 So there's a couple more graph transformations that are worth talking about. So let's take a look at the difference between the graph of y equals f of x and the graph of y equals c, f of x. So again, I'll start with my generic graph of y equals f of x, drop a point on the graph of y equals f of x, and if I'm at a point on the graph of y equals c, f of x, well that's going to be x coordinate whatever it is, y coordinate c times f of x. So here's my point on the graph of y equals c, f of x, and my horizontal component, my x coordinate hasn't changed, my vertical, my y coordinate is c times greater. So that suggests that my point is going to be c times farther away from the horizontal axis. Now it's important to recognize that this is a multiplicative change and not an additive change. So what that means is that all points are going to do that. The farther away you are, the greater the effect of that multiplication, the greater the actual distance. If you're really close to the x-axis, you don't get too much farther away. If you're an intermediate distance, if you're a farther distance, you get farthest away. And so this is true for all points on the graph of y equals f of x. So my final graph is going to look something like that, and this suggests that the graph of y equals c times f of x is going to be my original graph f of x stretched vertically, stretched vertically outwards by some factor c. Now if c is greater than 1, this will actually look like an expansion by some factor of c. While if c is less than 1, it's going to be compressed by some factor of 1 over c. So for example, if c is a half, then the compression 1 over a half is going to be compressed by a factor of 2. It's going to be squashed to half the size. Now how about y equals f of x and y equals f of c of x? So again, I have my generic graph. I have a point on the generic graph. And I want to get to some point on the graph of y equals f of c of x. So again, here's the point x, f of c of x. My y coordinate is my function value. And I have this point on the graph original. And I'm going to slide it over. So my cx horizontal coordinate becomes just plain x horizontal. And I'm going to be c times near to the vertical axis. Same vertical component, f of c of x in both cases. And again, I can do that for all points. And again, the farther away I start, the more in absolute terms that I'm shifted. If I start close by, I don't get too much closer and so on. And my graph looks like this. And that suggests the graph of f of c of x will be the graph of y equals f of x. And I'm going to undergo a horizontal compression by a factor of c. And again, if c is greater than 1, this will look like a compression. If c is less than 1, this looks like an expansion. So for example, let's take a look at the graph of y equals f of x. So again, here's our graph, y equals f of x. And I want to sketch the graph, y equals 1 half f of 1 half x plus 4. So again, we'll follow the order of operations and sketch several intermediate graphs. The expressions inside the parentheses have to be evaluated first. So that means I have to take care of 1 half x plus 4. And I have both a product, 1 half x, and a sum. And order of operations says product gets handled first. So 1 half x. So I want to take a look at the graph of y equals f of 1 half x. Well, that's the same as the graph of y equals f of x. And this is going to be stretched out horizontally by a factor of 2. It's going to be pulled made wider by this factor of 2. And it'll look something like that. Now, the next thing inside the parentheses is this plus 4. So I have the graph of y equals f of 1 half x. I want to find the graph of f of 1 half x plus 4. And that's going to be a horizontal shift by 4 units. And the thing to remember, the tricky part about this, is to remember that horizontal shift, because this is plus 4. That horizontal shift is going to be to the left. So I'm going to do that horizontal shift to the left, to there. And now I have the graph of y equals f of 1 half x plus 4. I do want the graph of y equals 1 half f of x plus 4. So I need to multiply everything by 1 half. And so my final graph, what I want, f of 1 half x plus 4, is going to be the graph of y equals f of 1 half x plus 4. This graph here, compressed vertically by a factor of 1 half. I'm going to squash this down, make it half as big. And there's my final graph.