Welcome back, everyone. Up to this point, we've been talking about transformations of functions. We've looked at reflections and shifts, and in this video, we're going to focus on the stretch transformation, or to be a bit more specific, stretches and shrinks. The nice thing about the stretch and shrink transformation is, unlike shifts where we deal with two numbers in our function, the stretch and shrink only involve one number. With stretches and shrinks, there are a few things you'll have to keep track of and remember for how the graph is going to behave, but in this video, we're going over a bunch of different scenarios and examples that will hopefully make this topic super clear. So let's get into this.

A stretch or a shrink occurs when some constant is multiplied either inside or outside of the function. To understand this better, let’s look at these two cases down here. In this example, on the left, we have a vertical stretch or compression. I want to mention right off the bat whenever you see the word 'compression' or 'shrink,' these words mean the exact same thing. So that's just something to keep in mind because these will often be used interchangeably.

In this example here, we have a function f(x), which is plotted on this graph; it's the dotted line here, and we're looking at what this function would look like if it went through a vertical stretch and a vertical shrink. Imagine taking this function and vertically stretching it. What that would do is cause the graph that we have right here, which is the vertical stretch. Now, if you were to take this function that we have in the middle and compress it, you would end up with a vertical shrink, which looks like this graph in here.

But now let's take a look at the horizontal case. For the horizontal situation, we have the same function that we had before, which is the dotted curve that you see right here. When going through a horizontal stretch, you can imagine taking the graph and stretching it horizontally. If you were to do that, you would get a stretch which looks like this. So this is the function after you've stretched the graph horizontally, whereas if you were to perform a horizontal compression, the graph would end up looking like this. Notice how it looks like we just took our graph and squeezed it closer to the y-axis.

The idea of a horizontal stretch or compression depends on where the constant is multiplied. If you see a vertical stretch or compression, this means the constant is multiplied outside the function. Whereas if you see a horizontal stretch or compression, the constant is multiplied on the inside of the function. In the vertical case, whenever you have a constant between 0 and 1, the graph is going to vertically shrink. You're going to see a vertical shrink if this happens, whereas if your constant is greater than 1, it means the graph is going to stretch.

In the case of the horizontal stretch or compression, if your constant is between 0 and 1, then your graph is going to do a horizontal stretch, whereas if the constant is greater than 1, the graph is going to horizontally shrink. Notice how the vertical stretch or compression is the opposite of the horizontal stretch or compression when it comes to what constant stretches versus shrinks the graph. This is something important to keep in mind when solving problems as well.

Now, let's actually see if we can try an example where we have this type of situation. In this example, we are given the function f(x), which is plotted on this graph to the right, and we're asked to sketch the graphs of the following functions where we have some kind of stretch or compression happening to this function. Let’s first start with the case that we see on the left, which is a. For case a, we have 2 times the function, and the 2 is multiplied outside f(x). Whenever you have your constant multiplied outside of the function, this corresponds to the vertical case and, since our constant we see is 2, that's greater than 1, which means we're going to have a vertical stretch. What this means is our graph is going to vertically stretch by a factor of 2, so we can see right here that we have the point (2,2), but our y values are going to stretch vertically, so we're going to end up actually at (2,4), and at this point where we have (1, -1), we would stretch down here to (1, -2). Likewise, at this point (-1,1), we would stretch up here to (-1,2), and then at this point, we would stretch down here to (-1, -4). So our vertical stretch will look something like this, where our graph would be stretched vertically. But now let's take a look at the second case where we have 0.5 being multiplied on the outside of the function. Since this constant is outside of the function, we're still going to have the vertical case for the stretch or compression, but notice that the constant is now between 0 and 1, which means we're going to have a vertical shrink. So everything is going to shrink by a factor of 0.5; rather than being at this point (2,2), we're going to be at (2,1) because we're shrinking our graph. Rather than being here at one, we're actually going to be at (1, -0.5). Likewise, we'd be at a y-value of positive 0.5 there, and we'd be at a y-value of negative 1 right here. So notice that in this case, we have a vertical shrink.

Now, the last case we're going to take a look at is this situation where we have a 0.5 multiplied inside the function. Since we're on the inside of the function this corresponds to the horizontal stretch or compression, and since the constant that we have is between 0 and 1, this corresponds to a horizontal stretch. This one can be a little bit tricky to do just by looking at the graph, but what you can basically imagine is since we have this x value here that's being multiplied by a 0.5 on the inside, the way that we would get back to our original function is if we doubled all the x values because multiplying this by 2 would cancel the 2 that we have there, so we actually want to double all the x-values that we see. So, rather than our original function, which is in here, rather than it being at 2, it would be over here at 4, and down here rather than being at an x value of 1 we'd be at an x value of 2, and rather than being at -1 we would be here at -2, and then rather than being at -2 we would be over here at -4. So in this case, our graph is going to horizontally stretch. Notice how we basically stretched it on the horizontal axis.

This is the basic idea behind the stretch transformation. Hopefully, you found this video helpful, and let me know if you have any questions.