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And we know from our exponent rules that 2 to the third to the third is the same thing as 2 to the ninth. And actually, it's this exponent property where you can multiply. When you take something to an exponent and then take that to an exponent, and you can essentially just multiply the exponents, that's the exponent p... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
But I'm not going to dwell on that too much in this presentation. There's a whole video on proving it a little bit more formally. The next logarithm property I'm going to show you, and then I'll review everything and maybe do some examples. This is probably the single most useful logarithm property if you are a calcula... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
This is probably the single most useful logarithm property if you are a calculator addict. And I'll show you why. So let's say I have log base B of A is equal to log base C of A divided by log base C of B. Now why is this a useful property if you are a calculator addict? Well, let's say you go to class and there's a qu... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
Now why is this a useful property if you are a calculator addict? Well, let's say you go to class and there's a quiz. The teacher says, you can use your calculator. And using your calculator, I want you to figure out the log base 17 of 357. And you will scramble and look for the log base 17 button on your calculator an... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
And using your calculator, I want you to figure out the log base 17 of 357. And you will scramble and look for the log base 17 button on your calculator and not find it. Because there is no log base 17 number, a button on your calculator. You'll probably either have a log button or you'll have an LN button. And just so... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
You'll probably either have a log button or you'll have an LN button. And just so you know, the log button on your calculator is probably base 10. And your LN number, your LN button on your calculator is going to be base E. For those of you who aren't familiar with E, don't worry about it. But it's 2.71 something somet... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
But it's 2.71 something something. It's a number. It's nothing. It's an amazing number, but we'll talk more about that. In a future presentation. So there's only two bases you have on your calculator. So if you want to figure out another base logarithm, you use this property. | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
It's an amazing number, but we'll talk more about that. In a future presentation. So there's only two bases you have on your calculator. So if you want to figure out another base logarithm, you use this property. So if you're given this on an exam, you can very confidently say, oh, well, that is just the same thing as ... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
So if you want to figure out another base logarithm, you use this property. So if you're given this on an exam, you can very confidently say, oh, well, that is just the same thing as you would have to switch to your yellow color in order to act with confidence. We could do either E or 10. But you could say that's the s... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
But you could say that's the same thing as log base 10 of 357 divided by log base 10 of 17. So you literally could just say, type in 357 in your calculator and press the log button and you're going to get bam, bam, bam, bam. Then you can clear it or if you know how to use a parenthesis on your calculator, you can do th... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
But then you say, type 17 in your calculator, press the log button, you get bam, bam, bam, bam. And then you just divide them and you get your answer. So this is a super useful property for calculator addicts. And once again, I'm not going to go into a lot of depth of how this one to me is the most useful, but it doesn... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
And once again, I'm not going to go into a lot of depth of how this one to me is the most useful, but it doesn't completely fall out of, obviously, of the exponent properties, but it's hard for me to describe the intuition simply, so you probably want to watch the proof on it if you don't believe why this happens. But ... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
Let's do some examples. So let's just rewrite a bunch of things in simpler forms. So if I wanted to write the log base 2 of the square root of 32 divided by the square root of 8, how can I rewrite this so it's reasonably not messy? Well, let's think about this. This is the same thing. This is equal to, I don't know if ... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
Well, let's think about this. This is the same thing. This is equal to, I don't know if I move vertically or horizontally, but I'll move vertically. This is the same thing as the log base 2 of 32 over the square root of 8 to the 1 half power, right? And we know from our logarithm properties, the third one we learned, t... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
This is the same thing as the log base 2 of 32 over the square root of 8 to the 1 half power, right? And we know from our logarithm properties, the third one we learned, that that is the same thing as 1 half times the logarithm of 32 divided by the square root of 8, right? I just took the exponent and made that the coe... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
And now we have a little quotient here, right? Logarithm of 32 divided by logarithm of square root of 8. Well, we can use our other logarithm. Let's keep the 1 half out. That's going to equal, oops, parentheses, logarithm, oh, I forgot my base, logarithm base 2 of 32 minus, right, because this is in the quotient, minus... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
Let's keep the 1 half out. That's going to equal, oops, parentheses, logarithm, oh, I forgot my base, logarithm base 2 of 32 minus, right, because this is in the quotient, minus the logarithm base 2 of the square root of 8, right? Let's see. Well, here, once again, we have a square root here, so we could say that this ... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
Well, here, once again, we have a square root here, so we could say that this is equal to 1 half times log base 2 of 32 minus, this 8 to the 1 half, which is the same thing as 1 half log base 2 of 8. We learned that property at the beginning of this presentation. And then if we want, we can distribute this original 1 h... | Introduction to logarithm properties (part 2) Logarithms Algebra II Khan Academy.mp3 |
Now in order to really use the quadratic equation, or to figure out what our a's, b's, and c's are, we have to have our equation in the form a x squared plus bx plus c is equal to 0. And then if we know our a's, b's, and c's, we'll say that the solutions to this equation are x is equal to negative b plus or minus the s... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
And on one side of the equation, we have a negative x squared plus 8x. So that looks like the first two terms. But our constant's on the other side. So let's get the constant on the left-hand side and get a 0 here on the right-hand side. So let's subtract 1 from both sides of this equation. The left-hand side of the eq... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So let's get the constant on the left-hand side and get a 0 here on the right-hand side. So let's subtract 1 from both sides of this equation. The left-hand side of the equation will become negative x squared plus 8x minus 1. And then the right-hand side, 1 minus 1 is 0. Now we have it in that form. We have ax squared.... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
And then the right-hand side, 1 minus 1 is 0. Now we have it in that form. We have ax squared. a is negative 1. So let me write this down. a is equal to negative 1. a is right here. a is equal to negative 1. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
a is negative 1. So let me write this down. a is equal to negative 1. a is right here. a is equal to negative 1. It's implicit there. You could put a 1 here if you like, a negative 1. Negative x squared is the same thing as negative 1x squared. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
a is equal to negative 1. It's implicit there. You could put a 1 here if you like, a negative 1. Negative x squared is the same thing as negative 1x squared. b is equal to 8. So b is equal to 8. That's the 8 right there. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
Negative x squared is the same thing as negative 1x squared. b is equal to 8. So b is equal to 8. That's the 8 right there. And c is equal to negative 1. And c is equal to negative 1. That's the negative 1 right there. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
That's the 8 right there. And c is equal to negative 1. And c is equal to negative 1. That's the negative 1 right there. So now we can just apply the quadratic formula. The solutions to this equation are x is equal to negative b. So negative b plus or minus the square root of b squared, of 8 squared, minus 4ac. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
That's the negative 1 right there. So now we can just apply the quadratic formula. The solutions to this equation are x is equal to negative b. So negative b plus or minus the square root of b squared, of 8 squared, minus 4ac. Let me do that in that green color. Minus 4. The green's the part of the formula. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So negative b plus or minus the square root of b squared, of 8 squared, minus 4ac. Let me do that in that green color. Minus 4. The green's the part of the formula. The colored parts are the things that we're substituting into the formula. Minus 4 times a, which is negative 1, times c, which is also negative 1. And the... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
The green's the part of the formula. The colored parts are the things that we're substituting into the formula. Minus 4 times a, which is negative 1, times c, which is also negative 1. And then all of that, let me extend the square root sign a little bit further, all of that is going to be over 2 times a. In this case,... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
And then all of that, let me extend the square root sign a little bit further, all of that is going to be over 2 times a. In this case, a is negative 1. So let's simplify this. So this becomes negative 8. This is negative 8 plus or minus the square root of 8 squared is 64. And then you have a negative 1 times a negativ... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So this becomes negative 8. This is negative 8 plus or minus the square root of 8 squared is 64. And then you have a negative 1 times a negative 1. These just cancel out just to be a 1. So 64 minus 4, that's just that 4 over there. All of that over negative 2. So this is equal to negative 8 plus or minus the square roo... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
These just cancel out just to be a 1. So 64 minus 4, that's just that 4 over there. All of that over negative 2. So this is equal to negative 8 plus or minus the square root of 60. All of that over negative 2. Now let's see if we can simplify the radical expression here, the square root of 60. Let's see, 60 is equal to... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So this is equal to negative 8 plus or minus the square root of 60. All of that over negative 2. Now let's see if we can simplify the radical expression here, the square root of 60. Let's see, 60 is equal to 2 times 30. 30 is equal to 2 times 15. And then 15 is 3 times 5. So we do have a perfect square here. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
Let's see, 60 is equal to 2 times 30. 30 is equal to 2 times 15. And then 15 is 3 times 5. So we do have a perfect square here. We do have a 2 times 2 in there. It is 2 times 2 times 15, or 4 times 15. So we could write the square root of 60 is equal to the square root of 4 times the square root of 15. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So we do have a perfect square here. We do have a 2 times 2 in there. It is 2 times 2 times 15, or 4 times 15. So we could write the square root of 60 is equal to the square root of 4 times the square root of 15. Right? Square root of 4 times the square root of 15. That's what 60 is, 4 times 15. | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So we could write the square root of 60 is equal to the square root of 4 times the square root of 15. Right? Square root of 4 times the square root of 15. That's what 60 is, 4 times 15. And so this is equal to, square root of 4 is 2 times the square root of 15. So we can rewrite this expression right here as being equa... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
That's what 60 is, 4 times 15. And so this is equal to, square root of 4 is 2 times the square root of 15. So we can rewrite this expression right here as being equal to negative 8 plus or minus 2 times the square root of 15, all of that over negative 2. Now, we can divide both of these terms right here are divisible b... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
Now, we can divide both of these terms right here are divisible by either 2 or negative 2. So let's divide it. So we have negative 8 divided by negative 2, which is positive 4. So let me write it over here. Negative 8 divided by negative 2 is positive 4. And then you have this weird thing, plus or minus 2 divided by ne... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So let me write it over here. Negative 8 divided by negative 2 is positive 4. And then you have this weird thing, plus or minus 2 divided by negative 2. And really what we have here is two expressions. But if we're plus 2 and we divide by negative 2, it will be negative 1. And if we take negative 2 and divide by negati... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
And really what we have here is two expressions. But if we're plus 2 and we divide by negative 2, it will be negative 1. And if we take negative 2 and divide by negative 2, we're going to have positive 1. So instead of plus or minus, you could imagine it now as being minus or plus. But it's really the same thing, right... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
So instead of plus or minus, you could imagine it now as being minus or plus. But it's really the same thing, right? It's really now minus or plus. If it was plus, it's not going to be a minus. Now if it was a minus, it's not going to be a plus. Minus or plus 2 times the square root of 15. Or another way to view it is ... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
If it was plus, it's not going to be a minus. Now if it was a minus, it's not going to be a plus. Minus or plus 2 times the square root of 15. Or another way to view it is that the two solutions here are 4 minus 2 roots of 15 and 4 plus 2 roots of 15. These are both values of x that'll satisfy this equation. And if thi... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
Or another way to view it is that the two solutions here are 4 minus 2 roots of 15 and 4 plus 2 roots of 15. These are both values of x that'll satisfy this equation. And if this confused you, what I did, turning a plus or minus into minus plus, let me just take a little bit of a side there. I could write this expressi... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
I could write this expression up here as two expressions. That's what the plus or minus really is. There's a negative 8 plus 2 roots of 15 over negative 2. And then there's a negative 8 minus 2 roots 15 over negative 2. This one simplifies to negative 8 divided by negative 2 is 4. 2 divided by negative 2 is negative 1 ... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
And then there's a negative 8 minus 2 roots 15 over negative 2. This one simplifies to negative 8 divided by negative 2 is 4. 2 divided by negative 2 is negative 1 times 4 minus the square root of 15. And then over here, you have negative 8 divided by negative 2, which is 4, and then negative 2 divided by negative 2, w... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
And then over here, you have negative 8 divided by negative 2, which is 4, and then negative 2 divided by negative 2, which is plus the square root of 15. And I just realized I made a mistake up here. When we're dividing it 2 divided by negative 2, we don't have this 2 over here. This is just a plus or minus the root o... | Example 2 Using the quadratic formula Quadratic equations Algebra I Khan Academy.mp3 |
By now we're used to seeing functions defined like h of y is equal to y squared, or f of x is equal to the square root of x. But what we're now going to explore is functions that are defined piece by piece over different intervals. And functions like this are, you'll sometimes view them as a piecewise, or these types o... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
And so let's take a look at this graph right over here. This graph, you can see that the function is constant over this interval for x, and then it jumps up to this interval for x, and then it jumps back down for this interval for x. So let's think about how we would write this using our function notation. So if we say... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
So if we say that this right over here is the x-axis, and if this is the y is equal to f of x axis, then let's see, our function f of x is going to be equal to, let's see, there's three different intervals. So let me give myself some space for the three different intervals. Now this first interval is from, not includin... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
So I could write that as negative nine is less than x, less than or equal to negative five. That's this interval. And what is the value of the function over this interval? Well we see the value of the function is negative nine. It's a constant negative nine over that interval. It's a little confusing because the value ... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
Well we see the value of the function is negative nine. It's a constant negative nine over that interval. It's a little confusing because the value of the function is actually also the value of the lower bound on this interval right over here. And it's very important to look at, this says negative nine is less than x, ... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
And it's very important to look at, this says negative nine is less than x, not less than or equal. If it was less than or equal, then the function would have been defined at x equals negative nine, but it's not. We have an open circle right over there. But now let's look at the next interval. The next interval is from... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
But now let's look at the next interval. The next interval is from x is greater, or negative five is less than x, which is less than or equal to negative one. And over that interval, the function is equal to, the function is a constant six. It jumps up here. Sometimes people call this a step function. It steps up. It l... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
It jumps up here. Sometimes people call this a step function. It steps up. It looks like stairs to some degree. Now it's very important here that at x equals negative five, for it to be defined only one place. Here it's defined by this part. It's only defined over here. | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
It looks like stairs to some degree. Now it's very important here that at x equals negative five, for it to be defined only one place. Here it's defined by this part. It's only defined over here. And so that's why it's important that this isn't a negative five is less than or equal to, because then if you put negative ... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
It's only defined over here. And so that's why it's important that this isn't a negative five is less than or equal to, because then if you put negative five into the function, this thing would be filled in, and then the function would be defined to both places, and that's not cool for a function. It wouldn't be a func... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
So it's very important that this, that when you input negative five in here, you know which of these intervals you are in. You can't be in two of these intervals. If you are in two of these intervals, the interval should give you the same value, so that the function maps from one input to the same output. Now let's kee... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
Now let's keep going. We have this last interval where we're going from negative one, we're going from negative one to nine, from negative one to positive nine, and x, it starts off with negative one less than x, because you have an open circle right over here, and that's good because x equals negative one is defined u... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
Well, you see the value of our function is a constant negative seven. A constant negative seven. And we're done. We have just constructed a piece-by-piece definition of this function. And actually, when you see this type of function notation, it becomes a lot clearer why function notation is useful, even. And hopefully... | Piecewise function formula from graph Functions and their graphs Algebra II Khan Academy.mp3 |
So the equation for line A is y is equal to 3 4ths x minus 4. Line B is 4y minus 20 is equal to negative 3x. And then line C is negative 3x plus 4y is equal to 40. So to figure out if any of these lines are parallel to any of the other lines, we just have to compare their slopes. If any two of these lines have the same... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
So to figure out if any of these lines are parallel to any of the other lines, we just have to compare their slopes. If any two of these lines have the same slope and they're different lines, they have different y-intercepts, then they're going to be parallel. Now line A, it's very easy to figure out its slope. It's al... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
It's already in slope-intercept form. This is mx plus b. The slope is 3 4ths and the y-intercept, which isn't as relevant when you're figuring out parallel lines, is negative 4. So let's see what the other character's slopes are. This isn't in any kind of standard form. It's not in standard form, slope-intercept, or po... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
So let's see what the other character's slopes are. This isn't in any kind of standard form. It's not in standard form, slope-intercept, or point-slope form. But let's see what the slope of this line is. So to get it into slope-intercept form, which is really the easiest one to pick out the slope from, let's add 20 to ... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
But let's see what the slope of this line is. So to get it into slope-intercept form, which is really the easiest one to pick out the slope from, let's add 20 to both sides of this equation. So let's add 20 to both sides. The left-hand side, those cancel out, that was the whole point. You get 4y is equal to negative 3x... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
The left-hand side, those cancel out, that was the whole point. You get 4y is equal to negative 3x plus 20. And now we can divide everything by 4. Just dividing both sides of this equation by 4. We are left with y is equal to negative 3 4ths x plus 5. So in this case, y-intercept is 5, but most importantly, the slope i... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
Just dividing both sides of this equation by 4. We are left with y is equal to negative 3 4ths x plus 5. So in this case, y-intercept is 5, but most importantly, the slope is negative 3 4ths. So it's different than this guy. This is negative 3 4ths, this is positive 3 4ths. So these two guys definitely aren't parallel.... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
So it's different than this guy. This is negative 3 4ths, this is positive 3 4ths. So these two guys definitely aren't parallel. Let's move on to this guy. This guy written in standard form. Let's get the x term on the other side. So let's add 3x to both sides of this equation. | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
Let's move on to this guy. This guy written in standard form. Let's get the x term on the other side. So let's add 3x to both sides of this equation. Left-hand side, these cancel out. We're just left with 4y is equal to 3x plus 40, or 40 plus 3x, either way. Now we can divide both sides by 4. | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
So let's add 3x to both sides of this equation. Left-hand side, these cancel out. We're just left with 4y is equal to 3x plus 40, or 40 plus 3x, either way. Now we can divide both sides by 4. We have to divide every term by 4. The left-hand side, you're left with y. The right-hand side, you have 3 4ths x plus 10. | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
Now we can divide both sides by 4. We have to divide every term by 4. The left-hand side, you're left with y. The right-hand side, you have 3 4ths x plus 10. So here, our slope is 3 4ths, and our y-intercept, if we care about it, is 10. So this line and this line have the exact same slope, 3 4ths, and they're different... | Parallel lines from equation (example 2) Mathematics I High School Math Khan Academy.mp3 |
So if we want to write the same information, really, in logarithmic form, we could say that the power, the power that I need to raise 10 to, to get to 100 is equal to two, or log base 10, log base 10 of 100 is equal to two. Notice, these are equivalent statements. This is just an exponential form. This is in logarithmi... | Writing in logarithmic and exponential form Logarithms Algebra II Khan Academy.mp3 |
This is in logarithmic form. This is saying the power that I need to raise 10 to, to get to 100, is equal to two, which is the same thing as saying that 10 to the second power is 100. 10 to the second power is 100. And the way that I specified the base is by doing this underscore right over here. So underscore 10, log ... | Writing in logarithmic and exponential form Logarithms Algebra II Khan Academy.mp3 |
And the way that I specified the base is by doing this underscore right over here. So underscore 10, log base 10 of 100 is equal to two. Now they, here they ask us to rewrite the following equation in exponential form. So this is log base five of one over 125 is equal to negative three. This is, one way to think about ... | Writing in logarithmic and exponential form Logarithms Algebra II Khan Academy.mp3 |
So this is log base five of one over 125 is equal to negative three. This is, one way to think about it is saying the power that I need to raise five to, to get to one over 125 is equal to negative three, or that five to the negative three power, five to the negative three power is equal to one over 125. And we can ver... | Writing in logarithmic and exponential form Logarithms Algebra II Khan Academy.mp3 |
It's not like memorizing how to solve problems. It's learning the tools of how to solve problems and then using them and building them up in creative ways. So it's kind of like, it really does remind me of art because if you are doing like a painting or something, you have like a specific tools about like maybe paint b... | Creativity break Why is creativity important in algebra Algebra 1 Khan Academy.mp3 |
And what we're going to start off doing is just graph a plain vanilla function, f of x is equal to x squared. That looks as we would expect it to look. But now let's think about how we could shift it up or down. Well, one thought is, well, to shift it up, we just have to make the value of f of x higher so we could add ... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Well, one thought is, well, to shift it up, we just have to make the value of f of x higher so we could add a value. And that does look like it shifted it up by one. Whatever f of x was before, we're now adding one to it, so it shifts the graph up by one. That's pretty intuitive. If we subtract one, or actually let's s... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
That's pretty intuitive. If we subtract one, or actually let's subtract three, notice it shifted it down. The vertex was right over here at zero, zero. Now it is at zero, negative three, so it shifted it down. And we can set up a slider here to make that a little bit clearer. So if I just replace this with, if I just r... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Now it is at zero, negative three, so it shifted it down. And we can set up a slider here to make that a little bit clearer. So if I just replace this with, if I just replace this with the variable k, then, let me delete this little thing here, that little subscript thing that happened. Then we can add a slider k here.... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Then we can add a slider k here. And this is just allowing us to set what k is equal to. So here k is equal to one. So this is x squared plus one. And notice, we have shifted up. And if we increase the value of k, notice how it shifts the graph up. And as we decrease the value of k, if k is zero, we're back where our v... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
So this is x squared plus one. And notice, we have shifted up. And if we increase the value of k, notice how it shifts the graph up. And as we decrease the value of k, if k is zero, we're back where our vertex is right at the origin. And then as we decrease the value of k, it shifts our graph down. And that's pretty in... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
And as we decrease the value of k, if k is zero, we're back where our vertex is right at the origin. And then as we decrease the value of k, it shifts our graph down. And that's pretty intuitive, because we're adding or subtracting that amount to x squared so it changes, we could say, the y value. It shifts it up or do... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
It shifts it up or down. But how do we shift to the left or to the right? So what's interesting here is, to shift to the left or to the right, we can replace our x with an x minus something. So let's see how that might work. So I'm going to replace our x with an x minus, let's replace it with an x minus one. What do yo... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
So let's see how that might work. So I'm going to replace our x with an x minus, let's replace it with an x minus one. What do you think's going to happen? Do you think that's going to shift it one to the right or one to the left? So let's just put the one in. Well, that's interesting. Before, our vertex was at zero, z... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Do you think that's going to shift it one to the right or one to the left? So let's just put the one in. Well, that's interesting. Before, our vertex was at zero, zero. Now, our vertex is at one, zero. So by replacing our x with an x minus one, we actually shifted one to the right. Now, why does that make sense? | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Before, our vertex was at zero, zero. Now, our vertex is at one, zero. So by replacing our x with an x minus one, we actually shifted one to the right. Now, why does that make sense? Well, one way to think about it, before we put this x, before we replaced our x with an x minus one, the vertex was when we were squaring... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Now, why does that make sense? Well, one way to think about it, before we put this x, before we replaced our x with an x minus one, the vertex was when we were squaring zero. Now, in order to square zero, squaring zero happens when x is equal to one. When x is equal to one, you do one minus one, you get zero, and then ... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
When x is equal to one, you do one minus one, you get zero, and then that's when you are squaring zero. So it makes sense that you have a similar behavior of the graph at the vertex now when x equals one as before you had when x equals zero. And to see how this can be generalized, let's put another variable here and le... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
And then if h increases, we're replacing our x with x minus a larger value, that's shifting to the right, and then as h decreases, as it becomes negative, that shifts to the left. Now, right here, h is equal to negative five. You typically won't see x minus negative five. You would see that written as x plus five. So i... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
You would see that written as x plus five. So if you replace your x's with an x plus five, that actually shifts everything five units to the left. And of course, we can shift both of them together like this. So here, we're shifting it up, and then we can get back to our neutral horizontal shift, and then we can shift i... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
So here, we're shifting it up, and then we can get back to our neutral horizontal shift, and then we can shift it to the right like that. And everything we did just now is with the x squared function as our core function, but you could do it with all sorts of functions. You could do it with an absolute value function. ... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
Let's do absolute value. That's always a fun one. So instead of squaring all this business, let's have an absolute value here. So I'm gonna put an absolute, whoops, absolute value, and there you have it. You can start at, let me make both of these variables equal to zero. So that would just be the graph of f of x is eq... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
So I'm gonna put an absolute, whoops, absolute value, and there you have it. You can start at, let me make both of these variables equal to zero. So that would just be the graph of f of x is equal to the absolute value of x. But let's say you wanted to shift it so that this point right over here that's at the origin is... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
But let's say you wanted to shift it so that this point right over here that's at the origin is at the point negative five, negative five, which is right over there. So what you would do is you would replace your x with x plus five, or you would make this h variable to negative five right over here, because notice, if ... | Shifting functions introduction Transformations of functions Algebra 2 Khan Academy.mp3 |
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