 In this video, I'm going to talk about stretching horizontal. So I'm going to take a function. I'm going to stretch it horizontally. This is one of the types of transformations that we can do to a function. Now, the previous videos that I had, I just used simple points. This one, for stretching horizontally, it's much, much easier. And we better understand if I actually use functions, if I actually use lines and curves, as opposed to just simple points. So it's a little easier to see this way. So for this example, what I'm going to do is I'm going to perform a horizontal stretch of this function by a factor of two. So hold on a second. Let's get an idea of what horizontal stretch actually means. So I'm going to take this function, and I'm going to stretch it horizontally, horizontally. Now, by stretching horizontally, what I'm going to do is I'm actually going to widen. That's one way to describe it. I'm going to widen this graph. So actually, after I widen it, it might look something like this. I'm going to widen that. And then this curve over here is going to be widened also. So that gives you a general idea of what stretching a function horizontally is going to look like. So this would be a little bit more exact because I'm stretching by a factor of two. So now, the first thing that I need to do is I need to find a couple of points that are on this function so I can be a little bit more exact by stretching by a factor of two. So one of the more obvious points is right down here, the very bottom, which is going to be 0, negative 2. And then let's find a couple of other points. I think I only probably need three points here. So let's find another point. There's an easy one to use right there. That one is negative 1, 0. And then let's find a point on this curve. I've got to be careful. Find a point that looks like it's at an intersection. That's pretty close to there. It looks like we've got another one up here. This one looks pretty good. This one looks pretty good. That one is going to be 1, 2, 3, 1, 2. So that one's going to be 3, 2, 3, 2. So I'm going to use these three points to help me stretch this function by a factor of two. Now, as we saw earlier with that picture, we're going to widen everything horizontally. We're going to widen everything. Now, when we do that, widen horizontal. Those are all descriptors of my x-axis. Those are all descriptions of my x-axis. So what we're going to do is we're going to stretch everything by a factor of two. We're going to change all of the x-axis, the x-coordinates, all these points. We're going to change all these points by a factor of two. So this is what it's going to look like. Let's take this first point right here. So 3, 2. Now we are widening. We are stretching horizontally. So what that's going to do is we're going to take the x-coordinate, again, widening along the x-axis horizontally. That's going to take the x-coordinate, 3. And we're going to multiply by a factor of two. But the y-coordinate, we're going to leave that alone. It's only the x-coordinate that's going to be affected. When we are widening, widening, the only thing that gets affected is the x-axis. Horizontal stretching is widening, which only affects the x-coordinate. So I'm going to take the x-coordinate and multiply it by two, which is going to give me a new coordinate of 6, 2. It's going to give me a new coordinate of 6, 2. OK, now what I'm going to do is I'm going to do the same thing for all of my other points, all other two points that I have here. So my other point here is 0, negative 2, 0, negative 2, which if I multiply by a factor of two, well, coincidentally, nothing actually happens, because 2 times 0 is still going to give me 0, negative 2. So actually, that point didn't change at all. That point didn't change at all. After I graph this, we'll see how the function actually changes. So let's get to the last point, which is negative 1, 0, negative 1, 0. So change it so that I multiply the x-coordinate by 2, and that's going to give me a new coordinate of negative 2, 0, negative 2, 0. So these are the three new points that I have. Then I'm going to use to stretch this function horizontally. So I'm going to graph these. So 6, 2, OK, so here we go. We have 6, 2, so we have 1, 2, 3, 4, 5, 6, 1, 2, right here. 6, 2, right here, there's one of my new points. And then after that, my next point is going to be 0, negative 2. So actually, 0, negative 2 just stays right there. Actually, 0, negative 2 just stays right there. That point doesn't even move. And that was a nice point to use, because I don't have to really draw anything else. I don't have to draw anything new. My next point is going to be negative 2, 0. Negative 2, right here, negative 2, 0. All right, so those are my three points that I'm going to use to stretch this. So I'm going to start here, and I'm going to draw this arm out to this side. OK, very handy to have a ruler right here. Notice how it's really close down here, but then as we get farther and farther away, the space in here gets a lot wider, gets a lot wider. Stretching horizontally, we are widening this graph. And then also, OK, and then now I have my curve that's going to end up up here. So I'm going to curve this, little arrow. There we go. Notice how it's real close together, then we widen, and widen, and widen, and widen. We are widening because we are stretching horizontally. All right. So that's what it looks like with the numbers. In general, what did we do? What we did is we took our x, y coordinates. Now, as we saw earlier, since we're widening, we're only affecting the x-coordinate. So what do we do? We multiplied by that factor. So say, for example, that factor is, we'll call it A. That factor is A. We're going to take the x-coordinate. We're going to multiply by that A, and we're going to leave the y-coordinate alone. So we're going to just take that A, and we're going to multiply times the x, and we're going to leave the y-coordinate alone. Then we can also say the factor is A. We can say that the factor is A. Notice up here, it's by a factor of 2. So the factor is A. All right. So that's in general. That's kind of using algebra, lots of variables, to kind of explain that. All right, next, we have stretching. We have stretching. Oh, too far, too far. We have stretching vertically. What we're going to do is we're going to stretch this vertically now. So we're actually going to use the other act. We're going to use a different axis. Instead of using the y-axis, instead of using the y-axis, instead of the x-axis, instead of the x-axis to widen everything, we're actually going to use, we're going to stretch vertically. We're actually going to stretch everything using the y-axis. So a little bit different from what we just did. We're going to perform a vertical stretch of this function by a factor of 2. So it's going to look very similar. So I'm going to actually use, I'm going to use the same points. So this is going to be 0, negative 2. And this point over here was 3, 2. And then this point that I used over here was negative 1, 0. So I'm actually going to use those three same points to perform a vertical stretch. So I'm actually going to write down those points first. So I got 3, 2. I have 0, negative 2. And I have negative 1, 0. So now I'm going to change each one of these points. They're very similar to last time. Since I'm stretching vertically, I'm actually going to make this taller. That's one way to look at it. I'm going to make this thing taller. So what I can do is I'm getting an idea of what this is going to look like. If I make this line taller, if I stretch it up, it's going to look something like that. If I take this curve right here and I stretch it up, it's going to curve up a lot faster. So it might look something like that. So actually everything is coming. It looks like everything's going to come inside a little bit. It's going to come inside just a little bit to make this look a little bit thinner. We're stretching it vertically. We're stretching it vertically. So we can call this getting taller. We can call this getting taller. I think there's a good descriptor of it. OK, since that is getting taller, oh, let me actually get rid of that. Get rid of those lines there. And we'll call this getting taller. Put that back. Put that back. All right, so what we're going to do is, since we're getting taller, getting taller, what that tells us is that we're going to affect the y-coordinates. We're just going to affect the y-coordinates of all these points. So actually the x-coordinate is going to stay the same. We're going to change the y-coordinate by a factor of 2. So we're going to take 2 times the y-coordinate. We're going to take 2 times the y-coordinate, which is going to get us a new point, which is 2, 4. It's going to give us a new point, which is 2, 4. So now let's do that for the rest of the points. So the x-coordinate is going to stay the same. My y-coordinate is going to be multiplied by a factor of 2. So now this new coordinate is going to be 0, negative 4. And then my last point is negative 1 is the x-coordinate, which is going to stay the same. And my y-coordinate is 0, which actually is just going to stay negative 1, 0. The y-coordinate actually is not going to change at all. So that point looks like it doesn't change at all. All right, so let's take all three of these points. Let's take all three of these points and see what this function is going to look like after I have stretched this vertically. So my net, my first point is 2, 4. So 1, 2. Oh, what? That looks like I made a little bit of an error, because I'm looking at this point here. Look at this point here. Looks like I made an error. That's not going to be a 2 over there. Let me change that. That's a mistake right there. Let's change that. There we go. So it's not going to be 2, 4. Excuse me, it is going to be 3, 4. That's why we do all of our math notes with pencils. Mistakes happen every once in a while. All righty, so now that I've caught that, so 1, 2, 3, and then 1, 2, 3, 4. Right there. There's one of my new points. One of my new points. Okay, so then another point I have is 0, negative 4. Take my math real quick. Make sure I didn't make another mistake. Nope, we look good. So 0 and then negative 4. 1, 2, 3, 4. Okay, so it looks like this point actually moves, we go down with this point. Interesting, interesting. Okay, so now we have the last point, which actually doesn't move at all. It stays right there at 1, 0. So now what I'm going to do is I'm going to use those three points to create my new function. So I'm going to start here, start here, and I'm actually going to go through this point up here. Interesting, I can do a better line than that. I can do a better line than that. Let's try this one more time. One more time. There we go, a little bit better, a little bit better. Not too terribly straight, but a lot better. And then now I have this point up here, so I start here and I'm going to curve, curve to get to that point up there. Don't necessarily need to go through this point here. I want to concentrate on going through this new point. That's the old point, 3, 2, this is the new point, 3, 4. Okay, so that's what it looks like when I stretch vertically. It looks like everything is taller, but notice that this line is going more up and notice this point had to be stretched down. So not only are we stretching vertically, are we going up with everything, we're also stretching vertically down. Okay, so points that are down here are going to get farther down, points that are up here are going to get farther up. It looks like points that are actually on the x-axis, this point right here is not going to move at all. Okay, so in general, what did we do? In general, what did we do? We took our xy coordinate and since we are stretching vertically, vertically is just up and down, is just up and down, we took, let's call it b, we used a last time, we took some factor b and we multiplied it times the y coordinate, factor, factor of b. Okay, factor of b. So what we do is when we stretch vertically, we are going to take the y coordinate, we're going to take the y coordinate and we're going to multiply by that factor, whatever it might be. All right, that was two examples of stretching horizontally and stretching vertically.