 In this video, I'm going to talk about stretching and compressing functions. I'm going to go over just a couple of examples of how to horizontally and vertically stretch a function. Okay, so little directions here. Let g of x be the indicated transformation of f of x. So this is down here is going to be the transformation that we're going to do. And write the rule for g of x. Okay, so what we're going to do is we're going to write the new equation for g of x. Okay, so we're going to take this function and we're going to horizontally stretch it by a factor of 2. Okay, now what you can well imagine though is that when we horizontally stretch something, it's actually going to get smaller. The slope of it is going to go down. Keep that in mind as we go through this problem because when we check our problem at the end to see if we did things correctly, that's what we're going to base everything off of is that when we horizontally stretch something, if you horizontally stretch actually the slope is going to go down. Okay? All right, so what I'm going to do is I'm going to show you with the notation and then I'm going to show you with the notation first, then graph it. Just to show you kind of two ways to do this. All right, so what I'm going to do is I'm going to take my function. I'm going to change it by... Now if I do something horizontally to a function, it only affects the x portion of the function. It only affects the x. So what I'm going to do is I'm going to horizontally stretch by a factor of 2. It's only going to affect the x. Now here's my choice though. I can either multiply times 2 or divide by 2. That's basically my two choices there. So now I've got to think to myself, is this function going to get bigger or smaller? Now if I horizontally stretch it, the slope of it is going to get smaller. So over here on the graph, if I have a function that looks like this and then I horizontally stretch it, if I stretch everything out, it's actually going to end up looking like this second one here. So I'm going to go from 1 to 2. So that right there gives you kind of an idea of what's going to happen. That's what happens when you horizontally stretch. The slope of your function is actually going to go down. The slope of the function is actually going to go down. I'll do a little bit more exact drawing here in a minute, or graph here in a minute. So what does that mean? The slope is going to go down, which means I have to divide by 2. So I'm going to take the function and divide by 2. Or you could also say multiply by 1 half, it does the same thing. So for my new function, g of x, for the new one, I'm going to take the old one, take the old one, and multiply the x portion of it times 2, or divide by 2, or multiply by 1 half, same difference. So again, I'm going to take the x portion of it, so 3. And I'm going to replace, now notice here x, and then 1 half x. Take the x and replace it with a 1 half x. Take the x and replace it with a 1 half x. That's basically what we're doing. OK, so then my new function g of x, my new function g of x is going to be, so 3 divided by 2, that's just going to be 3 halves x minus 2. 3 halves x minus 2. So that's my new rule. That's the new function g of x. That's what it's going to look like. Now notice the slope went from 3 to 3 halves. So the slope went down. Now you can also think of 3 halves. 3 halves is 1 and 1 half. So it's exactly half of 3. So you can see there that it just went down. So now let's do that same thing. Let's do it with graphing though, in case you have a different way to do this. I'm going to take this function and graph it. Negative 2 for my y intercept and a slope of 3, 3 over 1. 1, 2, 3 over 1. And 1, 2, 3 over 1. And then here's my line. There we go. All right, this is my f function. Make sure you label it so I know which one is which. And then now I'm going to draw. Now what I'm going to do is I'm going to take these points. And if I'm horizontally stretching by a factor of 2, I take the x coordinates. Now again, remember, horizontal. So I take the x coordinates and I multiply them times 2. So the x coordinate here is 2. Make that a 4. x coordinate here is a 1. Make it a 2. So you have just multiplying times 2, doing it very quickly. This right here is an x coordinate of 0. So 0 times 2 is still going to be 0. So that point stays right where it's at. And then this is an x coordinate of negative 1 times 2 would be a negative 2. So these are my new points for g of x. These are my new points for g of x, which I had a ruler on this. There we go. All right, so that's my new function. Now notice that right there. You can visually see we stretched our function to get from f to g. We stretched everything out. And now what you can also see is the y intercept is going to be negative 2, which that's what it is. And then my slope is going to be 1, 2, 3, 1, 2. 1, 2, 3, 1, 2. So it's going to be 2 thirds, positive 2 thirds x. So we did do that correctly. So there's just two different ways to see it. You can either see it with the graphing or with the notation. There's two different ways to do it. All right, now let's do another example. Stretching and compressing. But this time we're going to vertically stretch by a factor of 2. So same deal. Let g of x be the indicated transformation of f of x. Write the rule for g of x. So what we're going to do is we're going to write the new equation. But this time what we're going to do is instead of horizontally stretch, we're actually going to vertically stretch by a factor of 2. Now what I'm going to do first is I'm actually going to do this backwards from what I did last time. I'm going to graph it first, figure out what the equation is, and then I'm going to show it again, do the problem again, but showing you how to do it with a notation. Okay, so I'm going to graph this first. So negative 2 for my y intercept. And then 1, 2, 3, 1 for my slope. 1, 2, 3, 1 for my slope. 1, 2, 3, 1 for my slope. And there is my f function. F function. Okay, now what I'm going to do is I'm going to vertically stretch by a factor of 2, which means I take now vertical stretch. Vertical means I'm going to change the y coordinates. So I'm going to take the y coordinates and multiply them times 2. And that's causing a little bit of trouble here in your CY in a second. Okay, so right here I have a y coordinate of 1. So take that times 2 is just going to be 2. All right, and then here I have a y coordinate of 1, 2, 3, 4. So I have a y coordinate of 4. Times 2 is going to be 8, which is going to be way up here. So I can't really graph that. I don't have a big enough graph to do that. Okay, so let's find something else. Right here I have a y intercept of negative 2. Negative 2 times 2 is negative 4. So negative 4 is down here. All right, very good, very good. And then, now this point here is at negative 1, negative 2, negative 3, negative 4, negative 5. So negative 5 times 2 is negative 10, which is going to be way down here. Again, I can't graph that point. But I have two points here. That's exactly what I need. I need two points to be able to write the equation of a line. All I need is two points, okay? So right there, that is going to be my g function. Now notice we are vertically stretching. So notice everything is getting taller. Everything's getting taller. The slope is getting bigger. The y intercept is getting more negative. You can call that getting bigger. Little things like that, okay? All right, so then my g of x function, my g of x function, what's it going to be? Okay, well I have to find the slope and the y intercept. My y intercept is negative 4. So right there. And then my slope, I just got to count that out. One, two, three, four, five, six, and one. Six over one, which just reduces to six. And then it's a positive slope, so I don't have to change anything. So there it is. There is the new rule for g of x, okay? There's my new function. Six x minus four. Okay, now notice compare that to the old one, three x minus two. All we did was if we vertically stretched by a factor of two, you multiply the entire function times two. Okay. That's one way of seeing it with the graphing. I'm gonna show you again with the notation. So f of x, we're going to change it by multiplying everything times two. If you vertically stretch something, if you vertically stretch something, you're gonna multiply the entire function times that number, okay? Or divide by that number if you vertically compress something because everything's gonna get smaller. All right, so my new g of x, my new function is going to be, the old function is going to be the old function, except for I'm just going to multiply times two. So I take the old function, the old function and just multiply times two, okay? Take the old function and multiply times two. And then this is the result that we would get, okay? All right, so there we go. There's two examples and two ways to do both of examples. You can either graph or you can use the notation to figure out what your new equation is going to be.