Welcome back everyone. So up to this point, we've been talking about transformations of functions. We've taken a look 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. Now, the nice thing about the stretch and shrink transformation is unlike shifts where we're dealing with two numbers in our function, the stretch and shrink only deals with one number. Now with stretches and shrinks. There are a few things that you'll have to track of and remember for how the graph is going to behave. But in this video, we're going to be 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. Now, to understand this a bit better, let's take a look at these two cases down here. Now, for this example, we have on the left, this is an example of a vertical stretch or compression and something I want to mention right off the bat is 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. Now, in this example, here we have a function F of X and this function is plotted on this graph. It's this 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. So if this function went through a vertical stretch, you can 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. And 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 looked like this. So this is the function after you've stretched the graph horizontally. Whereas if you were to do 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. So that's the idea of a horizontal stretch or compression. Now, whether you see the vertical case or the horizontal case is going to depend on where the constant is multiplied. So 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 in side of the function. And when it comes to the vertical case, whenever you have a situation where the constant that you have is between zero and one, the graph is going to vertically shrink. So you're going to see a vertical shrink if this happens. Whereas if you see your constant is greater than one, then this means the graph is going to stretch. Now, in the case for the horizontal stretcher compression, if you see that your constant is between zero and one, then your graph is actually going to do a horizontal stretch. Whereas if the constant is greater than one, the graph is going to horizontally shrink. So notice how the vertical stretch or compression is opposite of the horizontal stretch or compression when it comes to what constant stretches versus shrinks the graph. So that's 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 of 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. So let's first start with this case that we see on the left here, which is a for case A, we have two times the function and the two is multiplied outside F of X. Whenever you have your constant multiplied outside of the function, this corresponds to the vertical case. And since our constant, we see is two, that's greater than one, which means we're going to have a vertical stretch. So what this means is our graph is going to vertically stretch by a factor of two. So we can see right here that we have the 0.22 but our Y values are going to stretch vertically. So we're going to end up actually at 24. And at this point where we have one negative one, we would stretch down here to one, negative two, right? Because we're vertically stretching. Likewise, at this point, negative 11, we would stretch up here to, to negative 12. And then at this point, we would stretch down here to negative four as a Y value. So our vertical stretch would 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 one half 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 stretcher compression. But notice that the constant is now between zero and one, which means we're going to have a vertical shrink. So everything is going to shrink by a factor of one half. Rather than being at this 0.22 we're going to be at 21 because we're shrinking our graph. And rather than being here at uh at one negative one, we're actually going to be at one negative one half. Likewise, we'd be at a Y value of positive one half there. And then we would be at a Y value of negative one 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 one half multiplied inside the function. So in this case, since we're on the inside of the function, this corresponds to the horizontal stretcher compression. And since the constant that we have is between zero and one, this corresponds to a horizontal stretch. Now 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 one half 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 two would cancel the two 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 two, it would be over here at four. And down here, rather than being at an X value of one, we'd be at an X value of two. And rather than being at negative one, we would be here at negative two. And then rather than being at negative two, we would be over here at negative four. So in this case, our graph is going to horizontally stretch, notice how we basically stretched it on the horizontal axis. So this is the basic idea behind the stretch transformation. Hopefully, you found this video helpful and let me know if you have any questions.