> 03:47:08,009
then I just have to remember that any polynomial\n
2094
03:47:08,010 --> 03:47:11,950
has the same end behavior as x squared.
2095
03:47:11,950 --> 03:47:18,020
And similarly, any polynomial with even degree\n
2096
03:47:18,020 --> 03:47:25,729
end behavior as negative x squared. And similar\n
2097
03:47:25,728 --> 03:47:30,738
We can use facts about turning points and\n
2098
03:47:30,738 --> 03:47:34,270
of a polynomial just by looking at this graph.
2099
03:47:34,270 --> 03:47:42,699
In this example, because of the end behavior,\n
2100
03:47:42,699 --> 03:47:46,569
we know that the leading coefficient
2101
03:47:49,389 --> 03:47:58,099
And finally, since there are 1234, turning\n
2102
03:48:01,299 --> 03:48:06,288
That's because the number of turning points\n
2103
03:48:06,289 --> 03:48:10,529
one. And in this case, the number of turning\n
2104
03:48:10,529 --> 03:48:18,238
And so solving that inequality, we get the\n
2105
03:48:18,238 --> 03:48:23,129
Put in some of that information together,\n
2106
03:48:23,129 --> 03:48:31,059
five, or seven, or nine, or any odd number\n
2107
03:48:31,059 --> 03:48:38,939
be for example, three or six. Because even\n
2108
03:48:41,129 --> 03:48:46,500
This video gave a lot of definitions, including\n
2109
03:48:54,930 --> 03:48:59,908
We saw that knowing the degree and the leading\n
2110
03:48:59,908 --> 03:49:06,189
about the number of turning points and the\n
2111
03:49:06,189 --> 03:49:11,068
This video is about exponential functions\nand their graphs.
2112
03:49:11,068 --> 03:49:16,908
an exponential function is a function that\n
2113
03:49:16,908 --> 03:49:24,420
times b to the x, where a and b are any real\n
2114
03:49:26,549 --> 03:49:33,170
It's important to notice that for an exponential\n
2115
03:49:33,170 --> 03:49:37,609
This is different from many other functions\n
2116
03:49:37,609 --> 03:49:44,309
function like f of x equals 3x squared has\n
2117
03:49:45,500 --> 03:49:53,010
For exponential functions, f of x equals a\n
2118
03:49:53,010 --> 03:50:00,100
to zero, because otherwise, we would have\n
2119
03:50:00,100 --> 03:50:04,850
Which just means that f of x equals zero.\n
2120
03:50:04,850 --> 03:50:11,149
an exponential function. Because f of x is\nalways equal to zero
2121
03:50:11,148 --> 03:50:17,389
in an exponential function, but we require\n
2122
03:50:17,389 --> 03:50:27,578
for example, if b is equal to negative one,\n
2123
03:50:27,578 --> 03:50:34,038
one to the x. Now, this would make sense for\n
2124
03:50:34,039 --> 03:50:40,600
something like f of one half, with our Bs,\n
2125
03:50:40,600 --> 03:50:45,340
as a times the square root of negative one,\n
2126
03:50:45,340 --> 03:50:52,158
we'd get the same problem for other values\n
2127
03:50:52,158 --> 03:50:59,299
if we tried b equals zero, we'd get a kind\n
2128
03:50:59,299 --> 03:51:03,828
which again is always zero. So that wouldn't\n
2129
03:51:03,828 --> 03:51:09,728
use any negative basis, and we can't use zero\n
2130
03:51:09,728 --> 03:51:18,448
The number A in the expression f of x equals\n
2131
03:51:18,449 --> 03:51:21,029
And the number B is called the base.
2132
03:51:21,029 --> 03:51:28,998
The phrase initial value comes from the fact\n
2133
03:51:28,998 --> 03:51:34,529
times b to the zero, well, anything to the\n
2134
03:51:34,529 --> 03:51:40,920
other words, f of zero equals a. So if we\nthink of starting out
2135
03:51:40,920 --> 03:51:44,700
when x equals zero, we get the y value
2136
03:51:44,700 --> 03:51:51,629
of a, that's why it's called the initial value.
2137
03:51:51,629 --> 03:51:59,890
Let's start out with this example, where y\n
2138
03:51:59,889 --> 03:52:06,049
and we've set a equal to one and B equals\n
2139
03:52:06,049 --> 03:52:09,340
when x is zero, is going to be one.
2140
03:52:09,340 --> 03:52:20,318
If I change my a value, my initial value,\n
2141
03:52:21,318 --> 03:52:27,379
If I make the value of a go to zero, and then\nnegative
2142
03:52:27,379 --> 03:52:35,219
then my initial value becomes negative, and\n
2143
03:52:35,219 --> 03:52:41,778
back to an a value of say one, and see what\n
2144
03:52:41,779 --> 03:52:52,869
value the basis two, if I increase B, my y\n
2145
03:52:52,869 --> 03:53:03,020
steeper and steeper. If I put B back down\n
2146
03:53:03,020 --> 03:53:05,959
exactly one, my graph is just a constant.
2147
03:53:05,959 --> 03:53:14,779
As B gets into fractional territory, point\n
2148
03:53:14,779 --> 03:53:20,119
other way, it's decreasing now instead of\n
2149
03:53:20,119 --> 03:53:26,199
still hasn't changed, I can get it more and\n
2150
03:53:26,199 --> 03:53:33,550
away from one of course, when B goes to negative\n
2151
03:53:33,549 --> 03:53:44,418
So a changes the y intercept, and B changes\n
2152
03:53:44,418 --> 03:53:50,949
whether it's increasing for B values bigger\n
2153
03:53:55,078 --> 03:53:58,978
we'll summarize all these observations on\nthe next slide.
2154
03:53:58,978 --> 03:54:04,568
We've seen that for an exponential function,\n
2155
03:54:04,568 --> 03:54:13,159
or number a gives the y intercept, the parameter\n
2156
03:54:13,159 --> 03:54:19,889
decreasing. Specifically, if b is greater\n
2157
03:54:19,889 --> 03:54:23,868
And if b is less than one, the graph is decreasing.
2158
03:54:23,869 --> 03:54:29,459
The closer B is to the number one, the flatter\nthe graph.
2159
03:54:29,459 --> 03:54:37,600
So for example, if I were to graph y equals\n
2160
03:54:37,600 --> 03:54:44,379
four to the x, they would both be decreasing\n
2161
03:54:44,379 --> 03:54:52,209
less than one. But point two five is farther\n
2162
03:54:52,209 --> 03:54:58,560
one. So point four is going to be flatter.\n
2163
03:54:58,559 --> 03:55:03,418
So in this picture,\nThis red graph would correspond to point two
2164
03:55:03,418 --> 03:55:11,728
five to the x, and the blue graph would correspond\n
2165
03:55:11,728 --> 03:55:17,458
functions, whether the graphs are decreasing\n
2166
03:55:17,459 --> 03:55:25,460
asymptote along the x axis. In other words,\n
2167
03:55:25,459 --> 03:55:34,799
always from negative infinity to infinity,\n
2168
03:55:34,799 --> 03:55:43,309
because the range is always positive y values.\n
2169
03:55:43,309 --> 03:55:52,109
zero, if a is less than zero, then our graph\n
2170
03:55:52,109 --> 03:55:59,890
same, but our range becomes negative infinity\n
2171
03:55:59,889 --> 03:56:04,868
is f of x equals e to the x. This function\n
2172
03:56:04,869 --> 03:56:13,420
x of x. The number E is Oilers number as approximately\n
2173
03:56:13,420 --> 03:56:21,100
to calculus and to some compound interest\n
2174
03:56:21,100 --> 03:56:28,879
functions, functions of the form a times b\n
2175
03:56:28,879 --> 03:56:37,720
we saw that they all have the same general\n
2176
03:56:37,719 --> 03:56:45,929
like this, unless a is negative, in which\n
2177
03:56:45,930 --> 03:56:55,859
a horizontal asymptote at y equals zero, the\n
2178
03:56:55,859 --> 03:57:02,748
functions to model real world examples. Let's\n
2179
03:57:02,748 --> 03:57:10,039
salary is $40,000. With a guaranteed annual\n
2180
03:57:10,039 --> 03:57:16,899
salary be after one year, two years, five\n
2181
03:57:16,898 --> 03:57:23,349
me chart out the information. The left column\n
2182
03:57:23,350 --> 03:57:30,930
And the right column will be your salary.\n
2183
03:57:30,930 --> 03:57:40,970
hired, your salary will be $40,000. After\n
2184
03:57:40,969 --> 03:57:52,459
your salary will be the original 40,000 plus\n
2185
03:57:52,459 --> 03:58:00,528
of this first number as one at times 40,000.\n
2186
03:58:00,529 --> 03:58:15,659
terms, to get 40,000 times one plus 0.03.\n
2187
03:58:15,659 --> 03:58:26,119
is your original salary multiplied by a growth\n
2188
03:58:26,119 --> 03:58:33,418
a 3% raise from your previous year salary,\n
2189
03:58:36,668 --> 03:58:45,559
3% of that, again, I can think of the first\n
2190
03:58:45,559 --> 03:58:53,828
I can factor out the common factor of 40,000\n
2191
03:58:53,828 --> 03:59:08,920
times 1.03 times one plus 0.03. Let me rewrite\n
2192
03:59:08,920 --> 03:59:19,068
times 1.3 squared. We can think of this as\n
2193
03:59:19,068 --> 03:59:26,799
growth factor of 1.03. After three years,\n
2194
03:59:26,799 --> 03:59:37,019
new salary is your previous year salary times\n
2195
03:59:37,020 --> 03:59:45,720
as 40,000 times 1.03 cubed. And in general,\n
2196
03:59:45,719 --> 03:59:53,760
years, your salary should be 40,000 times\n
2197
03:59:53,760 --> 04:00:06,260
salary after two years is your original salary\n
2198
04:00:06,260 --> 04:00:15,818
to the t power, let me write this as a formula\n
2199
04:00:15,818 --> 04:00:24,969
to 40,000 times 1.03 to the T. This is an\n
2200
04:00:24,969 --> 04:00:38,760
the form a times b to the T, where your initial\n
2201
04:00:38,760 --> 04:00:45,050
Notice that your base is the amount that your\n
2202
04:00:45,050 --> 04:00:49,868
this formula, we can easily figure out what\n
2203
04:00:49,869 --> 04:01:01,279
years by plugging in five for T. I worked\n
2204
04:01:01,279 --> 04:01:08,459
to the nearest cent. exponential functions\n
2205
04:01:08,459 --> 04:01:15,209
The United Nations estimated that the world\n
2206
04:01:15,209 --> 04:01:20,739
at a rate of 1.1% per year. Assuming that\n
2207
04:01:20,738 --> 04:01:26,309
to stay the same, we'll write an equation\n
2208
04:01:26,309 --> 04:01:39,109
2010 1.1%, written as a decimal is 0.011.\n
2209
04:01:39,109 --> 04:01:46,998
that after zero years since 2010, we have\n
2210
04:01:46,998 --> 04:01:59,680
one year, we'll take that 6.7 9 billion and\n
2211
04:01:59,680 --> 04:02:15,898
This works out to 6.79 times one plus point\n
2212
04:02:15,898 --> 04:02:27,599
initial population of 6.7 9 billion, and our\n
2213
04:02:27,600 --> 04:02:35,640
population got multiplied by in one year.\n
2214
04:02:35,639 --> 04:02:45,010
years, our population becomes 6.79 times 1.011\n
2215
04:02:45,010 --> 04:02:56,488
twice, and after two years, it'll be 6.79\n
2216
04:02:56,488 --> 04:03:09,520
that models population is going to be 6.79\n
2217
04:03:09,520 --> 04:03:21,220
in years, since 2010. Just for fun, I'll plug\n
2218
04:03:21,219 --> 04:03:31,408
So that's the year 2050. And I get 6.79 times\n
2219
04:03:31,408 --> 04:03:40,368
10 point 5 billion. That's the prediction\n
2220
04:03:40,369 --> 04:03:45,629
The previous two examples were examples of\n
2221
04:03:45,629 --> 04:03:51,869
example of exponential decay. The drugs Seroquel\n
2222
04:03:51,869 --> 04:03:58,959
at a rate of 11% per hour. If 400 milligrams\n
2223
04:03:58,959 --> 04:04:09,090
hours later, I'll chart out my information\n
2224
04:04:09,090 --> 04:04:15,998
since the dose was given, and the right column\n
2225
04:04:15,998 --> 04:04:22,329
still on the body. zero hours after the dose\n
2226
04:04:22,329 --> 04:04:30,668
in the body. One hour later, we have the formula\n
2227
04:04:30,668 --> 04:04:40,078
point one one times 400. If I factor out the\n
2228
04:04:40,078 --> 04:04:53,219
0.11 or 400 times point eight nine. The 400\n
2229
04:04:53,219 --> 04:04:58,608
nine I'll call the growth factor, even though\n
2230
04:04:58,609 --> 04:05:08,440
growing. So really it's kind of a shrink factor.\n
2231
04:05:08,440 --> 04:05:15,550
I'll have 400 times 0.89 my previous amount,\n
2232
04:05:15,549 --> 04:05:22,469
nine, so that's going to be 400 times 0.89\n
2233
04:05:22,469 --> 04:05:29,528
have 400 multiplied by this growth or shrinkage\n
2234
04:05:29,529 --> 04:05:36,270
t power. Since each hour, the amount of Seroquel\n
2235
04:05:36,270 --> 04:05:44,289
number less than one. All right, my exponential\n
2236
04:05:44,289 --> 04:05:54,560
0.89 to the T, where f of t represents the\n
2237
04:05:54,559 --> 04:06:01,958
And t represents the number of hours since\n
2238
04:06:01,959 --> 04:06:13,239
in the body after 24 hours, I just plug in\n
2239
04:06:13,238 --> 04:06:17,788
I hope you notice the common form for the\n
2240
04:06:17,789 --> 04:06:26,418
examples. The functions are always in the\n
2241
04:06:26,418 --> 04:06:36,449
a represented the initial amount, and B represented\n
2242
04:06:36,449 --> 04:06:45,449
B, we started with the percent increase or\n
2243
04:06:45,449 --> 04:06:50,800
we either added or subtracted it from one,\n
2244
04:06:50,799 --> 04:06:58,738
or decreasing. Let me show you that as a couple\n
2245
04:06:58,738 --> 04:07:09,010
increase of 3% on the race as a decimal, I'll\n
2246
04:07:09,010 --> 04:07:16,619
get the growth factor, we added that to one\n
2247
04:07:16,619 --> 04:07:29,640
we had a 1.1% increase, we wrote this as point\n
2248
04:07:29,639 --> 04:07:42,349
drug example, we had a decrease of 11%. We\n
2249
04:07:42,350 --> 04:07:50,988
the point one one from one to get 0.89. In\n
2250
04:07:50,988 --> 04:07:58,838
B as one plus the percent change written as\n
2251
04:07:58,838 --> 04:08:04,539
percent change, negative when the quantity\n
2252
04:08:04,539 --> 04:08:09,189
is increasing. Since here one plus negative\npoint 11
2253
04:08:09,189 --> 04:08:14,210
gives us the correct growth factor of point\n
2254
04:08:14,209 --> 04:08:23,599
check. Remember that if your quantity is increasing,\n
2255
04:08:23,600 --> 04:08:30,529
if the quantity is decreasing, then B should\nbe less than one.
2256
04:08:30,529 --> 04:08:35,129
exponential functions can also be used to\n
2257
04:08:35,129 --> 04:08:42,229
interest, as we'll see in another video. This\n
2258
04:08:42,228 --> 04:08:49,728
An antique car is worth $50,000 now, and its\n
2259
04:08:49,728 --> 04:09:00,448
an equation to model its value x years from\n
2260
04:09:00,449 --> 04:09:10,459
plus 0.07 times the 50,000. That's because\n
2261
04:09:10,459 --> 04:09:20,199
times 50,000. this can be written as 50,000\n
2262
04:09:20,199 --> 04:09:28,359
adding 7% to the original value is the same\n
2263
04:09:28,359 --> 04:09:41,770
point oh seven or by 1.07. After two years,\n
2264
04:09:41,770 --> 04:09:51,748
squared, or 50,000 times 1.07 squared. That's\n
2265
04:09:51,748 --> 04:10:06,408
again by 1.7. In general, after x years We\n
2266
04:10:06,408 --> 04:10:16,728
times 1.07 to the x. That's because the original\n
2267
04:10:16,728 --> 04:10:24,019
one time for each year. If we dissect this\n
2268
04:10:24,020 --> 04:10:29,709
comes from the original value of the car.\n
2269
04:10:29,709 --> 04:10:41,949
factor comes from one plus point 707. The\n
2270
04:10:41,949 --> 04:10:50,010
written as a decimal. So the form of this\n
2271
04:10:50,010 --> 04:10:57,738
x equals a times b to the x, where A is the\n
2272
04:10:57,738 --> 04:11:06,478
But we could also write this as a times one\n
2273
04:11:08,478 --> 04:11:15,709
R is the percent increase written as a decimal.\n
2274
04:11:15,709 --> 04:11:22,559
example, here, my Toyota Prius is worth only\n
2275
04:11:22,559 --> 04:11:34,889
5% each year. So after one year, its value\n
2276
04:11:34,889 --> 04:11:48,738
3000 times one minus 0.05. I can also write\n
2277
04:11:48,738 --> 04:12:00,020
by 5% is like multiplying the value by one\n
2278
04:12:00,020 --> 04:12:08,658
After two years, the value will be multiplied\n
2279
04:12:08,658 --> 04:12:17,908
be 3000 times point nine, five squared. And\n
2280
04:12:17,908 --> 04:12:26,998
point nine five to the x. So my equation for\n
2281
04:12:26,998 --> 04:12:36,408
the x. This is again, an equation of the form\n
2282
04:12:36,408 --> 04:12:47,588
a is 3000, the initial value, and B is point\n
2283
04:12:47,588 --> 04:12:53,738
factor, even though we're actually declining\n
2284
04:12:53,738 --> 04:13:01,788
point nine five came from, it came from taking\n
2285
04:13:01,789 --> 04:13:09,129
5%, decrease in value, so I can again, write\n
2286
04:13:09,129 --> 04:13:19,829
time times one minus r to the x, where R is\n
2287
04:13:19,828 --> 04:13:26,920
written as a decimal. Please take a moment\n
2288
04:13:26,920 --> 04:13:35,408
They say that when you have an exponential\n
2289
04:13:35,408 --> 04:13:42,088
If it's written in this form, B is your growth\n
2290
04:13:42,088 --> 04:13:50,510
one minus r, where r is the percent decrease,\n
2291
04:13:50,510 --> 04:13:55,898
In this example, we're given a function f\n
2292
04:13:55,898 --> 04:14:03,510
petri dish x hours after 12 o'clock noon,\n
2293
04:14:03,510 --> 04:14:09,168
at noon, and by what percent, the number of\n
2294
04:14:09,168 --> 04:14:14,408
see from the equation that the number of bacteria\n
2295
04:14:14,408 --> 04:14:21,458
the base of the exponential function 1.45\n
2296
04:14:21,459 --> 04:14:30,189
f of x equals 12 times 1.45 to the X has the\n
2297
04:14:30,189 --> 04:14:43,389
of it as a times one plus r to the x. Here\n
2298
04:14:43,389 --> 04:14:49,158
this familiar form, we can recognize that\n
2299
04:14:49,158 --> 04:15:00,949
be 12 12,000. Since those are our units and\n
2300
04:15:00,949 --> 04:15:08,859
the number of bacteria is multiplied by each\n
2301
04:15:08,859 --> 04:15:18,529
the rate of increase, in other words, a 45%\n
2302
04:15:18,529 --> 04:15:29,359
questions are 12,045%. In this example, the\n
2303
04:15:29,359 --> 04:15:37,529
exponential function, where x is the number\n
2304
04:15:37,529 --> 04:15:42,850
of salamanders is decreasing, because the\n
2305
04:15:42,850 --> 04:15:50,100
eight is less than one. So if we recognize\n
2306
04:15:50,100 --> 04:15:59,579
b to the x, or we can think of this as a times\n
2307
04:15:59,579 --> 04:16:08,998
value, and r is our percent decrease written\nas a decimal.
2308
04:16:08,998 --> 04:16:18,779
Our initial value is 3000. So that's the number\n
2309
04:16:18,779 --> 04:16:27,529
growth factor B is 0.78. But if I write that\n
2310
04:16:27,529 --> 04:16:40,720
minus 0.78, or 0.22. In other words, our population\n
2311
04:16:40,719 --> 04:16:47,028
we saw that exponential functions can be written\n
2312
04:16:47,029 --> 04:16:58,050
x, where A is the initial value. And B is\n
2313
04:16:58,049 --> 04:17:05,878
be written in the form a times one plus r\n
2314
04:17:05,879 --> 04:17:16,609
as a times one minus r to the x when the amount\n
2315
04:17:16,609 --> 04:17:28,350
increase or the percent decrease written as\n
2316
04:17:28,350 --> 04:17:43,690
of 0.15 and a growth factor B of 1.15. Whereas\n
2317
04:17:43,690 --> 04:17:55,779
one, two, and a B value of one minus point\n
2318
04:17:55,779 --> 04:18:03,439
help us quickly interpret exponential functions.\n
2319
04:18:03,439 --> 04:18:17,889
of 100 and a 15% increase. And here, we have\n
2320
04:18:17,889 --> 04:18:23,118
exponential functions can be used to model\n
2321
04:18:23,119 --> 04:18:30,291
Suppose you invest $200 in a bank account\n
2322
04:18:30,290 --> 04:18:37,418
make no deposits or withdrawals, how much\n
2323
04:18:37,418 --> 04:18:41,939
because 3% of the money that's in the bank\n
2324
04:18:41,939 --> 04:18:51,359
bank gets multiplied by 1.03 each year. So\n
2325
04:18:51,359 --> 04:19:02,180
1.3, after two years 200 times one point O\n
2326
04:19:02,180 --> 04:19:08,479
one point O three to the t power. So the function\n
2327
04:19:08,478 --> 04:19:20,418
P of t is given by 200 times 1.03 to the T.\n
2328
04:19:20,418 --> 04:19:32,628
at an annual interest rate of our for t years.\n
2329
04:19:37,078 --> 04:19:45,039
here r needs to be written as a decimal, so\n
2330
04:19:45,040 --> 04:19:51,909
rate. Going back to our specific example,\n
2331
04:19:51,908 --> 04:20:03,038
to be P of 10 which is 200 times 1.03 to the\n
2332
04:20:03,039 --> 04:20:10,239
cent. In this problem, we've assumed that\n
2333
04:20:10,238 --> 04:20:14,510
in the next few examples, we'll see what happens\n
2334
04:20:14,510 --> 04:20:23,079
twice a year, or every month. For example,\n
2335
04:20:23,079 --> 04:20:31,059
4.5% annual interest compounded semi annually,\n
2336
04:20:31,059 --> 04:20:40,748
A 4.5% annual interest rate compounded two\n
2337
04:20:40,748 --> 04:20:53,039
4.5 over 2% interest, every time the interest\n
2338
04:20:53,040 --> 04:21:05,890
2.25% interest every half a year. Note that\n
2339
04:21:05,889 --> 04:21:17,128
So every time we earn interest, our money\n
2340
04:21:17,129 --> 04:21:25,399
a chart of what happens. After zero years,\n
2341
04:21:25,398 --> 04:21:33,760
original $300. for half a year, that's one\n
2342
04:21:33,760 --> 04:21:42,078
time, so we multiply the 300 by 1.02 to five,\n
2343
04:21:42,078 --> 04:21:52,770
money earns interest two times. So we multiply\n
2344
04:21:52,770 --> 04:22:01,680
after 1.5 years, that's three half years,\n
2345
04:22:01,680 --> 04:22:09,750
after two years or four half years, we have\n
2346
04:22:09,750 --> 04:22:20,158
after t years, which is to T half years, our\n
2347
04:22:20,158 --> 04:22:28,299
to the two t power. Because we've compounded\n
2348
04:22:28,299 --> 04:22:35,328
amount of money is P of t equals 300 times\n
2349
04:22:35,328 --> 04:22:45,639
t is the number of years. To finish the problem,\n
2350
04:22:45,639 --> 04:22:59,849
is 300 times 1.02 to five to the two times\n
2351
04:22:59,850 --> 04:23:06,470
to the nearest cent. In this next example,\n
2352
04:23:06,469 --> 04:23:14,719
in annual interest rate of 6% compounded monthly.\n
2353
04:23:14,719 --> 04:23:19,130
different, they're mathematically the same.\n
2354
04:23:19,130 --> 04:23:25,278
investing money in you and getting interest\n
2355
04:23:25,279 --> 04:23:36,050
out with the same kind of math 6% annual interest\n
2356
04:23:36,049 --> 04:23:42,250
12 times a year. So each time you compound\n
2357
04:23:42,250 --> 04:23:56,148
12% interest. That's point 5% interest. And\n
2358
04:23:56,148 --> 04:24:05,689
again, what happens. Time is zero, of course,\n
2359
04:24:05,689 --> 04:24:15,809
After one year, that's 12 months, your loan\n
2360
04:24:15,809 --> 04:24:22,510
gets multiplied by 1.05 to the 12th.
2361
04:24:22,510 --> 04:24:30,648
After two years, that's 24 months, it's had\n
2362
04:24:30,648 --> 04:24:39,108
multiplied by 1.05 to the 24th power. Similarly,\n
2363
04:24:39,109 --> 04:24:49,890
amount will be 1200 times 1.05 to the 36th\n
2364
04:24:49,889 --> 04:24:59,939
12 t months. So the interest will be compounded\n
2365
04:24:59,939 --> 04:25:08,120
To the 12 t power. This gives us the general\n
2366
04:25:08,120 --> 04:25:20,399
1200 times 1.05 to the 12 T, where T is the\n
2367
04:25:20,398 --> 04:25:31,398
years, we'll have to pay back a total of 1200\n
2368
04:25:31,398 --> 04:25:42,799
which works out to $1,436.02 to the nearest\n
2369
04:25:42,799 --> 04:25:51,938
pattern. If A is the initial amount of the\n
2370
04:25:51,939 --> 04:26:03,529
interest rate, compounded n times per year,\n
2371
04:26:03,529 --> 04:26:13,620
going to be a times one plus r over n to the\n
2372
04:26:13,620 --> 04:26:20,680
did in this problem. First, we took the interest\n
2373
04:26:20,680 --> 04:26:25,838
number of compounding periods each year 12.\n
2374
04:26:25,838 --> 04:26:35,059
it. That's where we got the 1.05 from. We\n
2375
04:26:35,059 --> 04:26:39,818
but to 12 times the number of years. That's\n
2376
04:26:39,818 --> 04:26:46,628
times the number of years. And we multiplied\n
2377
04:26:46,629 --> 04:26:52,979
was 1200. This formula for compound interest\n
2378
04:26:52,978 --> 04:26:57,408
to be able to reason your way through it,\n
2379
04:26:57,408 --> 04:27:02,908
type of compound interest. And that's interest\n
2380
04:27:02,908 --> 04:27:09,950
continuous compounding as the limit of compounding\n
2381
04:27:09,950 --> 04:27:14,969
100 times a year 1000 times a year a million\n
2382
04:27:14,969 --> 04:27:24,809
compounding. The formula for continuous compounding\n
2383
04:27:24,809 --> 04:27:37,269
p of t is the amount of money, t is the time\n
2384
04:27:37,270 --> 04:27:49,120
And R is the annual interest rate written\n
2385
04:27:49,120 --> 04:27:57,000
the 2.5% annual interest rate. He represents\n
2386
04:27:57,000 --> 04:28:12,059
is about 2.718. So in this problem, we have\n
2387
04:28:12,059 --> 04:28:21,549
after five years, we'll have P of five, which\n
2388
04:28:21,549 --> 04:28:32,228
which works out to $4,532.59 to the nearest\n
2389
04:28:32,228 --> 04:28:43,398
interest rate written as a decimal, that is\n
2390
04:28:43,398 --> 04:28:50,719
of years and a represents the initial amount\n
2391
04:28:50,719 --> 04:29:00,608
compounded once a year. Our formula is P of\n
2392
04:29:00,609 --> 04:29:08,309
interest compounded n times per year. Our\n
2393
04:29:08,309 --> 04:29:16,930
n to the n T. And for compound interest compounded\n
2394
04:29:18,430 --> 04:29:24,670
In this video, we looked at three kinds of\n
2395
04:29:24,670 --> 04:29:33,908
interest, interest compounded and times per\n
2396
04:29:33,908 --> 04:29:42,118
This video introduces logarithms. logarithms\n
2397
04:29:42,119 --> 04:29:52,949
log base a of B equals c means that a to the\n
2398
04:29:52,949 --> 04:30:02,119
B is the exponent that you raise a to to get\n
2399
04:30:02,119 --> 04:30:07,489
logarithm. It's also called the base when\n
2400
04:30:07,488 --> 04:30:14,139
students find it helpful to remember this\n
2401
04:30:14,139 --> 04:30:17,778
a to the C equals b, by drawing arrows
2402
04:30:20,510 --> 04:30:30,779
Other students like to think of it in terms\n
2403
04:30:30,779 --> 04:30:41,439
What power do you raise a to in order to get\n
2404
04:30:41,439 --> 04:30:50,408
of eight is three, because two to the three\n
2405
04:30:50,408 --> 04:30:59,359
y is asking you the question, What power do\n
2406
04:30:59,359 --> 04:31:08,130
log base two of 16 is four, because it's asking\n
2407
04:31:08,129 --> 04:31:15,128
And the answer is four. Please pause the video\n
2408
04:31:15,129 --> 04:31:22,079
base two of two is asking, What power do you\n
2409
04:31:22,079 --> 04:31:34,369
one. Two to the one equals two. log base two\n
2410
04:31:34,369 --> 04:31:40,790
you one half? Well, to get one half, you need\n
2411
04:31:40,790 --> 04:31:49,699
would be two to the negative one. So the answer\n
2412
04:31:49,699 --> 04:31:58,658
what power do we raise to two in order to\n
2413
04:31:58,658 --> 04:32:06,418
we have to raise two to the negative three\n
2414
04:32:06,418 --> 04:32:13,689
is negative three. And that's our answer to\n
2415
04:32:13,689 --> 04:32:21,979
of one is asking to what power equals one.\n
2416
04:32:21,978 --> 04:32:28,918
us one, so this log expression evaluates to\n
2417
04:32:28,918 --> 04:32:36,299
and zero answers for our logarithm expressions.\n
2418
04:32:36,299 --> 04:32:46,929
these logs evaluate to. to work out log base\n
2419
04:32:46,930 --> 04:32:54,078
10 to the sixth power. Now we're asking the\n
2420
04:32:54,078 --> 04:32:59,748
get a million? So that is what power do we\n
2421
04:32:59,748 --> 04:33:11,030
course, the answer is going to be six. Similarly,\n
2422
04:33:11,030 --> 04:33:16,271
this log expression is the same thing as asking,\n
2423
04:33:16,271 --> 04:33:22,859
three? Well, what power do you have to raise\n
2424
04:33:22,859 --> 04:33:31,020
the answer is negative three. Log base 10\n
2425
04:33:31,020 --> 04:33:38,189
10 to to get zero. If you think about it,\n
2426
04:33:38,188 --> 04:33:43,468
get zero. Raising 10 to a positive exponent\n
2427
04:33:43,469 --> 04:33:49,840
10 to a negative exponent is like one over\n
2428
04:33:49,840 --> 04:33:55,000
but they're still positive numbers, we're\n
2429
04:33:55,000 --> 04:33:59,919
10 to the zero power, we'll just get one.\n
2430
04:33:59,919 --> 04:34:05,599
base 10 of zero does not exist. If you try\n
2431
04:34:05,599 --> 04:34:11,159
button, you'll get an error message. Same\n
2432
04:34:11,159 --> 04:34:17,109
100. We're asking 10 to what power equals\n
2433
04:34:17,109 --> 04:34:23,170
will work. And more generally, it's possible\n
2434
04:34:23,169 --> 04:34:29,929
than zero, but not for numbers that are less\n
2435
04:34:29,930 --> 04:34:36,309
domain of the function log base a of x, no\n
2436
04:34:36,309 --> 04:34:44,131
is going to be all positive numbers. A few\n
2437
04:34:44,131 --> 04:34:50,729
called natural log, and it means the log base\n
2438
04:34:50,729 --> 04:34:59,041
about 2.718. When you see log of x with no\n
2439
04:34:59,041 --> 04:35:05,770
base 10 of x And it's called the common log.\n
2440
04:35:05,770 --> 04:35:13,449
natural log, and for common log. Let's practice\n
2441
04:35:13,449 --> 04:35:20,949
base three of one nine is negative two, can\n
2442
04:35:26,099 --> 04:35:34,750
Log of 13 is shorthand for log base 10 of\n
2443
04:35:34,750 --> 04:35:45,760
1.11394 equals 13. Finally, in this last expression,\n
2444
04:35:45,760 --> 04:35:52,878
can rewrite this equation as log base e of\n
2445
04:35:52,879 --> 04:36:00,898
means the same thing as e to the negative\n
2446
04:36:00,898 --> 04:36:05,559
let's go the opposite direction. We'll start\n
2447
04:36:05,559 --> 04:36:14,600
as logs. Remember that log base a of B equals\n
2448
04:36:14,599 --> 04:36:23,979
b, the base stays the same in both expressions.\n
2449
04:36:23,979 --> 04:36:29,317
the exponential equation, that's going to\n
2450
04:36:29,317 --> 04:36:33,438
just have to figure out what's in the argument\n
2451
04:36:33,438 --> 04:36:40,519
of the equal sign. Remember that the answer\n
2452
04:36:40,520 --> 04:36:48,067
goes in this box should be my exponent for\n
2453
04:36:48,067 --> 04:36:57,919
And I'll put the 9.78 as the argument of my\n
2454
04:36:57,919 --> 04:37:06,108
9.78 equals view means the same thing as three\n
2455
04:37:06,109 --> 04:37:13,831
started with. In the second example, the base\n
2456
04:37:13,830 --> 04:37:22,051
of my log is going to be the answer to my\n
2457
04:37:22,051 --> 04:37:32,430
3x plus seven. And the other expression, the\n
2458
04:37:32,430 --> 04:37:41,638
Let me check, log base e of four minus y equals\n
2459
04:37:41,638 --> 04:37:47,159
equals four minus y, which is just what I\n
2460
04:37:47,159 --> 04:37:56,409
e as natural log. This video introduced the\n
2461
04:37:56,409 --> 04:38:07,618
of B equal c means the same thing as a to\n
2462
04:38:07,618 --> 04:38:16,669
you the question, What power exponent Do you\n
2463
04:38:16,669 --> 04:38:22,968
we'll work out the graph, so some log functions\n
2464
04:38:22,969 --> 04:38:29,240
first example, let's graph a log function\n
2465
04:38:29,240 --> 04:38:36,648
we're working with is y equals log base two\n
2466
04:38:36,648 --> 04:38:41,229
Since we're working this out by hand, I want\n
2467
04:38:41,229 --> 04:38:47,879
log base two of x. So I'll start out with\n
2468
04:38:47,879 --> 04:38:56,789
one is zero, log base anything of one is 02\n
2469
04:38:56,789 --> 04:39:03,809
log base two of two, that's asking, What power\n
2470
04:39:03,809 --> 04:39:11,969
is one. Power other powers of two are easy\n
2471
04:39:11,969 --> 04:39:18,340
of four that saying what power do I raise\n
2472
04:39:18,340 --> 04:39:26,349
Similarly, log base two of eight is three\n
2473
04:39:26,349 --> 04:39:33,919
work with some fractional values for X. If\n
2474
04:39:33,919 --> 04:39:39,329
that saying what power do I raise to two to\n
2475
04:39:39,330 --> 04:39:48,430
negative one. It's also easy to compute by\n
2476
04:39:48,430 --> 04:39:57,740
base two of 1/4 is negative two since two\n
2477
04:39:57,740 --> 04:40:04,850
log base two of one eight is negative f3 I'll\n
2478
04:40:04,849 --> 04:40:10,817
pause the video and take a moment to plot\n
2479
04:40:10,817 --> 04:40:17,579
one zero, that's here to one that's here,\nfor two
2480
04:40:17,580 --> 04:40:26,190
that is here, and then eight, three, which\n
2481
04:40:26,189 --> 04:40:33,457
half goes with negative one, and 1/4 with\n
2482
04:40:33,457 --> 04:40:39,739
if I connect the dots, I get a graph that\n
2483
04:40:39,740 --> 04:40:44,700
and smaller fractions, I would keep getting\n
2484
04:40:44,700 --> 04:40:51,200
log base two of them, so my graph is getting\n
2485
04:40:51,200 --> 04:40:56,920
more and more negative, as x is getting close\n
2486
04:40:56,919 --> 04:41:02,189
graph over here with negative X values, I\n
2487
04:41:02,189 --> 04:41:08,599
that omission is no accident. Because if you\n
2488
04:41:08,599 --> 04:41:15,829
of a negative number, like say negative four\n
2489
04:41:15,830 --> 04:41:24,951
exist because there's no power that you can\n
2490
04:41:24,951 --> 04:41:29,360
are no points on the graph for negative X\n
2491
04:41:29,360 --> 04:41:35,549
on the graph where x is zero, because you\n
2492
04:41:35,549 --> 04:41:42,378
power you can raise to two to get zero. I\n
2493
04:41:42,378 --> 04:41:51,297
graph. First of all, the domain is x values\n
2494
04:41:51,297 --> 04:41:58,029
can write that as a round bracket because\n
2495
04:41:58,029 --> 04:42:05,039
the range is going to be the y values, while\n
2496
04:42:05,040 --> 04:42:11,069
of the negative numbers. And the graph gradually\n
2497
04:42:11,069 --> 04:42:17,279
So the range is actually all real numbers\n
2498
04:42:17,279 --> 04:42:25,590
to infinity. Finally, I want to point out\n
2499
04:42:25,590 --> 04:42:35,887
the y axis, that is at the line x equals zero.\n
2500
04:42:35,887 --> 04:42:42,569
A vertical asymptote is a line that our functions\n
2501
04:42:42,569 --> 04:42:50,189
the graph of y equals log base two of x. But\n
2502
04:42:50,189 --> 04:42:57,250
10 of x, it would look very similar, it would\n
2503
04:42:57,250 --> 04:43:02,878
zero, a range of all real numbers and a vertical\n
2504
04:43:02,878 --> 04:43:10,360
through the point one zero, but it would go\n
2505
04:43:10,360 --> 04:43:20,159
log base 10 of 10 is one, it would look pretty\n
2506
04:43:20,159 --> 04:43:24,930
But even though it doesn't look like it with\n
2507
04:43:24,930 --> 04:43:34,950
goes up to n towards infinity. In fact, the\n
2508
04:43:34,950 --> 04:43:42,387
bigger than one looks pretty much the same,\n
2509
04:43:42,387 --> 04:43:46,969
we know what the basic log graph looks like,\n
2510
04:43:46,970 --> 04:43:52,968
log functions without plotting points. Here\n
2511
04:43:52,968 --> 04:43:57,750
five. And again, I'm just going to draw a\n
2512
04:43:57,750 --> 04:44:02,477
graph, I probably would want to plot some\n
2513
04:44:02,477 --> 04:44:09,930
if it was just like y equals ln of x, that\n
2514
04:44:09,930 --> 04:44:17,271
go through the point one zero, with a vertical\n
2515
04:44:17,271 --> 04:44:23,389
a graph, ln of x plus five, that just shifts\n
2516
04:44:23,389 --> 04:44:27,450
the same vertical asymptote. Since the vertical\n
2517
04:44:27,450 --> 04:44:33,119
a vertical line, but instead of going through\n
2518
04:44:33,119 --> 04:44:42,680
five. So I'll draw a rough sketch here. Let's\n
2519
04:44:42,680 --> 04:44:51,080
x and the transformed version y equals ln\n
2520
04:44:51,080 --> 04:44:59,750
and the vertical asymptote. Our original function\n
2521
04:44:59,750 --> 04:45:08,659
Since adding five on the outside affects the\n
2522
04:45:08,659 --> 04:45:16,340
this transformation doesn't change the domain.\n
2523
04:45:16,340 --> 04:45:22,529
Now the range of our original y equals ln\n
2524
04:45:22,529 --> 04:45:28,680
Shifting up by five does affect the y values,\n
2525
04:45:28,680 --> 04:45:33,260
But since the original range was all real\n
2526
04:45:33,259 --> 04:45:38,147
real numbers, you still get the set of all\n
2527
04:45:38,148 --> 04:45:43,958
change either. And finally, we already saw\n
2528
04:45:43,957 --> 04:45:49,647
y axis x equals zero, when we shift that up\n
2529
04:45:49,648 --> 04:45:57,490
x equals zero. In this next example, we're\n
2530
04:45:57,490 --> 04:46:02,978
since the plus two is on the inside, that\n
2531
04:46:02,977 --> 04:46:09,718
I'll draw our basic log function. Here's our\n
2532
04:46:09,718 --> 04:46:15,317
as y equals log of x going through the point\none, zero
2533
04:46:15,317 --> 04:46:23,000
here's its vertical asymptote. Now I need\n
2534
04:46:23,000 --> 04:46:28,610
asymptote shifts left, and now it's at the\n
2535
04:46:28,610 --> 04:46:37,090
x equals zero, and my graph, let's see my\n
2536
04:46:37,090 --> 04:46:45,270
negative one zero, since I'm subtracting two\n
2537
04:46:45,270 --> 04:46:54,069
the resulting graph. Let's compare the features\n
2538
04:46:54,069 --> 04:47:01,207
about domains, the original had a domain of\n
2539
04:47:01,207 --> 04:47:07,119
that left. So I've subtracted two from all\n
2540
04:47:07,119 --> 04:47:15,227
I can also verify just by looking at the picture.\n
2541
04:47:15,227 --> 04:47:20,218
to infinity, well, shifting left only affects\n
2542
04:47:20,218 --> 04:47:26,690
range. So my range is still negative infinity\n
2543
04:47:26,689 --> 04:47:32,967
at x equals zero. And since I subtract two\n
2544
04:47:32,968 --> 04:47:38,190
negative two. In this last problem, I'm not\n
2545
04:47:38,189 --> 04:47:44,567
just use algebra to compute its domain. So\n
2546
04:47:44,567 --> 04:47:50,079
taking the logs of things? Well, you can't\n
2547
04:47:50,080 --> 04:47:56,010
So whatever's inside the argument of the log\n
2548
04:47:56,009 --> 04:48:03,269
better be greater than zero. So I'll write\n
2549
04:48:03,270 --> 04:48:08,797
than zero. Now it's a matter of solving an\n
2550
04:48:08,797 --> 04:48:15,759
3x. So two thirds is greater than x. In other\n
2551
04:48:15,759 --> 04:48:23,340
our domain is all the x values from negative\n
2552
04:48:23,340 --> 04:48:31,420
thirds. It's a good idea to memorize the basic\n
2553
04:48:31,419 --> 04:48:36,759
something like this, go through the point\n
2554
04:48:36,759 --> 04:48:44,798
the y axis. Also, if you remember that you\n
2555
04:48:44,798 --> 04:48:52,878
zero, then that helps you quickly compute\n
2556
04:48:52,878 --> 04:49:01,779
the log function, you set that greater than\n
2557
04:49:01,779 --> 04:49:07,500
logs and exponents. Please pause the video\n
2558
04:49:07,500 --> 04:49:15,707
evaluate the following four expressions. Remember,\n
2559
04:49:15,707 --> 04:49:24,817
log button. While log base e on your calculator\n
2560
04:49:24,817 --> 04:49:34,409
that the log base 10 of 10 cubed is three.\n
2561
04:49:34,409 --> 04:49:45,387
the log base 10 of 1000 is 1000. And eat the\n
2562
04:49:45,387 --> 04:49:51,718
log and the exponential function with the\n
2563
04:49:51,718 --> 04:49:59,420
the exponent. In fact, it's true that for\n
2564
04:49:59,419 --> 04:50:05,289
equal to x, the same sort of cancellation\n
2565
04:50:05,290 --> 04:50:10,770
in the log function with the same base in\n
2566
04:50:10,770 --> 04:50:15,869
to the power of log base 10 of 1000, the 10\n
2567
04:50:15,869 --> 04:50:21,770
other, and we're left with the 1000s. This\n
2568
04:50:21,770 --> 04:50:31,067
a of x is equal to x. We can describe this\n
2569
04:50:31,067 --> 04:50:40,169
a log function with the same base undo each\n
2570
04:50:40,169 --> 04:50:46,869
of inverse functions, the exponential function\n
2571
04:50:46,869 --> 04:50:55,289
these roles hold for the first log role. log\n
2572
04:50:55,290 --> 04:51:03,270
power do we raise a two in order to get a\n
2573
04:51:03,270 --> 04:51:11,430
is a dx? Well, the answer is clearly x. And\n
2574
04:51:15,290 --> 04:51:25,120
notice that the log base a of x means the\n
2575
04:51:25,119 --> 04:51:30,137
is saying that we're supposed to raise a to\n
2576
04:51:30,137 --> 04:51:37,128
need to raise a two to get x, then we'll certainly\n
2577
04:51:37,128 --> 04:51:44,610
examples. If we want to find three to the\n
2578
04:51:44,610 --> 04:51:51,860
base three undo each other, so we're left\n
2579
04:51:51,860 --> 04:51:59,060
Remember that ln means log base e. So we're\n
2580
04:51:59,060 --> 04:52:05,398
functions undo each other, and we're left\n
2581
04:52:05,398 --> 04:52:12,600
three z, remember that log without a base\n
2582
04:52:12,599 --> 04:52:18,750
we want to take 10 to the log base 10 of three\n
2583
04:52:18,750 --> 04:52:26,409
each other. So we're left with a three z.\n
2584
04:52:26,409 --> 04:52:36,419
10 to the x equal to x, well, ln means log\n
2585
04:52:36,419 --> 04:52:41,159
the x, notice that the base of the log and\n
2586
04:52:41,159 --> 04:52:47,939
the same. So they don't undo each other. And\n
2587
04:52:47,939 --> 04:52:53,919
usually equal to x, we can check with one\n
2588
04:52:53,919 --> 04:52:59,629
e of 10 to the one, that's log base e of 10.\n
2589
04:52:59,630 --> 04:53:06,558
equal to 2.3. And some more decimals, which\n
2590
04:53:06,558 --> 04:53:13,900
is false, it does not hold. We need the basis\n
2591
04:53:13,900 --> 04:53:23,000
each other. In this video, we saw that logs\n
2592
04:53:26,250 --> 04:53:33,779
log base a of a to the x is equal to x and\n
2593
04:53:34,779 --> 04:53:42,860
for any values of x and any base a. This video\n
2594
04:53:42,860 --> 04:53:46,779
log rules are closely related to the exponent\n
2595
04:53:46,779 --> 04:53:51,378
the exponent rules. To keep things simple,\n
2596
04:53:51,378 --> 04:53:57,297
two. Even though the exponent rules hold for\n
2597
04:53:57,297 --> 04:54:04,039
the zero power, we get one, we have a product\n
2598
04:54:04,040 --> 04:54:12,120
the M times two to the n is equal to two to\n
2599
04:54:12,119 --> 04:54:18,590
two numbers, then we add the exponents. We\n
2600
04:54:18,590 --> 04:54:26,590
to the M divided by n to the n is equal to\n
2601
04:54:26,590 --> 04:54:34,520
that if we divide two numbers, then we subtract\n
2602
04:54:34,520 --> 04:54:43,317
that says if we take a power to a power, then\n
2603
04:54:43,317 --> 04:54:50,270
rules can be rewritten as a log rule. The\n
2604
04:54:50,270 --> 04:54:57,990
be rewritten in terms of logs as log base\n
2605
04:54:57,990 --> 04:55:05,090
base two of one equals zero mean To the zero\n
2606
04:55:05,090 --> 04:55:14,990
can be rewritten in terms of logs by saying\n
2607
04:55:14,990 --> 04:55:21,990
of y. I'll make these base two to agree with\n
2608
04:55:21,990 --> 04:55:28,860
In words, that says the log of the product\n
2609
04:55:28,860 --> 04:55:34,637
represent exponent, this is saying that when\n
2610
04:55:34,637 --> 04:55:41,329
their exponents, which is just what we said\n
2611
04:55:41,330 --> 04:55:48,830
for exponents can be rewritten in terms of\n
2612
04:55:48,830 --> 04:55:58,160
equal to the log of x minus the log of y.\n
2613
04:55:58,159 --> 04:56:04,968
is equal to the difference of the logs. Since\n
2614
04:56:04,968 --> 04:56:11,770
saying the same thing is that when you divide\n
2615
04:56:11,770 --> 04:56:18,600
That's how we described the exponent rule\n
2616
04:56:18,599 --> 04:56:27,317
can be rewritten in terms of logs by saying\n
2617
04:56:27,317 --> 04:56:34,090
log of x. Sometimes people describe this rule\n
2618
04:56:34,090 --> 04:56:40,148
with an exponent, you can bring down the exponent\n
2619
04:56:40,148 --> 04:56:45,409
power of two, this is really saying when we\n
2620
04:56:45,409 --> 04:56:50,340
exponents. That's exactly how we described\n
2621
04:56:50,340 --> 04:56:56,819
if you multiply this exponent on the left\n
2622
04:56:56,819 --> 04:57:03,189
traditional to multiply it on the left side.\n
2623
04:57:03,189 --> 04:57:07,887
but they actually work for any base. To help\n
2624
04:57:07,887 --> 04:57:17,509
write out the log roles using a base of a\n
2625
04:57:17,509 --> 04:57:23,137
use the log rules to rewrite the following\n
2626
04:57:23,137 --> 04:57:30,369
In the first expression, we have a log base\n
2627
04:57:30,369 --> 04:57:39,307
of the quotient as the difference of the logs.\n
2628
04:57:39,308 --> 04:57:47,727
can rewrite the log of a product as the sum\n
2629
04:57:47,727 --> 04:57:55,099
of z. When I put things together, I have to\n
2630
04:57:55,099 --> 04:58:02,529
entire log expression. So I need to subtract\n
2631
04:58:02,529 --> 04:58:08,270
do that by putting them in parentheses. Now\n
2632
04:58:08,270 --> 04:58:16,340
the negative sign. And here's my final answer.\n
2633
04:58:16,340 --> 04:58:21,909
product. So I can rewrite that as the sum\nof two logs.
2634
04:58:21,909 --> 04:58:31,042
I can also use my power rule to bring down\n
2635
04:58:31,042 --> 04:58:38,909
That gives me the final expression log of\n
2636
04:58:38,909 --> 04:58:47,968
on this problem is to rewrite this expression\n
2637
04:58:47,968 --> 04:58:54,218
those two expressions are not equal. Because\n
2638
04:58:54,218 --> 04:58:59,729
whole five times two, we can't just bring\n
2639
04:58:59,729 --> 04:59:08,349
all, the power rule only applies to a single\n
2640
04:59:08,349 --> 04:59:14,457
to a product like this. And these next examples,\n
2641
04:59:14,457 --> 04:59:18,729
we're given sums and differences of logs.\n
2642
04:59:18,729 --> 04:59:24,989
log expression. By look at the first two pieces,\n
2643
04:59:24,990 --> 04:59:34,978
rewrite it as the log of a quotient. Now I\n
2644
04:59:34,977 --> 04:59:43,889
that as the log of a product. I'll clean that\n
2645
04:59:43,889 --> 04:59:54,610
five of a times c over B. In my second example,\n
2646
04:59:54,610 --> 05:00:03,369
of a product now, I will Like to rewrite this\n
2647
05:00:03,369 --> 05:00:09,817
But I can't do it yet, because of that factor\n
2648
05:00:09,817 --> 05:00:15,669
the power rule backwards to put that two back\n
2649
05:00:15,669 --> 05:00:23,909
So I will copy down the ln of x plus one times\n
2650
05:00:23,909 --> 05:00:31,968
as ln of x squared minus one squared. Now\n
2651
05:00:31,968 --> 05:00:40,990
logs, which I can rewrite as the log of a\n
2652
05:00:40,990 --> 05:00:50,030
more. Since x plus one times x minus one is\n
2653
05:00:50,029 --> 05:01:00,119
cancel factors to get ln of one over x squared\n
2654
05:01:00,119 --> 05:01:07,329
for logs that are related to exponent rules.\n
2655
05:01:07,330 --> 05:01:15,330
one is equal to zero. Second, we saw the product\n
2656
05:01:15,330 --> 05:01:23,968
sum of the logs. We saw the quotient rule,\n
2657
05:01:23,968 --> 05:01:30,639
the logs. And we saw the power rule. When\n
2658
05:01:30,639 --> 05:01:37,200
in it, you can bring down the exponent and\n
2659
05:01:37,200 --> 05:01:44,137
no log rule that helps you split up the log\n
2660
05:01:44,137 --> 05:01:52,489
is not equal to the sum of the logs. If you\n
2661
05:01:52,490 --> 05:02:00,740
together, this kind of makes sense, because\n
2662
05:02:00,740 --> 05:02:04,450
of two exponential expressions.
2663
05:02:04,450 --> 05:02:10,308
Log rules will be super handy, as we start\n
2664
05:02:10,308 --> 05:02:17,760
have an equation like this one that has variables\n
2665
05:02:17,759 --> 05:02:23,387
for getting those variables down where you\n
2666
05:02:23,387 --> 05:02:29,797
a few examples of solving equations with variables\n
2667
05:02:29,797 --> 05:02:37,250
solve for x the equation five times two to\n
2668
05:02:37,250 --> 05:02:42,707
I'm going to isolate the difficult spot, the\n
2669
05:02:42,707 --> 05:02:47,889
In this example, I can do that by dividing\n
2670
05:02:47,889 --> 05:02:56,250
x plus one equals 17 over five. Next, I'm\n
2671
05:02:56,250 --> 05:03:00,919
possible to take the log with any base, but\n
2672
05:03:00,919 --> 05:03:06,149
base e for the simple reason that my calculator\n
2673
05:03:06,150 --> 05:03:12,610
I'll just take the log base 10. So I can omit\n
2674
05:03:12,610 --> 05:03:19,500
here. And that gives me this expression. As\n
2675
05:03:19,500 --> 05:03:24,547
bring down my exponent and multiply it on\n
2676
05:03:24,547 --> 05:03:31,217
here because the entire x plus one needs to\n
2677
05:03:31,218 --> 05:03:37,409
third step using the log roles. Now all my\n
2678
05:03:37,409 --> 05:03:42,040
I can work with them, but I still need to\n
2679
05:03:42,040 --> 05:03:47,100
parentheses. So I'm going to free it from\n
2680
05:03:47,099 --> 05:03:55,887
x log two plus log two equals log of 17 fifths.\n
2681
05:03:55,887 --> 05:04:01,520
my terms with x's in them to one side, and\n
2682
05:04:01,520 --> 05:04:08,850
side. Finally, I factor out my x. Well, it's\n
2683
05:04:08,849 --> 05:04:17,797
to isolate it. So I read out what I did. So\n
2684
05:04:17,797 --> 05:04:24,739
on one side, and the terms without x's on\n
2685
05:04:24,740 --> 05:04:31,830
by factoring out and dividing. We have an\n
2686
05:04:31,830 --> 05:04:36,830
maybe not so useful if you want a decimal\n
2687
05:04:36,830 --> 05:04:46,870
into your calculator using parentheses liberally\n
2688
05:04:46,869 --> 05:04:51,829
always a good idea to check your work by typing\n
2689
05:04:51,830 --> 05:04:57,540
checks out. This next example is trickier\n
2690
05:04:57,540 --> 05:05:05,180
in two places with two different bases. First\n
2691
05:05:05,180 --> 05:05:10,207
by isolating the tricky stuff. But in this\n
2692
05:05:10,207 --> 05:05:15,590
or simplifier, or no way to isolate anything\n
2693
05:05:15,590 --> 05:05:20,400
to the next step and take the log of both\n
2694
05:05:20,400 --> 05:05:27,548
10. But we couldn't use log base e instead.\n
2695
05:05:27,547 --> 05:05:35,789
exponents. This gives me 2x minus three in\n
2696
05:05:35,790 --> 05:05:43,138
five. Now I'm going to distribute things out\n
2697
05:05:43,137 --> 05:05:52,950
gives me 2x log two minus three log two equals\n
2698
05:05:52,950 --> 05:05:59,350
to group the x terms on one side, and the\n
2699
05:05:59,349 --> 05:06:08,099
side. So I'll keep the 2x log two on the left,\n
2700
05:06:08,099 --> 05:06:13,759
and that gives me the minus two log five that\n
2701
05:06:13,759 --> 05:06:22,799
on the right. Finally, I need to isolate x\n
2702
05:06:22,799 --> 05:06:27,009
mean I factor out the x from all the terms\n
2703
05:06:27,009 --> 05:06:37,989
quantity to log two minus log five. And that\n
2704
05:06:37,990 --> 05:06:48,540
the right side by the quantity on the left\n
2705
05:06:48,540 --> 05:06:53,850
I encourage you to type the whole thing in\n
2706
05:06:53,849 --> 05:06:58,159
if you round off, you'll get a less accurate\n
2707
05:06:58,159 --> 05:07:06,648
at once. In this example, when I type it in,\n
2708
05:07:06,648 --> 05:07:12,670
In this equation, we have the variable t in\n
2709
05:07:12,669 --> 05:07:19,500
not a variable, it represents the number e\n
2710
05:07:19,500 --> 05:07:24,148
Because there's already an E and the expression,\n
2711
05:07:24,148 --> 05:07:29,468
in this problem instead of log base 10. But\n
2712
05:07:29,468 --> 05:07:35,468
clean things up. by isolating the tricky parts,\n
2713
05:07:35,468 --> 05:07:42,409
And that will give us either the negative\n
2714
05:07:42,409 --> 05:07:47,799
T. One way to proceed would now be to clean\n
2715
05:07:47,799 --> 05:07:51,898
by either the point two t, but I'm going to\n
2716
05:07:51,898 --> 05:08:01,290
take the natural log of both sides. That gives\n
2717
05:08:01,290 --> 05:08:08,479
ln of three fifths e to the 0.2 t. Now on\n
2718
05:08:08,479 --> 05:08:18,099
rule to bring down my exponent and get minus\n
2719
05:08:18,099 --> 05:08:23,519
can't bring down the exponent yet because\n
2720
05:08:23,520 --> 05:08:28,797
three fifths. So before I can bring down the\n
2721
05:08:28,797 --> 05:08:36,110
using the product rule. So I can rewrite this\n
2722
05:08:36,110 --> 05:08:47,887
T. And now I can bring down the exponent.\n
2723
05:08:47,887 --> 05:08:56,128
ultimately bring down my exponents. Now ln\n
2724
05:08:56,128 --> 05:09:02,431
log base e of E. So that's asking what power\n
2725
05:09:02,431 --> 05:09:10,570
answer is one. So anytime I have ln of E,\n
2726
05:09:10,570 --> 05:09:14,990
using natural log is a little bit handier\n
2727
05:09:14,990 --> 05:09:23,770
that simplification. Next, I'm ready to solve\n
2728
05:09:23,770 --> 05:09:29,610
But I do need to bring my T terms to one side\n
2729
05:09:29,610 --> 05:09:37,779
let's see. I'll put my T terms on the left\n
2730
05:09:37,779 --> 05:09:45,619
And finally I'm going to isolate t by factoring\n
2731
05:09:45,619 --> 05:09:57,147
out my T \nand now I can divide. Using my calculator
2732
05:09:57,148 --> 05:10:04,718
I can get a decimal answer of 2.04 Three,\n
2733
05:10:04,718 --> 05:10:10,477
equations with variables in the exponent.\n
2734
05:10:10,477 --> 05:10:21,270
sides and use the log properties to bring\n
2735
05:10:21,270 --> 05:10:26,718
examples of equations with logs in them like\n
2736
05:10:26,718 --> 05:10:32,540
like this one, we have to free the variable\n
2737
05:10:32,540 --> 05:10:38,830
functions. My first step in solving pretty\n
2738
05:10:38,830 --> 05:10:44,000
and isolate the tricky part. In this case,\n
2739
05:10:44,000 --> 05:10:51,599
it. So I can isolate it by first adding three\n
2740
05:10:51,599 --> 05:11:01,180
five equals four, and then I can divide both\n
2741
05:11:01,180 --> 05:11:07,520
part, I still need to solve for x, but x is\n
2742
05:11:07,520 --> 05:11:13,200
I need to somehow undo the log function. Well\n
2743
05:11:13,200 --> 05:11:18,727
each other. And since this is a log base e,\n
2744
05:11:18,727 --> 05:11:26,270
e also. So I'm going to take e to the power\n
2745
05:11:26,270 --> 05:11:33,420
to the ln 2x plus five, and that's going to\n
2746
05:11:33,419 --> 05:11:40,189
Now e to the ln of anything, I'll just write\n
2747
05:11:40,189 --> 05:11:46,859
e of a, that you the power and log base e\n
2748
05:11:46,860 --> 05:11:53,200
to use that principle over here, e to the\n
2749
05:11:53,200 --> 05:11:59,030
e undo each other. And we're left with 2x\n
2750
05:11:59,029 --> 05:12:04,360
is equal to E squared. And from there, it's\n
2751
05:12:04,360 --> 05:12:11,047
by subtracting five from both sides and then\n
2752
05:12:11,047 --> 05:12:17,967
step here is to just say finish solving for\n
2753
05:12:17,968 --> 05:12:25,909
solving equations with logs in them. And that's\n
2754
05:12:25,909 --> 05:12:31,529
solutions. an extraneous solution is a solution\n
2755
05:12:31,529 --> 05:12:36,807
doesn't actually satisfy the original equation.\n
2756
05:12:36,808 --> 05:12:42,317
in them, because we might get a solution that\n
2757
05:12:42,317 --> 05:12:52,898
zero, and we can't take the log of a negative\n
2758
05:12:52,898 --> 05:13:01,020
solution of E squared minus five over two.\n
2759
05:13:01,020 --> 05:13:09,840
and see if that works. So let's see the twos\n
2760
05:13:09,840 --> 05:13:17,049
five plus five, minus three, I want that to\n
2761
05:13:17,049 --> 05:13:23,648
I have two ln e squared minus three that I\n
2762
05:13:23,649 --> 05:13:34,900
to work out because let's see, ln is log base\n
2763
05:13:34,900 --> 05:13:38,890
the question, What power do I raise e two\n
2764
05:13:38,889 --> 05:13:46,750
to the power of two to get e squared. So this\n
2765
05:13:46,750 --> 05:13:53,080
that equal one, four minus three does equal\n
2766
05:13:53,080 --> 05:13:58,780
have any problem with taking the log of negative\n
2767
05:13:58,779 --> 05:14:04,619
solutions. So this is our solution. The second\n
2768
05:14:04,619 --> 05:14:10,637
there's a log into places. Now notice that\n
2769
05:14:10,637 --> 05:14:16,340
So a base 10 is implied. So I'm already thinking\n
2770
05:14:16,340 --> 05:14:23,080
to want to take a 10 to the power of both\n
2771
05:14:23,080 --> 05:14:29,060
isolate the tricky part, but there's nothing\n
2772
05:14:29,060 --> 05:14:37,298
can't do it here. So we'll just jump right\n
2773
05:14:37,297 --> 05:14:46,887
of both sides. Okay, so that's going to give\n
2774
05:14:46,887 --> 05:14:52,389
plus log x, that whole thing is in the exponent\n
2775
05:14:52,389 --> 05:14:58,750
to do with the right side 10 to the one is\n
2776
05:15:00,930 --> 05:15:06,869
while remembering my exponent rules, I know\n
2777
05:15:06,869 --> 05:15:12,419
what happens when you multiply two things.\n
2778
05:15:12,419 --> 05:15:18,369
x plus three times 10 to the log x, right,\n
2779
05:15:18,369 --> 05:15:24,439
add the exponent, so these are the same. Okay,\n
2780
05:15:24,439 --> 05:15:31,797
base 10, those undo each other. And so this\n
2781
05:15:31,797 --> 05:15:38,967
three. Similarly, 10 to the log base, 10 of\n
2782
05:15:38,968 --> 05:15:46,030
by x, that's equal to 10. Now I have an equation\n
2783
05:15:46,029 --> 05:15:52,397
going to first multiply out to make it look\n
2784
05:15:52,398 --> 05:15:59,409
side. So is equal to zero. And, and now I\n
2785
05:15:59,409 --> 05:16:06,500
I think this one factors, it looks like X\n
2786
05:16:06,500 --> 05:16:14,060
to get x is negative five, or x is two. So\n
2787
05:16:14,060 --> 05:16:20,860
for x. Finally, we need to check our solutions\n
2788
05:16:20,860 --> 05:16:27,378
ones. So let's see if x equals negative five.\n
2789
05:16:27,378 --> 05:16:36,047
that says, I'm checking that log of negative\n
2790
05:16:36,047 --> 05:16:40,879
checking that's equal to one. Well, this is\n
2791
05:16:40,880 --> 05:16:48,540
giving you a queasy feeling too, because log\n
2792
05:16:48,540 --> 05:16:53,750
can't take the log of a negative number. Same\n
2793
05:16:53,750 --> 05:17:00,297
negative five is an extraneous solution, it\n
2794
05:17:00,297 --> 05:17:08,029
Let's check the other solution, x equals two.\n
2795
05:17:08,029 --> 05:17:15,360
plus three plus log of two is equal to one.\n
2796
05:17:15,360 --> 05:17:20,670
of negative numbers, or zero here, this should\n
2797
05:17:20,669 --> 05:17:27,423
we can see let's see this is log of five plus\n
2798
05:17:27,423 --> 05:17:34,540
my log rules, the sum of two logs is the log\n
2799
05:17:34,540 --> 05:17:41,148
we want that to equal one. And that's just\n
2800
05:17:41,148 --> 05:17:46,909
one because log base 10 of 10 says, What power\n
2801
05:17:46,909 --> 05:17:56,079
is one. So the second solution x equals two\n
2802
05:17:56,080 --> 05:18:00,350
Before I leave this problem, I do want to\n
2803
05:18:00,349 --> 05:18:05,659
approach. Some people like to start with the\n
2804
05:18:05,659 --> 05:18:12,790
to combine everything into one log expression.\n
2805
05:18:12,790 --> 05:18:18,388
that's the same as the log of a product, right,\n
2806
05:18:18,387 --> 05:18:25,849
x plus three times x, that equals one, then\n
2807
05:18:25,849 --> 05:18:32,717
of both sides. And as before, the 10 to the\n
2808
05:18:32,718 --> 05:18:40,690
and we get x plus three times x equals 10,\n
2809
05:18:40,689 --> 05:18:45,639
we ended up using exponent rules to rewrite\n
2810
05:18:45,639 --> 05:18:50,689
we use log rules to rewrite things. So the\n
2811
05:18:50,689 --> 05:18:54,849
equivalent, and they certainly will get us\n
2812
05:18:54,849 --> 05:19:02,137
examples of equations with logs in them and\n
2813
05:19:02,137 --> 05:19:13,279
use exponential functions to undo the log.\n
2814
05:19:13,279 --> 05:19:25,090
sides to undo natural log, and take 10 to\n
2815
05:19:27,560 --> 05:19:33,030
In this video, we'll use exponential equations\n
2816
05:19:33,029 --> 05:19:40,217
growth and radioactive decay. I'll also introduce\n
2817
05:19:40,218 --> 05:19:47,350
In this first example, let's suppose we invest\n
2818
05:19:47,349 --> 05:19:52,889
interest compounded once a year. How many\n
2819
05:19:52,889 --> 05:19:59,450
in it if you don't make any further deposits\n
2820
05:19:59,450 --> 05:20:05,540
six point 5% interest each year, that means\n
2821
05:20:05,540 --> 05:20:18,817
by 1.065. So after t years, my 1600 gets multiplied\n
2822
05:20:18,817 --> 05:20:30,308
function notation as f of t equals 1600 times\n
2823
05:20:30,308 --> 05:20:41,080
of money after t years. Now we're trying to\n
2824
05:20:41,080 --> 05:20:48,580
$2,000 is a amount of money. So that's an\n
2825
05:20:48,580 --> 05:20:55,290
for T the amount of time. So let me write\n
2826
05:20:55,290 --> 05:21:04,840
for t. Now to solve for t, I want to first\n
2827
05:21:04,840 --> 05:21:08,790
tricky part is the part with the exponential\n
2828
05:21:08,790 --> 05:21:18,190
1600. That gives me 2000 over 1600 equals\n
2829
05:21:18,189 --> 05:21:24,317
bit further as five fourths. Now that I've\n
2830
05:21:24,317 --> 05:21:29,727
going to be to take the log of both sides.\n
2831
05:21:29,727 --> 05:21:33,829
And I know that if I log take the log of both\n
2832
05:21:33,830 --> 05:21:40,850
down where I can solve for it. I think I'll\n
2833
05:21:40,849 --> 05:21:48,759
force equals ln of 1.065 to the T. Now by\n
2834
05:21:48,759 --> 05:21:56,919
I can bring that exponent t down and multiply\n
2835
05:21:56,919 --> 05:22:07,329
t just by dividing both sides by ln of 1.065.\n
2836
05:22:07,330 --> 05:22:17,308
t is approximately 3.54 years. And the next\n
2837
05:22:17,308 --> 05:22:24,290
that initially contains 1.5 million bacteria,\n
2838
05:22:24,290 --> 05:22:34,659
find the doubling time, the doubling time\n
2839
05:22:34,659 --> 05:22:41,110
to double in size. For example, the amount\n
2840
05:22:41,110 --> 05:22:47,540
million bacteria to 3 million bacteria would\n
2841
05:22:47,540 --> 05:22:56,558
an equation for the amount of bacteria. So\n
2842
05:22:56,558 --> 05:23:08,968
in millions, then my equation and T represents\n
2843
05:23:08,968 --> 05:23:19,887
by the initial amount of bacteria times the\n
2844
05:23:19,887 --> 05:23:25,360
my population of bacteria is growing by 12%\n
2845
05:23:25,360 --> 05:23:30,760
gets multiplied by one point 12. Since we're\n
2846
05:23:30,759 --> 05:23:39,309
for the t value when P of D will be twice\n
2847
05:23:39,310 --> 05:23:56,298
solve for t. As before, I'll start by isolating\n
2848
05:23:56,297 --> 05:24:08,237
bringing the T down. And finally solving for\n
2849
05:24:08,238 --> 05:24:19,450
write this as ln two over ln 1.12. Using my\n
2850
05:24:19,450 --> 05:24:32,387
interesting fact that doubling time only depends\n
2851
05:24:32,387 --> 05:24:37,137
initial population. In fact, I could have\n
2852
05:24:37,137 --> 05:24:41,647
knowing how many bacteria were in my initial\n
2853
05:24:41,648 --> 05:24:48,100
work. If I didn't know how many I started\n
2854
05:24:48,099 --> 05:24:55,989
times 1.12 to the t where a is our initial\n
2855
05:24:55,990 --> 05:25:01,121
Then if I want to figure out how long it takes\n
2856
05:25:01,120 --> 05:25:08,869
with a and double that I get to a. So I'll\n
2857
05:25:08,869 --> 05:25:19,297
solve for t. Notice that my A's cancel. And\n
2858
05:25:19,297 --> 05:25:32,099
the T down and solve for t, I get the exact\n
2859
05:25:32,099 --> 05:25:38,009
the initial population was, I didn't even\n
2860
05:25:38,009 --> 05:25:42,217
we're told the initial population, and we're\n
2861
05:25:42,218 --> 05:25:49,218
what percent the population increases each\n
2862
05:25:49,218 --> 05:25:53,440
multiply the population by each minute. So\n
2863
05:25:53,439 --> 05:25:59,340
that I want to use an equation of the form\n
2864
05:25:59,340 --> 05:26:04,920
to be the number of minutes, and y is going\n
2865
05:26:04,919 --> 05:26:11,459
my initial amount a is 350. So I can really\n
2866
05:26:11,459 --> 05:26:19,119
doubling time tells me that when 15 minutes\n
2867
05:26:19,119 --> 05:26:28,619
twice as big, or 700. plugging that into my\n
2868
05:26:28,619 --> 05:26:34,789
the 15. Now I need to solve for b. Let me\n
2869
05:26:34,790 --> 05:26:43,798
sides by 350. That gives me 700 over 350 equals\n
2870
05:26:43,797 --> 05:26:51,479
b to the 15th. To solve for B, I don't have\n
2871
05:26:51,479 --> 05:26:57,270
in the base not in the exponent, so I don't\n
2872
05:26:57,270 --> 05:27:03,439
the easiest way to solve this is just by taking\n
2873
05:27:03,439 --> 05:27:12,467
The 1/15 power. That's because if I take B\n
2874
05:27:12,468 --> 05:27:19,690
that gives me B to the one is equal to two\n
2875
05:27:19,689 --> 05:27:29,789
1/15, which as a decimal is approximately\n
2876
05:27:29,790 --> 05:27:34,170
if I'm doing a decimal approximation and these\n
2877
05:27:34,169 --> 05:27:38,467
of course, the most accurate thing is just\n
2878
05:27:38,468 --> 05:27:50,190
rewrite my equation is y equals 350 times\n
2879
05:27:50,189 --> 05:27:56,079
work this problem one more time. And this\n
2880
05:27:56,080 --> 05:28:04,370
y equals a times e to the RT. This is called\n
2881
05:28:04,369 --> 05:28:09,750
but it's actually an equivalent form to this\n
2882
05:28:09,750 --> 05:28:14,450
about why these two forms are equivalent at\n
2883
05:28:14,450 --> 05:28:21,950
to solve in this form. So I know that my initial\n
2884
05:28:21,950 --> 05:28:33,530
t is 15, my Y is 700. So I plug in 700 here\n
2885
05:28:33,529 --> 05:28:44,869
Again, I'm going to simplify things by dividing\n
2886
05:28:44,869 --> 05:28:51,840
e to the r times 15. This time my variable\n
2887
05:28:51,840 --> 05:28:56,830
want to take the log of both sides, I'm going\n
2888
05:28:56,830 --> 05:29:01,818
an E and my problem. So natural log and E\n
2889
05:29:01,817 --> 05:29:09,259
then common log with base 10 and E. So I take\n
2890
05:29:09,259 --> 05:29:14,789
exponent down. So that's our times 15 times\n
2891
05:29:14,790 --> 05:29:19,540
Because Elena V is asking what power do I\n
2892
05:29:19,540 --> 05:29:27,218
I get 15 r equals ln two. So r is equal to\n
2893
05:29:27,218 --> 05:29:35,250
in to my original equation as e to the ln\n
2894
05:29:35,250 --> 05:29:40,279
two equations were actually the same thing\n
2895
05:29:40,279 --> 05:29:49,099
that is if I start with this equation, and\n
2896
05:29:50,099 --> 05:29:58,090
to the tee. Well, I claim that this quantity\n
2897
05:29:58,090 --> 05:30:06,727
the 1/15. And in fact One way to see that\n
2898
05:30:06,727 --> 05:30:10,509
Right, that's the same, because every time\n
2899
05:30:10,509 --> 05:30:18,840
But what's Ed Oh, and two, E and ln undo each\n
2900
05:30:18,840 --> 05:30:24,887
1/15 to the T TA, the equations are really\n
2901
05:30:24,887 --> 05:30:30,897
to find two different versions of an exponential\n
2902
05:30:30,898 --> 05:30:38,738
b to the T, or the continuous growth one,\n
2903
05:30:38,738 --> 05:30:43,569
we're going to work with half life, half life\n
2904
05:30:43,569 --> 05:30:55,099
means the amount of time that it takes for\n
2905
05:30:55,099 --> 05:31:02,509
we originally started with, we're told that\n
2906
05:31:02,509 --> 05:31:11,009
5750 years. So that means it takes that long\n
2907
05:31:11,009 --> 05:31:16,807
decay, so that you just have half as much\n
2908
05:31:16,808 --> 05:31:21,718
So we're told a sample of bone that originally\n
2909
05:31:21,718 --> 05:31:29,670
14 now contains only 40 grams, we're supposed\n
2910
05:31:29,669 --> 05:31:36,957
carbon dating. Let's use the continuous growth\n
2911
05:31:36,957 --> 05:31:49,099
is our amount of radioactive c 14 is going\n
2912
05:31:49,099 --> 05:31:54,250
times e to the RT, we could have used the\n
2913
05:31:54,250 --> 05:31:59,468
a times b to the T, but I just want to use\n
2914
05:31:59,468 --> 05:32:13,659
that our half life is 5750. So what that means\n
2915
05:32:13,659 --> 05:32:19,290
be one half of what we started with. Let me\n
2916
05:32:19,290 --> 05:32:26,968
figure out use that to figure out what r is.\n
2917
05:32:26,968 --> 05:32:35,350
So I plug in one half a for the final amount,\n
2918
05:32:35,349 --> 05:32:46,279
and I have 5750 I can cancel my A's. And now\n
2919
05:32:46,279 --> 05:32:50,987
So I do need to take the log of both sides\n
2920
05:32:50,988 --> 05:32:57,120
e since I already have an E and my problem,\n
2921
05:32:57,119 --> 05:33:04,227
log base 10 is okay. Now, on the left side,\n
2922
05:33:04,227 --> 05:33:09,887
of one half on the right side, ln n e to a\n
2923
05:33:09,887 --> 05:33:20,649
R times 5750. Now I can solve for r, it's\n
2924
05:33:20,650 --> 05:33:25,909
decimal, but it's actually more accurate just\n
2925
05:33:25,909 --> 05:33:40,707
my equation, I have f of t equals a times\n
2926
05:33:40,707 --> 05:33:46,750
I can use that to figure out my problem. And\n
2927
05:33:46,750 --> 05:33:53,468
grams, that's my a, I want to figure out when\n
2928
05:33:53,468 --> 05:34:00,030
my final amount. And so I need to solve for\n
2929
05:34:00,029 --> 05:34:10,501
by 200. Let's say 40 over 200 is 1/5. Now\n
2930
05:34:10,501 --> 05:34:19,029
ln and e to the power undo each other. So\n
2931
05:34:19,029 --> 05:34:27,889
divided by five 750 T. And finally I can solve\n
2932
05:34:27,889 --> 05:34:35,509
calculator gives me an answer of 13,351 years\napproximately.
2933
05:34:35,509 --> 05:34:41,397
That kind of makes sense in terms of the half\n
2934
05:34:41,398 --> 05:34:45,638
to decrease by half a little more than two\n
2935
05:34:45,637 --> 05:34:50,659
get you to 100 decreasing to half again would\n
2936
05:34:50,659 --> 05:34:57,349
half lives is getting pretty close to 13,000\n
2937
05:34:57,349 --> 05:35:04,797
things it introduced continue Less growth\n
2938
05:35:04,797 --> 05:35:16,939
writing an exponential function. The relationship\n
2939
05:35:16,939 --> 05:35:27,619
thing as EDR. In that version, it also introduced\n
2940
05:35:27,619 --> 05:35:32,409
amount of time it takes a quantity to double\n
2941
05:35:32,409 --> 05:35:42,701
model. Recall that a linear equation is equation\n
2942
05:35:42,701 --> 05:35:47,990
an equation without any x squared or y squared\n
2943
05:35:47,990 --> 05:35:55,120
the form y equals mx plus b, the equation\n
2944
05:35:55,119 --> 05:36:03,669
a collection of two or more linear equations.\n
2945
05:36:03,669 --> 05:36:10,089
A solution to a system of equations is that\n
2946
05:36:10,090 --> 05:36:18,590
of the equations. For example, the ordered\n
2947
05:36:18,590 --> 05:36:26,950
equals three is a solution to this system.\n
2948
05:36:26,950 --> 05:36:33,968
three into the first equation, it checks out\n
2949
05:36:33,968 --> 05:36:40,170
one. And if I plug in x equals two and y equals\n
2950
05:36:40,169 --> 05:36:49,279
out two plus three equals five. However, the\n
2951
05:36:49,279 --> 05:36:55,849
y equals four is not a solution to the system.\n
2952
05:36:55,849 --> 05:37:02,707
second equation, since one plus four does\n
2953
05:37:02,707 --> 05:37:10,567
because two times one minus four is not equal\n
2954
05:37:10,567 --> 05:37:17,229
methods to find the solutions to systems of\n
2955
05:37:17,229 --> 05:37:22,181
want to solve this system of equations, there\n
2956
05:37:22,181 --> 05:37:29,900
use the method of substitution, or we could\n
2957
05:37:29,900 --> 05:37:38,128
method of substitution, the main idea is to\n
2958
05:37:38,128 --> 05:37:44,440
then substitute it in to the other equation.\n
2959
05:37:44,439 --> 05:37:53,807
3x minus two y equals four, and isolate the\n
2960
05:37:53,808 --> 05:38:01,080
dividing both sides by three. Think I'll rewrite\n
2961
05:38:01,080 --> 05:38:07,900
into two fractions four thirds plus two thirds\n
2962
05:38:07,900 --> 05:38:16,530
equation 5x plus six y equals two. And I'm\n
2963
05:38:16,529 --> 05:38:24,887
That gives me five times four thirds plus\n
2964
05:38:24,887 --> 05:38:30,750
I've got an equation with only one variable\n
2965
05:38:30,750 --> 05:38:38,830
First, I'm going to distribute the five so\n
2966
05:38:38,830 --> 05:38:45,548
six y equals two. And now I'm going to keep\n
2967
05:38:45,547 --> 05:38:51,547
on the left side, but I'll move all my terms\n
2968
05:38:51,547 --> 05:38:56,110
this point, I could just add up all my fractions\n
2969
05:38:56,110 --> 05:39:00,360
like working with fractions, I think I'll\n
2970
05:39:00,360 --> 05:39:05,968
here. So I'm going to actually multiply both\n
2971
05:39:05,968 --> 05:39:10,637
to get rid of the denominators and not have\n
2972
05:39:10,637 --> 05:39:20,237
down. Distributing the three, I get 10 y plus\n
2973
05:39:20,238 --> 05:39:28,290
things together. So that's 28 y equals negative\n
2974
05:39:28,290 --> 05:39:32,110
14 over 28, which is negative one half.
2975
05:39:32,110 --> 05:39:38,680
So I've solved for y. And now I can go back\n
2976
05:39:38,680 --> 05:39:47,977
solve for x, I plug it into my first equation.\n
2977
05:39:47,977 --> 05:39:55,058
That gives me 3x plus one equals four. So\n
2978
05:39:55,058 --> 05:40:01,370
to one. I've solved my system of equations\n
2979
05:40:01,369 --> 05:40:09,680
one half, I can also write that as an ordered\n
2980
05:40:09,680 --> 05:40:14,297
Now let's go back and solve the same system,\n
2981
05:40:14,297 --> 05:40:23,750
elimination, the key idea to the method of\n
2982
05:40:23,750 --> 05:40:34,718
a constant to make the coefficients of one\n
2983
05:40:34,718 --> 05:40:42,080
my two equations. Say I'm trying to make the\n
2984
05:40:42,080 --> 05:40:48,730
is to multiply the first equation by five,\n
2985
05:40:48,729 --> 05:40:55,739
the coefficient of x will be 15 for both equations,\n
2986
05:40:55,740 --> 05:41:00,950
I'm going to multiply both sides by five.\n
2987
05:41:00,950 --> 05:41:09,979
multiply both sides by three. That gives me\n
2988
05:41:09,979 --> 05:41:19,110
20. And for the second equation, 15x plus\n
2989
05:41:19,110 --> 05:41:25,378
the coefficients of x match. So if I subtract\n
2990
05:41:25,378 --> 05:41:32,779
term will completely go away, it'll be zero\n
2991
05:41:32,779 --> 05:41:44,259
10 y minus 18, y is going to give me minus\n
2992
05:41:44,259 --> 05:41:52,237
to give me 14. solving for y, I get y is 14\n
2993
05:41:52,238 --> 05:41:59,228
like before. Now we can continue, like we\n
2994
05:41:59,227 --> 05:42:06,290
that value of y into either one of the equations.\n
2995
05:42:06,290 --> 05:42:13,280
proceeds as before. So once again, I get the\n
2996
05:42:13,279 --> 05:42:20,599
one half. Before we go on to the next problem,\n
2997
05:42:20,599 --> 05:42:29,147
Here I've graphed the equations 3x minus two,\n
2998
05:42:29,148 --> 05:42:36,488
And we can see that these two lines intersect\n
2999
05:42:36,488 --> 05:42:42,388
one half, just like we predicted by solving\n
3000
05:42:42,387 --> 05:42:49,297
at another system of equations. I'm going\n
3001
05:42:49,297 --> 05:42:55,329
is on the left side with the y term, and the\n
3002
05:42:55,330 --> 05:43:02,580
rewrite or copy down the second equation.\n
3003
05:43:02,580 --> 05:43:08,270
is minus four, and the second equation is\n
3004
05:43:08,270 --> 05:43:14,119
elimination and multiply the first equation\n
3005
05:43:14,119 --> 05:43:19,009
That'll give me a coefficient of x of negative\n
3006
05:43:19,009 --> 05:43:23,799
second equation, those are equal and opposite,\n
3007
05:43:23,799 --> 05:43:31,619
equations to cancel out access. So let's do\n
3008
05:43:31,619 --> 05:43:39,009
plus 24, y equals three, and I'll put everything\n
3009
05:43:39,009 --> 05:43:46,859
everything by four. So that's 12x minus 24,\n
3010
05:43:46,860 --> 05:43:53,369
has happened here, not only do the x coefficients\n
3011
05:43:53,369 --> 05:44:00,169
Y coefficients do also. So if I add together\n
3012
05:44:00,169 --> 05:44:06,457
x term, I'm also going to cancel out the y\n
3013
05:44:06,457 --> 05:44:12,369
equal to three plus eight is 11. Well, that's\n
3014
05:44:12,369 --> 05:44:19,817
to 11. And that shows that these two equations\n
3015
05:44:19,817 --> 05:44:25,628
Let's look at this situation graphically.\n
3016
05:44:25,628 --> 05:44:31,440
they're parallel lines with the same slope.\n
3017
05:44:31,439 --> 05:44:36,727
each equation, the first equation, I get y\n
3018
05:44:36,727 --> 05:44:43,329
the same thing as four eighths or one half\n
3019
05:44:43,330 --> 05:44:50,760
if I isolate y, let's say minus six y equals\n
3020
05:44:50,759 --> 05:44:59,487
y equals one half x minus 1/3. So indeed,\n
3021
05:44:59,488 --> 05:45:04,409
with different In intercepts, and so they\n
3022
05:45:04,409 --> 05:45:10,729
sense that we have no solution to our system\n
3023
05:45:10,729 --> 05:45:18,298
has no solution is called an inconsistent\n
3024
05:45:18,298 --> 05:45:24,797
behavior happens. This time, I think I'm going\n
3025
05:45:24,797 --> 05:45:30,579
have X with a coefficient of one. So it's\n
3026
05:45:30,580 --> 05:45:39,228
equation, and then plug in to the second equation\n
3027
05:45:39,227 --> 05:45:46,647
y equals 18. If I distribute out, I get the\n
3028
05:45:46,648 --> 05:45:54,488
and I just get 18 equals 18, which is always\n
3029
05:45:54,488 --> 05:46:00,850
of linear equations. If you look more closely,\n
3030
05:46:00,849 --> 05:46:06,169
just a constant multiple, the first equation\n
3031
05:46:06,169 --> 05:46:10,939
as big as the corresponding term and the first\n
3032
05:46:10,939 --> 05:46:16,539
the second equation, anything, any x and y\n
3033
05:46:16,540 --> 05:46:23,250
the second one. So this system of equations\n
3034
05:46:23,250 --> 05:46:32,580
pair x y, where X plus five y equals six,\n
3035
05:46:32,580 --> 05:46:40,318
will satisfy this system of equations. That\n
3036
05:46:40,317 --> 05:46:49,477
x value of six or a y value of one corresponding\n
3037
05:46:49,477 --> 05:46:55,259
Corresponding to an x value of 13 thirds just\n
3038
05:46:55,259 --> 05:47:00,679
Graphically, if I graph both of these equations,\n
3039
05:47:00,680 --> 05:47:06,099
so I'll just see one line. In this video,\n
3040
05:47:06,099 --> 05:47:13,199
using the method of substitution and the method\n
3041
05:47:13,200 --> 05:47:19,317
linear equations can have one solution. When\n
3042
05:47:19,317 --> 05:47:26,797
in one point, they can be inconsistent, and\n
3043
05:47:26,797 --> 05:47:33,029
lines, or they can be dependent and have infinitely\n
3044
05:47:33,029 --> 05:47:39,647
lying on top of each other. In this video,\n
3045
05:47:39,648 --> 05:47:46,317
rate and time. The key relationship to keep\n
3046
05:47:46,317 --> 05:47:51,559
distance traveled divided by the time it takes\n
3047
05:47:51,560 --> 05:47:57,888
60 miles an hour, that's your rate. And that's\n
3048
05:47:57,887 --> 05:48:02,750
in one hour. Sometimes it's handy to rewrite\n
3049
05:48:02,750 --> 05:48:10,020
by T time. And that gives us that R times\n
3050
05:48:10,020 --> 05:48:15,727
is equal to rate times time. There's one more\n
3051
05:48:15,727 --> 05:48:23,590
the idea that rates add. For example, if you\n
3052
05:48:23,590 --> 05:48:29,709
you're walking on a moving sidewalk, that's\n
3053
05:48:29,709 --> 05:48:34,919
your total speed of travel with respect to\n
3054
05:48:34,919 --> 05:48:43,179
be three plus two, or five miles per hour.\n
3055
05:48:43,180 --> 05:48:51,207
the first rate per second rate is equal to\n
3056
05:48:51,207 --> 05:48:59,000
distance equals rate times time, and rates\n
3057
05:48:59,000 --> 05:49:04,119
has a top speed of six miles per hour and\n
3058
05:49:04,119 --> 05:49:10,319
top speed. She went 10 miles upstream in the\n
3059
05:49:10,319 --> 05:49:15,579
we're supposed to find the rate of the river\n
3060
05:49:15,580 --> 05:49:17,568
in this problem into a chart.
3061
05:49:17,567 --> 05:49:23,110
During the course of Elsa stay, there were\n
3062
05:49:23,110 --> 05:49:27,779
one period of time she was going upstream.\n
3063
05:49:27,779 --> 05:49:35,439
downstream. For each of those, I'm going to\n
3064
05:49:35,439 --> 05:49:42,109
she went at and the time it took when she\n
3065
05:49:42,110 --> 05:49:49,547
of 10 miles. When she was going downstream\n
3066
05:49:49,547 --> 05:49:55,207
the times to travel those two distances were\n
3067
05:49:55,207 --> 05:50:02,930
was, I'll just give it a variable I'll call\n
3068
05:50:02,930 --> 05:50:07,189
rate she traveled upstream was slower because\n
3069
05:50:07,189 --> 05:50:11,889
when she was going downstream with the current.\n
3070
05:50:11,889 --> 05:50:17,628
is, that's what we're trying to figure out.\n
3071
05:50:17,628 --> 05:50:23,488
do know that in still water also can go six\n
3072
05:50:23,488 --> 05:50:29,260
since she's going with the direction of the\n
3073
05:50:29,259 --> 05:50:40,279
should be six plus R, that's her rate, and\n
3074
05:50:40,279 --> 05:50:46,707
On the other hand, when she's going upstream,\n
3075
05:50:46,707 --> 05:50:53,149
rate of six miles per hour, we need to subtract\n
3076
05:50:53,150 --> 05:50:57,250
we've charted out our information, we can\n
3077
05:50:57,250 --> 05:51:04,137
distance equals rate times time, we actually\n
3078
05:51:04,137 --> 05:51:12,047
T, and 30 is equal to six plus R times T.\n
3079
05:51:12,047 --> 05:51:17,149
a system of equations, our next job is to\n
3080
05:51:17,150 --> 05:51:23,580
I think the easiest way to proceed is to isolate\n
3081
05:51:23,580 --> 05:51:28,200
first equation, I'll divide both sides by\n
3082
05:51:28,200 --> 05:51:36,760
by six plus R. That gives me 10 over six minus\n
3083
05:51:36,759 --> 05:51:45,477
t. Now if I set my T variables equal to each\n
3084
05:51:45,477 --> 05:51:52,989
equal to 30 over six plus R. I'm making progress\n
3085
05:51:52,990 --> 05:51:58,670
single variable that I need to solve. Since\n
3086
05:51:58,669 --> 05:52:03,887
I'm going to proceed by clearing the denominator.\n
3087
05:52:03,887 --> 05:52:14,529
least common denominator, that is six minus\n
3088
05:52:14,529 --> 05:52:23,689
I get that the six plus r times 10 is equal\n
3089
05:52:23,689 --> 05:52:36,637
I'm going to get 60 plus xR equals 180 minus\n
3090
05:52:36,637 --> 05:52:43,909
going to be 40 r is equal to 120. So our,\n
3091
05:52:43,909 --> 05:52:51,099
miles per hour. This is all that the problem\n
3092
05:52:51,099 --> 05:52:57,579
wanted to solve for the other unknown time,\n
3093
05:52:57,580 --> 05:53:04,270
my equations and solving for T. In this video,\n
3094
05:53:04,270 --> 05:53:12,279
charting out my information for the two situations\n
3095
05:53:12,279 --> 05:53:17,930
to fill in some of my boxes, and then using\n
3096
05:53:17,930 --> 05:53:24,080
to build a system of equations. In this video,\n
3097
05:53:24,080 --> 05:53:30,340
we have to figure out what quantity of two\n
3098
05:53:30,340 --> 05:53:39,387
contains 6% sodium hypochlorite. The other\n
3099
05:53:39,387 --> 05:53:50,957
be combined with 70 liters of a weaker 1%\n
3100
05:53:50,957 --> 05:53:59,349
that's 2.5% sodium hypochlorite. I want to\n
3101
05:53:59,349 --> 05:54:05,977
So I'm asking myself what quantities are going\n
3102
05:54:05,977 --> 05:54:14,829
amount of sodium hypochlorite that has symbol\n
3103
05:54:14,830 --> 05:54:23,270
it should equal the total amount of sodium\n
3104
05:54:23,270 --> 05:54:30,378
amount of water before mixing should equal\n
3105
05:54:30,378 --> 05:54:37,790
there's just the total amount of solution.\n
3106
05:54:37,790 --> 05:54:47,290
with water should equal the total amount of\n
3107
05:54:47,290 --> 05:54:51,909
I'm looking for. But before I start reading\n
3108
05:54:51,909 --> 05:55:05,387
out my quantities. So I've got the 6% solution.\n
3109
05:55:05,387 --> 05:55:16,128
And I've got my Desired Ending 2.5% solution.\n
3110
05:55:16,128 --> 05:55:26,580
certain volume of sodium hypochlorite. I've\n
3111
05:55:26,580 --> 05:55:37,200
total volume of solution. Let me see which\n
3112
05:55:37,200 --> 05:55:45,640
that I'm adding 70 liters of the 1% solution.\n
3113
05:55:45,640 --> 05:55:53,509
here. I don't know what volume of the household\n
3114
05:55:53,509 --> 05:56:02,949
to find out. So I'm going to just call that\n
3115
05:56:02,950 --> 05:56:08,409
by combining my other two solutions, I know\n
3116
05:56:08,409 --> 05:56:16,680
two volumes, so I'll write 70 plus x in this\n
3117
05:56:16,680 --> 05:56:22,580
the volume of solution is, 6% of that is the\n
3118
05:56:22,580 --> 05:56:31,070
sodium hypochlorite is going to be 0.06 times\n
3119
05:56:31,069 --> 05:56:40,000
whatever's left, so that's going to be x minus\n
3120
05:56:40,000 --> 05:56:46,988
following the same reasoning for the 1% solution\n
3121
05:56:46,988 --> 05:56:56,530
So that's going to be 0.01 times 70. Or point\n
3122
05:56:56,529 --> 05:57:05,699
is going to be 99% or point nine, nine times\n
3123
05:57:05,700 --> 05:57:15,317
for the 2.5% solution, the volume of the sodium\n
3124
05:57:15,317 --> 05:57:22,599
the volume of solution 70 plus x and the volume\n
3125
05:57:22,599 --> 05:57:32,919
that's 0.975 times 70 plus x. Now I've already\n
3126
05:57:32,919 --> 05:57:39,359
before added up is the volume of solution\n
3127
05:57:39,360 --> 05:57:43,830
But I haven't yet used the fact that the volume\n
3128
05:57:43,830 --> 05:57:53,040
and after. So I can write that down as an\n
3129
05:57:53,040 --> 05:58:01,270
equal to 0.025 times 70 plus x. Now I've got\n
3130
05:58:01,270 --> 05:58:06,850
it. Since I don't like all these decimals,\n
3131
05:58:06,849 --> 05:58:15,057
by let's see, 1000 should get rid of all the\n
3132
05:58:15,058 --> 05:58:27,090
700 equals 25 times 70 plus x. Distributing\n
3133
05:58:27,090 --> 05:58:42,409
1750 plus 25x. So let's see 60 minus 25 is\n
3134
05:58:42,409 --> 05:58:49,139
x equals 30 liters of the household bleach.
3135
05:58:49,139 --> 05:58:54,610
Notice that I never actually had to use the\n
3136
05:58:54,610 --> 05:59:00,659
mixing is equal to the quantity of water after\n
3137
05:59:00,659 --> 05:59:08,450
column. In fact, that information is redundant.\n
3138
05:59:08,450 --> 05:59:15,520
hypochlorite add up, and the total volume\n
3139
05:59:15,520 --> 05:59:21,689
of waters add up is just redundant information.\n
3140
05:59:21,689 --> 05:59:28,039
involving solutions can be used to solve many\n
3141
05:59:28,040 --> 05:59:36,977
items. My favorite method is to first make\n
3142
05:59:36,977 --> 05:59:43,930
the types of items in your mixture. Fill in\n
3143
05:59:43,930 --> 05:59:53,159
fact that the quantities add. This video is\n
3144
05:59:53,159 --> 05:59:59,000
Recall that a rational function is a function\n
3145
05:59:59,000 --> 06:00:07,308
of two power. No Here's an example. The simpler\n
3146
06:00:07,308 --> 06:00:14,520
considered a rational function, you can think\n
3147
06:00:14,520 --> 06:00:20,110
graph of this rational function is shown here.\n
3148
06:00:20,110 --> 06:00:27,290
of a polynomial. For one thing, its end behavior\n
3149
06:00:27,290 --> 06:00:33,030
is the way the graph looks when x goes through\n
3150
06:00:33,029 --> 06:00:38,840
numbers, we've seen that the end behavior\n
3151
06:00:38,840 --> 06:00:44,700
cases. That is why marches off to infinity\n
3152
06:00:44,700 --> 06:00:50,200
big or really negative. But this rational\n
3153
06:00:50,200 --> 06:00:54,600
Notice, as x gets really big, the y values\nare leveling off
3154
06:00:54,599 --> 06:01:00,779
at about a y value of three. And similarly,\n
3155
06:01:00,779 --> 06:01:07,849
is leveling off near the line y equals three,\n
3156
06:01:07,849 --> 06:01:16,119
graph, that line is called a horizontal asymptote.\n
3157
06:01:16,119 --> 06:01:21,529
that our graph gets closer and closer to as\n
3158
06:01:21,529 --> 06:01:26,750
infinity, or both. There's something else\n
3159
06:01:26,750 --> 06:01:32,599
graph, look at what happens as x gets close\n
3160
06:01:32,599 --> 06:01:37,717
five with x values on the right, our Y values\n
3161
06:01:37,718 --> 06:01:42,690
And as we approach the x value of negative\n
3162
06:01:42,689 --> 06:01:49,340
up towards positive infinity. We say that\n
3163
06:01:49,340 --> 06:01:56,439
negative five. A vertical asymptote is a vertical\n
3164
06:01:56,439 --> 06:02:02,567
to. Finally, there's something really weird\n
3165
06:02:02,567 --> 06:02:10,000
open circle there, like the value at x equals\n
3166
06:02:10,000 --> 06:02:16,718
is a place along the curve of the graph where\n
3167
06:02:16,718 --> 06:02:19,600
identified some of the features of our rational\nfunctions graph
3168
06:02:19,599 --> 06:02:24,627
I want to look back at the equation and see\n
3169
06:02:24,628 --> 06:02:31,040
just by looking at the equation. To find horizontal\n
3170
06:02:31,040 --> 06:02:36,580
is doing when x goes through really big positive\n
3171
06:02:36,580 --> 06:02:42,990
our equation for our function, the numerator\n
3172
06:02:42,990 --> 06:02:47,659
term when x is really big, right, because\n
3173
06:02:47,659 --> 06:02:53,619
enormous compared to this negative 12. If\n
3174
06:02:53,619 --> 06:02:59,610
the denominator, the denominator will be dominated\n
3175
06:02:59,610 --> 06:03:04,920
big positive or negative number, like a million,\n
3176
06:03:04,919 --> 06:03:10,797
than three times a million or negative 10.\n
3177
06:03:10,797 --> 06:03:18,387
or the horizontal asymptote, for our function,\n
3178
06:03:18,387 --> 06:03:22,637
and the term on the denominator that have\n
3179
06:03:22,637 --> 06:03:29,468
dominate the expression in size. So as x gets\n
3180
06:03:29,468 --> 06:03:37,220
to be approximately 3x squared over x squared,\n
3181
06:03:37,220 --> 06:03:45,430
asymptote at y equals three. Now our vertical\n
3182
06:03:45,430 --> 06:03:51,308
denominator of our function is zero. That's\n
3183
06:03:51,308 --> 06:03:56,150
denominator is zero. And when we get close\n
3184
06:03:56,150 --> 06:04:01,680
we're going to be dividing by tiny, tiny numbers,\n
3185
06:04:01,680 --> 06:04:06,920
magnitude. So to check where our denominators\n
3186
06:04:06,919 --> 06:04:11,909
I'm going to go ahead and factor the numerator\n
3187
06:04:11,909 --> 06:04:18,468
let's see, pull out the three, I get x squared\n
3188
06:04:18,468 --> 06:04:25,040
factors into X plus five times x minus two,\n
3189
06:04:25,040 --> 06:04:34,950
further, that's three times x minus two times\n
3190
06:04:34,950 --> 06:04:41,319
when x is equal to negative five, my denominator\n
3191
06:04:41,319 --> 06:04:49,398
zero. That's what gives me the vertical asymptote\n
3192
06:04:49,398 --> 06:04:57,292
x equals two, the denominator is zero, but\n
3193
06:04:57,292 --> 06:05:01,909
cancelled the x minus two factor from the\n
3194
06:05:01,909 --> 06:05:11,279
form for my function that agrees with my original\n
3195
06:05:11,279 --> 06:05:16,760
That's because when x equals two, the simplified\n
3196
06:05:16,760 --> 06:05:23,637
does not, it's zero over zero, it's undefined.\n
3197
06:05:23,637 --> 06:05:29,829
near x equals to our original functions just\n
3198
06:05:29,830 --> 06:05:36,468
our function only has a vertical asymptote\n
3199
06:05:36,468 --> 06:05:41,069
two, because the x minus two factor is no\n
3200
06:05:41,069 --> 06:05:45,779
it does have a hole at x equals two, because\n
3201
06:05:45,779 --> 06:05:52,189
even though the simplified version is if we\n
3202
06:05:52,189 --> 06:05:59,599
just plug in x equals two into our simplified\n
3203
06:05:59,599 --> 06:06:06,769
of three times two plus two over two plus\n
3204
06:06:06,770 --> 06:06:15,477
thirds. So our whole is that to four thirds.\n
3205
06:06:15,477 --> 06:06:22,520
detail, let's summarize our findings. We find\n
3206
06:06:22,520 --> 06:06:29,477
where the denominator is zero. The holes happen\n
3207
06:06:29,477 --> 06:06:35,930
zero and those factors cancel out. The vertical\n
3208
06:06:35,930 --> 06:06:42,010
denominator is zero, we find the horizontal\n
3209
06:06:42,009 --> 06:06:47,477
term on the numerator and the denominator,\n
3210
06:06:47,477 --> 06:06:55,567
three examples. In the first example, if we\n
3211
06:06:55,567 --> 06:07:03,648
to 5x over 3x squared, which is five over\n
3212
06:07:03,648 --> 06:07:09,958
is going to be huge. So I'm going to be dividing\n
3213
06:07:09,957 --> 06:07:15,147
to be going very close to zero. And therefore\n
3214
06:07:15,148 --> 06:07:24,398
at y equals zero. In the second example, the\n
3215
06:07:24,398 --> 06:07:30,200
simplifies to two thirds. So as x gets really\n
3216
06:07:30,200 --> 06:07:36,950
thirds, and we have a horizontal asymptote\n
3217
06:07:36,950 --> 06:07:45,270
the highest power terms, x squared over 2x\n
3218
06:07:45,270 --> 06:07:51,887
big, x over two is getting really big. And\n
3219
06:07:51,887 --> 06:07:59,279
at all. This is going to infinity, when x\n
3220
06:07:59,279 --> 06:08:07,237
and is going to negative infinity when x goes\n
3221
06:08:07,238 --> 06:08:12,840
case, the end behavior is kind of like that\n
3222
06:08:12,840 --> 06:08:18,887
asymptote. In general, when the degree of\n
3223
06:08:18,887 --> 06:08:23,520
the denominator, we're in this first case\n
3224
06:08:23,520 --> 06:08:29,250
to the numerator and we go to zero. In the\n
3225
06:08:29,250 --> 06:08:34,297
and the degree of the dominant are equal,\n
3226
06:08:34,297 --> 06:08:42,750
asymptote at the y value, that's equal to\n
3227
06:08:42,750 --> 06:08:46,968
in the third case, when the degree of the\n
3228
06:08:46,968 --> 06:08:52,290
denominator, then the numerator is getting\n
3229
06:08:52,290 --> 06:08:57,840
we end up with no horizontal asymptote. Final\n
3230
06:08:57,840 --> 06:09:03,379
to one more example. Please pause the video\n
3231
06:09:03,379 --> 06:09:09,817
horizontal asymptotes and holes for this rational\n
3232
06:09:09,817 --> 06:09:15,547
and holes, we need to look at where the denominator\n
3233
06:09:15,547 --> 06:09:20,289
factor both the numerator and the denominator.\n
3234
06:09:20,290 --> 06:09:25,128
we might have a whole instead of a vertical\n
3235
06:09:25,128 --> 06:09:31,887
factor. Let's see that's 3x times x plus one\n
3236
06:09:31,887 --> 06:09:40,029
x. And then I'll factor some more using a\n
3237
06:09:40,029 --> 06:09:47,967
a 2x and an X to multiply together to the\n
3238
06:09:47,968 --> 06:09:56,030
one or alpha minus three and a one. Let's\n
3239
06:09:56,029 --> 06:10:02,860
one times x plus three that does get me back\n
3240
06:10:02,861 --> 06:10:08,590
checks out. Now I noticed that I have a common\n
3241
06:10:08,590 --> 06:10:14,270
denominator. So that's telling me I'm going\n
3242
06:10:14,270 --> 06:10:21,889
I could rewrite my rational function by cancelling\n
3243
06:10:21,889 --> 06:10:28,290
as long as x is not equal to zero. So the\n
3244
06:10:28,290 --> 06:10:35,628
zero into my simplified version, that would\n
3245
06:10:35,628 --> 06:10:42,797
zero minus one times zero plus three, which\n
3246
06:10:42,797 --> 06:10:49,619
So my whole is at zero minus one. Now all\n
3247
06:10:49,619 --> 06:10:55,239
make my denominator zero will get me vertical\n
3248
06:10:55,240 --> 06:11:04,780
when 2x minus one times x plus three equals\n
3249
06:11:04,779 --> 06:11:13,840
or x plus three is zero. In other words, when\n
3250
06:11:13,840 --> 06:11:20,299
Finally, to find my horizontal asymptotes,\n
3251
06:11:20,299 --> 06:11:28,759
term in the numerator and the denominator.\n
3252
06:11:28,759 --> 06:11:35,359
bottom heavy, right? When x gets really big,\n
3253
06:11:35,360 --> 06:11:41,718
means that we have a horizontal asymptote\n
3254
06:11:41,718 --> 06:11:48,150
of our graph, the whole, the vertical asymptotes\n
3255
06:11:48,150 --> 06:11:54,458
would give us a framework for what the graph\n
3256
06:11:54,457 --> 06:12:02,599
at y equals zero, vertical asymptotes at x\n
3257
06:12:02,599 --> 06:12:09,909
at a hole at the point zero minus one. plotting\n
3258
06:12:09,909 --> 06:12:17,950
of graphing program, we can see that our actual\n
3259
06:12:17,950 --> 06:12:24,159
Notice that the x intercept when x is negative\n
3260
06:12:24,159 --> 06:12:29,529
our rational function or reduced rational\n
3261
06:12:29,529 --> 06:12:33,949
a zero on the numerator that doesn't make\n
3262
06:12:33,950 --> 06:12:40,000
zero. And an X intercept is where the y value\n
3263
06:12:40,000 --> 06:12:44,797
we learned how to find horizontal asymptotes\n
3264
06:12:44,797 --> 06:12:50,779
highest power terms, we learned to find the\n
3265
06:12:50,779 --> 06:12:56,307
at the factored version of the functions.\n
3266
06:12:56,308 --> 06:13:03,317
make the numerator and denominator zero, his\n
3267
06:13:03,317 --> 06:13:09,099
asymptotes correspond to the x values that\n
3268
06:13:09,099 --> 06:13:13,957
any any common and in common factors in the\n
3269
06:13:13,957 --> 06:13:19,750
This video is about combining functions by\n
3270
06:13:19,750 --> 06:13:26,270
dividing them. Suppose we have two functions,\n
3271
06:13:26,270 --> 06:13:34,218
x squared. One way to combine them is by adding\n
3272
06:13:34,218 --> 06:13:43,909
x means the function defined by taking f of\n
3273
06:13:43,909 --> 06:13:50,957
that means we take x plus one and add x squared,\n
3274
06:13:50,957 --> 06:14:01,899
plus x plus one. So f plus g evaluated on\n
3275
06:14:01,900 --> 06:14:08,810
I wanted to evaluate f plus g, on the number\n
3276
06:14:08,810 --> 06:14:18,218
one, or seven. Similarly, the notation f minus\n
3277
06:14:18,218 --> 06:14:25,490
f of x and subtracting g of x. So that would\n
3278
06:14:25,490 --> 06:14:35,390
to take f minus g evaluated at one, that would\n
3279
06:14:35,389 --> 06:14:44,599
the notation F dot g of x, which is sometimes\n
3280
06:14:44,599 --> 06:14:52,969
we take f of x times g of x. In other words,\n
3281
06:14:52,970 --> 06:15:02,888
simplified as x cubed plus x squared. The\n
3282
06:15:02,887 --> 06:15:11,919
f of x and divided by g of x. So that would\n
3283
06:15:11,919 --> 06:15:19,717
figure, the blue graph represents h of x.\n
3284
06:15:19,718 --> 06:15:26,690
p of x, we're asked to find h minus p of zero.
3285
06:15:26,689 --> 06:15:32,259
We don't have any equations to work with,\n
3286
06:15:32,259 --> 06:15:40,877
minus p of x is defined as h of x minus p\n
3287
06:15:40,878 --> 06:15:48,968
is going to be h of zero minus p of zero.\n
3288
06:15:48,968 --> 06:15:55,830
finding the value of zero on the x axis, and\n
3289
06:15:55,830 --> 06:16:04,120
function h of x. So that's about 1.8. Now\n
3290
06:16:04,119 --> 06:16:10,090
for zero on the x axis, and finding the corresponding\n
3291
06:16:10,090 --> 06:16:18,939
a y value of one 1.8 minus one is 0.8. So\n
3292
06:16:18,939 --> 06:16:28,699
of zero. If we want to find P times h of negative\n
3293
06:16:28,700 --> 06:16:34,899
negative three times h of negative three.\n
3294
06:16:34,899 --> 06:16:45,190
value of negative three, the y value for P\n
3295
06:16:45,189 --> 06:16:50,180
corresponds to a y value of negative two for\nH.
3296
06:16:50,180 --> 06:16:55,610
Two times negative two is negative four. So\n
3297
06:16:56,610 --> 06:17:03,878
In this video, we saw how to add two functions,\n
3298
06:17:03,878 --> 06:17:13,290
and divide two functions in the following\n
3299
06:17:13,290 --> 06:17:19,940
the first function, and then you apply the\n
3300
06:17:19,939 --> 06:17:27,349
function. For example, the first function\n
3301
06:17:27,349 --> 06:17:35,807
years. So its input would be time in years,\n
3302
06:17:35,808 --> 06:17:45,620
of people in the population. The second function\n
3303
06:17:45,619 --> 06:17:53,919
of population size. So it will take population\n
3304
06:17:53,919 --> 06:18:00,189
costs. If you put these functions together,\n
3305
06:18:00,189 --> 06:18:07,770
way from time in years to healthcare costs.\n
3306
06:18:07,770 --> 06:18:16,409
F. The composition of two functions, written\n
3307
06:18:16,409 --> 06:18:26,919
as follows. g composed with f of x is G evaluated\n
3308
06:18:26,919 --> 06:18:36,009
and diagram f x on a number x and produces\n
3309
06:18:36,009 --> 06:18:45,137
f of x and produces a new number, g of f of\n
3310
06:18:45,137 --> 06:18:51,950
with F is the function that goes all the way\n
3311
06:18:51,950 --> 06:18:58,920
examples where our functions are defined by\n
3312
06:18:58,919 --> 06:19:08,189
with F of four, by definition, this means\n
3313
06:19:08,189 --> 06:19:15,029
we always work from the inside out. So we\n
3314
06:19:15,029 --> 06:19:23,739
f of four, using the table of values for f\n
3315
06:19:23,740 --> 06:19:33,128
so we can replace F of four with the number\n
3316
06:19:33,128 --> 06:19:39,420
seven becomes our new x value in our table\n
3317
06:19:39,419 --> 06:19:47,949
to the G of X value of 10. So g of seven is\n
3318
06:19:47,950 --> 06:19:56,387
F of four is equal to 10. If instead we want\n
3319
06:19:56,387 --> 06:20:04,547
can rewrite that as f of g of four Again work\n
3320
06:20:04,547 --> 06:20:10,250
g of four. So four is our x value. And we\n
3321
06:20:10,250 --> 06:20:20,430
g of four is one. So we replaced you a four\n
3322
06:20:20,430 --> 06:20:29,817
Using our table for F values, f of one is\n
3323
06:20:29,817 --> 06:20:36,907
f of four, we got a different answer than\n
3324
06:20:36,907 --> 06:20:43,509
g composed with F is not the same thing as\n
3325
06:20:43,509 --> 06:20:49,808
and take a moment to compute the next two\n
3326
06:20:49,809 --> 06:20:56,708
of two by the equivalent expression, f of\n
3327
06:20:56,707 --> 06:21:06,127
know that f of two is three, and f of three\n
3328
06:21:06,128 --> 06:21:17,010
g of six, rewrite that as f of g of six, using\n
3329
06:21:17,009 --> 06:21:25,109
of eight, eight is not on the table as an\n
3330
06:21:25,110 --> 06:21:35,148
there is no F of eight, this does not exist,\n
3331
06:21:35,148 --> 06:21:42,970
for F composed with g. Even though it was\n
3332
06:21:42,970 --> 06:21:49,780
the way through and get a value for F composed\n
3333
06:21:49,779 --> 06:21:54,399
to the composition of functions that are given\nby equations.
3334
06:21:54,400 --> 06:22:03,110
p of x is x squared plus x and q of x as negative\n
3335
06:22:03,110 --> 06:22:14,047
As usual, I can rewrite this as Q of P of\n
3336
06:22:14,047 --> 06:22:21,567
is one squared plus one, so that's two. So\n
3337
06:22:21,567 --> 06:22:28,009
of two is negative two times two or negative\n
3338
06:22:28,009 --> 06:22:35,237
In this next example, we want to find q composed\n
3339
06:22:35,238 --> 06:22:44,370
as usual as Q of p of x and work from the\n
3340
06:22:44,369 --> 06:22:52,047
for that. That's x squared plus x. So I can\n
3341
06:22:52,047 --> 06:22:59,919
I'm stuck with evaluating q on x squared plus\n
3342
06:22:59,919 --> 06:23:08,239
that thing. So q of x squared plus x is going\n
3343
06:23:08,240 --> 06:23:13,808
plus x, what I've done is I've substituted\n
3344
06:23:13,808 --> 06:23:20,010
where I saw the X in this formula for q of\n
3345
06:23:20,009 --> 06:23:24,297
So that will be multiplying negative two by\n
3346
06:23:24,297 --> 06:23:32,147
piece, I can simplify this a bit as negative\n
3347
06:23:32,148 --> 06:23:40,478
for Q composed with p of x. Notice that if\n
3348
06:23:40,477 --> 06:23:45,309
which I already did in the first problem,\n
3349
06:23:45,310 --> 06:23:52,740
two times one squared minus two and I get\n
3350
06:23:52,740 --> 06:24:00,340
try another one. Let's try p composed with\n
3351
06:24:00,340 --> 06:24:07,009
x. Working from the inside out, I can replace\n
3352
06:24:07,009 --> 06:24:15,419
P of negative 2x. Here's my formula for P.\n
3353
06:24:15,419 --> 06:24:23,657
plug in this expression everywhere I see an\n
3354
06:24:23,657 --> 06:24:29,819
2x squared plus negative 2x. Again, being\n
3355
06:24:29,819 --> 06:24:40,457
plug in the entire expression in forex. let\n
3356
06:24:40,457 --> 06:24:50,407
Notice that I got different expressions for\n
3357
06:24:50,407 --> 06:24:57,720
we see that q composed with P is not necessarily\n
3358
06:24:57,720 --> 06:25:06,850
video and try this last example yourself.\n
3359
06:25:06,849 --> 06:25:13,750
we're going to replace p of x with its expression\n
3360
06:25:13,750 --> 06:25:22,090
p on x squared plus x. That means we plug\n
3361
06:25:22,090 --> 06:25:29,657
x in this formula, so that's x squared plus\n
3362
06:25:29,657 --> 06:25:36,809
Once again, I can simplify by distributing\n
3363
06:25:36,810 --> 06:25:45,020
cubed plus x squared plus x squared plus x,\n
3364
06:25:45,020 --> 06:25:52,189
plus x. In this last set of examples, we're\n
3365
06:25:52,189 --> 06:25:57,919
for a function of h of x. But we're supposed\n
3366
06:25:57,919 --> 06:26:04,359
functions, F and G. Let's think for a minute,\n
3367
06:26:04,360 --> 06:26:13,619
first, f composed with g of x, let's see,\n
3368
06:26:13,619 --> 06:26:21,250
these expressions from the inside out, we\n
3369
06:26:21,250 --> 06:26:27,779
to figure out what what f and g could be,\n
3370
06:26:27,779 --> 06:26:32,779
my expression for H, so I'm going to draw\n
3371
06:26:32,779 --> 06:26:38,529
inside the box, that'll be my function, g\n
3372
06:26:38,529 --> 06:26:43,849
whatever happens to the box, in this case,\n
3373
06:26:43,849 --> 06:26:46,869
my outside function, my second function f.
3374
06:26:46,869 --> 06:26:54,750
So here, we're gonna say g of x is equal to\n
3375
06:26:54,750 --> 06:27:00,939
to the square root of x, let's just check\n
3376
06:27:00,939 --> 06:27:08,069
check that when I take the composition, f\n
3377
06:27:08,069 --> 06:27:17,029
as my original h. So let's see, if I do f\n
3378
06:27:17,029 --> 06:27:23,397
that's f of g of x, working from the inside\n
3379
06:27:23,398 --> 06:27:30,978
x squared plus seven. So I need to evaluate\n
3380
06:27:30,977 --> 06:27:37,500
in x squared plus seven, into the formula\n
3381
06:27:37,500 --> 06:27:44,128
of x squared plus seven to the it works because\n
3382
06:27:44,128 --> 06:27:48,180
a correct answer a correct way of breaking\n
3383
06:27:48,180 --> 06:27:54,680
But I do want to point out, this is not the\n
3384
06:27:54,680 --> 06:27:59,738
for H of X again, and this time, I'll put\n
3385
06:27:59,738 --> 06:28:08,128
the x squared. If I did that, then my inside\n
3386
06:28:08,128 --> 06:28:17,450
be x squared. And my second function is what\n
3387
06:28:17,450 --> 06:28:23,898
to the box, and the box gets added seven to\n
3388
06:28:23,898 --> 06:28:32,260
words, f of x is going to be the square root\n
3389
06:28:32,259 --> 06:28:39,727
works. If I do f composed with g of x, that's\n
3390
06:28:39,727 --> 06:28:46,390
I'm taking f of x squared. When I plug in\n
3391
06:28:46,390 --> 06:28:53,207
root of x squared plus seven. So this is that\n
3392
06:28:53,207 --> 06:29:00,590
we learn to evaluate the composition of functions.\n
3393
06:29:00,590 --> 06:29:10,170
out. We also learn to break apart a complicated\n
3394
06:29:10,169 --> 06:29:16,339
by boxing one piece of the function and letting\n
3395
06:29:16,340 --> 06:29:21,270
Let that be the inside of the box, and the\n
3396
06:29:21,270 --> 06:29:33,540
be whatever happens to the box.
3397
06:29:33,540 --> 06:29:38,580
The inverse of a function undoes what the\n
3398
06:29:38,580 --> 06:29:46,750
shoes would be to untie them. And the inverse\n
3399
06:29:46,750 --> 06:29:53,720
would be the function that subtracts two from\n
3400
06:29:53,720 --> 06:30:00,378
their properties. Suppose f of x is a function\n
3401
06:30:00,378 --> 06:30:08,790
two is three, f of three is five, f of four\n
3402
06:30:08,790 --> 06:30:17,817
function for F written f superscript. Negative\n
3403
06:30:17,817 --> 06:30:26,157
three, F inverse takes three, back to two.\n
3404
06:30:26,157 --> 06:30:38,430
of three is to. Similarly, since f takes three\n
3405
06:30:38,430 --> 06:30:46,860
since f takes four to six, f inverse of six\n
3406
06:30:46,860 --> 06:30:54,128
inverse of one is five. I'll use these numbers\n
3407
06:30:54,128 --> 06:31:00,450
of values when y equals f of x and the chart\n
3408
06:31:00,450 --> 06:31:07,208
closely related. They share the same numbers,\n
3409
06:31:07,207 --> 06:31:14,647
the y values for f inverse of x, and the y\n
3410
06:31:14,648 --> 06:31:21,138
for f inverse of x. That leads us to the first\n
3411
06:31:21,137 --> 06:31:29,439
of y and x. I'm going to plot the points for\n
3412
06:31:29,439 --> 06:31:35,849
points for y equals f inverse of x in red.\n
3413
06:31:35,849 --> 06:31:40,349
kind of symmetry you observe in this graph.\n
3414
06:31:40,349 --> 06:31:46,737
points, you might have noticed that the blue\n
3415
06:31:46,738 --> 06:31:55,250
over the mirror line, y equals x. So our second\n
3416
06:31:55,250 --> 06:31:58,047
of x can be obtained from the graph of y equals\nf
3417
06:31:59,047 --> 06:32:06,727
by reflecting over the line y equals x. This\n
3418
06:32:06,727 --> 06:32:15,770
roles of war annex. In the same example, let's\n
3419
06:32:15,770 --> 06:32:23,878
means composition. In other words, we're computing\n
3420
06:32:23,878 --> 06:32:33,350
the inside out. So that's f inverse of three.\n
3421
06:32:33,349 --> 06:32:44,717
three, we see as to similarly, we can compute\n
3422
06:32:44,718 --> 06:32:52,690
take f of f inverse of three. Since f inverse\n
3423
06:32:52,689 --> 06:33:02,039
computing F of two, which is three. Please\n
3424
06:33:02,040 --> 06:33:11,148
other compositions. You should have found\n
3425
06:33:11,148 --> 06:33:16,290
of f of a number, you get back to the very\n
3426
06:33:16,290 --> 06:33:20,738
if you take f of f inverse of any number,\n
3427
06:33:20,738 --> 06:33:29,909
with. So in general, f inverse of f of x is\n
3428
06:33:29,909 --> 06:33:37,430
equal to x. This is the mathematical way of\n
3429
06:33:37,430 --> 06:33:43,229
Let's look at a different example. Suppose\n
3430
06:33:43,229 --> 06:33:48,409
a moment, and guess what the inverse of f\n
3431
06:33:48,409 --> 06:33:57,180
work that F does. You might have guessed that\n
3432
06:33:57,180 --> 06:34:05,680
function, we can check that this is true by\n
3433
06:34:05,680 --> 06:34:09,200
the cube root of function, which means the\ncube root function
3434
06:34:09,200 --> 06:34:16,680
cubed, which gets us back to x. Similarly,\n
3435
06:34:19,610 --> 06:34:25,600
And we get back to excellence again. So the\n
3436
06:34:25,599 --> 06:34:30,699
the cubing function. When we compose the two\n
3437
06:34:30,700 --> 06:34:37,440
we started with. It'd be nice to have a more\n
3438
06:34:37,439 --> 06:34:44,547
besides guessing and checking. One method\n
3439
06:34:44,547 --> 06:34:50,520
of y and x. So if we want to find the inverse\n
3440
06:34:50,520 --> 06:34:59,840
x over 3x. We can write it as y equals five\n
3441
06:34:59,840 --> 06:35:09,957
x To get x equals five minus y over three\n
3442
06:35:09,957 --> 06:35:19,250
multiply both sides by three y. Bring all\n
3443
06:35:19,250 --> 06:35:28,369
alternate without wizened them to the right\n
3444
06:35:28,369 --> 06:35:40,047
why this gives us f inverse of x as five over\n
3445
06:35:40,047 --> 06:35:46,307
f and our inverse function, f inverse are\n
3446
06:35:46,308 --> 06:35:53,080
reciprocals of each other. And in general,\n
3447
06:35:53,080 --> 06:36:01,350
over f of x. This can be confusing, because\n
3448
06:36:01,349 --> 06:36:08,750
mean one of our two, but f to the minus one\n
3449
06:36:08,750 --> 06:36:16,047
reciprocal. It's natural to ask us all functions\n
3450
06:36:16,047 --> 06:36:23,119
you might encounter. Is there always a function\n
3451
06:36:23,119 --> 06:36:30,279
is no. See, if you can come up with an example\n
3452
06:36:30,279 --> 06:36:37,977
function. The word function here is key. Remember\n
3453
06:36:37,977 --> 06:36:47,559
x values and y values, such that for each\n
3454
06:36:47,560 --> 06:36:57,388
y value. One example of a function that does\n
3455
06:37:02,238 --> 06:37:09,840
the inverse of this function is not a function.\n
3456
06:37:09,840 --> 06:37:17,957
number two and the number negative two, both\n
3457
06:37:17,957 --> 06:37:27,029
you would have to send four to both two and\n
3458
06:37:27,029 --> 06:37:34,217
it might be easier to understand the problem,\n
3459
06:37:34,218 --> 06:37:40,878
Recall that inverse functions reverse the\n
3460
06:37:40,878 --> 06:37:47,840
line y equals x. But when I flipped the green\n
3461
06:37:47,840 --> 06:37:53,599
red graph. This red graph is not the graph\n
3462
06:37:53,599 --> 06:37:59,859
line test. The reason that violates the vertical\n
3463
06:37:59,860 --> 06:38:10,000
violates the horizontal line test, and has\n
3464
06:38:10,000 --> 06:38:14,950
a function f has an inverse function if and\n
3465
06:38:14,950 --> 06:38:21,690
line test, ie every horizontal line intersects\n
3466
06:38:21,689 --> 06:38:26,869
video for a moment and see which of these\n
3467
06:38:26,869 --> 06:38:34,579
In other words, which of the four corresponding\n
3468
06:38:34,580 --> 06:38:42,160
You may have found that graphs A and B violate\n
3469
06:38:42,159 --> 06:38:48,860
would not have inverse functions. But graph\n
3470
06:38:48,860 --> 06:38:54,740
So these graphs represent functions that do\n
3471
06:38:54,740 --> 06:39:02,399
horizontal line test are sometimes called\n
3472
06:39:02,399 --> 06:39:10,739
is one to one, if for any two different x\n
3473
06:39:10,739 --> 06:39:18,530
of x one and f of x two are different numbers.\n
3474
06:39:18,529 --> 06:39:27,509
whenever f of x one is equal to f of x two,\n
3475
06:39:27,509 --> 06:39:32,939
example, let's try to find P inverse of x,\n
3476
06:39:32,939 --> 06:39:40,707
two drawn here. If we graph P inverse on the\n
3477
06:39:40,707 --> 06:39:49,457
graph simply by flipping over the line y equals\n
3478
06:39:49,457 --> 06:39:56,728
we can write y equal to a squared of x minus\n
3479
06:39:56,728 --> 06:40:07,579
for y by squaring both sides adding two. Now\n
3480
06:40:07,580 --> 06:40:13,290
two, that would look like a parabola, it would\n
3481
06:40:13,290 --> 06:40:22,670
together with another arm on the left side.\n
3482
06:40:22,669 --> 06:40:31,779
consists only of this right arm, we can specify\n
3483
06:40:31,779 --> 06:40:40,897
that x has to be bigger than or equal to zero.\n
3484
06:40:40,898 --> 06:40:47,530
graph for the square root of x, y was only\n
3485
06:40:47,529 --> 06:40:55,820
closely at the domain and range of P and P\n
3486
06:40:55,820 --> 06:41:01,650
values of x such that x minus two is greater\n
3487
06:41:01,650 --> 06:41:07,250
the square root of a negative number. This\n
3488
06:41:07,250 --> 06:41:14,069
or equal to two, or an interval notation,\n
3489
06:41:14,069 --> 06:41:20,419
of P, we can see from the graph is all y value\n
3490
06:41:20,419 --> 06:41:29,047
from zero to infinity. Similarly, based on\n
3491
06:41:29,047 --> 06:41:34,137
is x values greater than or equal to zero,\n
3492
06:41:34,137 --> 06:41:40,500
range of P inverse is Y values greater than\n
3493
06:41:40,500 --> 06:41:45,907
to infinity. If you look closely at these\n
3494
06:41:45,907 --> 06:41:53,919
domain of P corresponds exactly to the range\n
3495
06:41:57,569 --> 06:42:04,387
This makes sense, because inverse functions\n
3496
06:42:04,387 --> 06:42:10,977
f inverse of x is the x values for F inverse,\n
3497
06:42:10,977 --> 06:42:18,349
of F. The range of f inverse is the y values\n
3498
06:42:18,349 --> 06:42:28,559
or the domain of f. In this video, we discussed\n
3499
06:42:28,560 --> 06:42:36,920
inverse functions, reverse the roles of y\n
3500
06:42:36,919 --> 06:42:47,349
x is the graph of y equals f of x reflected\n
3501
06:42:47,349 --> 06:42:56,099
F with F inverse, we get the identity function\n
3502
06:42:56,099 --> 06:43:04,217
f inverse with F, that brings x to x. In other\n
3503
06:43:04,218 --> 06:43:18,218
function f of x has an inverse function if\n
3504
06:43:18,218 --> 06:43:31,110
the horizontal line test. And finally, the\n
3505
06:43:31,110 --> 06:43:39,000
the range of f is the domain of f inverse.\n
3506
06:43:39,000 --> 06:43:45,457
be important when we study exponential functions\n
288930
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