Scaling functions horizontally: examples | Transformations of functions | Algebra 2 | Khan Academy

We are told this is the graph of function f. Fair enough. Function g is defined as g of x is equal to f of 2x. What is the graph of g? So, pause this video and try to figure that out on your own.

All right, now let's work through this. The way I will think about it, I'll set up a little table here, and I'll have an x column, and then I will have an actually just input g of x column. Of course, g of x is equal to f of 2x. So when x is, and actually let me see, when x is equal to—I could pick a point like x equaling 0.

So g of 0 is going to be f of 2 times 0. So it's going to be f of 2 times 0, which is still f of 0, which is going to be equal to a little bit over 4. So which is equal to f of 0. And so they're going to both have the same y-intercept. But interesting things are going to happen the further that we get from the y-axis, or as our x increases in either direction away or as x gets bigger in either direction from 0.

So let's think about what's going to happen at x equals 2. So at x equals 2, g of 2 is going to be equal to f of 2 times 2, which is equal to f of 4. And we know what f of 4 is; f of 4 is equal to 0. So g of 2 is equal to f of 4, which is equal to 0.

So notice the corresponding point is kind of gotten compressed in or squeezed in or squished in in the horizontal direction. And so what you see happening, at least on this side of the graph, is everything's happening a little bit faster. Whatever is happening at a certain x is now happening at half of that x. So this side of the graph is going to look something—I'll try to draw a little bit better than that—it's going to look something like this. Everything's happening twice as fast.

What happens when you go in the negative direction? Well, think about what g of negative 2 is. g of negative 2 is equal to f of 2 times negative 2, which is equal to f of negative 4, which we see is also equal to 0. So g of negative 2 is 0. And you might be thinking, why did you pick 2 or negative 2? Well, the intuition is that things are going to be squeezed in, and things are happening twice as fast.

So whatever was happening at x equals 4 is not going to happen at x equals 2. Whatever's happening at x equals negative 4 is never going to happen at x equals negative 2. And I saw that we were at very clear points at x equals negative 4 and x equals 4 on f, so I just took half of that to pick my x values right over here.

And then so what our graph is going to look like is something like this. It's going to look like it's been squished in from the right and the left. Now let's do another example. So now they've not only given the graph of f, they've given an expression for it. What is the graph of g of x, which is equal to this business? So pause this video and try to figure that out.

All right, the key is to figure out the relationship between f of x and g of x. And what we can see—the main difference is that instead of an x here and f of x, we have an x over 2. So everywhere there was an x, we have been replaced with an x over 2.

So another way of thinking about it is g of x is equal to f of not x but f of x over two. Or another way of thinking about it, g of x is equal to f of one half x. And then we can do a similar type of exercise, and they've given us some interesting points: the points 2, the point x equals 2, the point x equals 4, and the point x equals 6.

So let's think about this: last time when it was g of x is equal to 2x, things were happening twice as fast. Now, things are going to happen half as fast. And so what I would do—let me just set up a little table here. The interesting x values for me are the ones that if I take half of them, then I'm going to get one of these points.

So actually, let me write this half, one half x, and then I can think about what g of x is equal to f of one half x is going to be. So I want my one half x to be, let's see, it could be 2, 4, and 6. 2, 4, and 6. And why did I pick those again? Well, it's very clear what values f takes on at those points.

So if one half x is 2, then x is equal to 4. If one half x is 4, then x is equal to 8. If x is equal to 12, then one half x is 6. And so then we could say, all right, g of 4 is equal to f of 2, which is equal to 0. That's why I picked 2, 4, and 6. It's very easy to evaluate f of 2, f of 4, and f of 6.

It gave us those points very clearly, so g of 8 is going to be equal to f of one half of 8 or f of 4, which is equal to negative 4. And then g of 12 is equal to f of 6, which is half of 12, which is equal to 0 again. So then we can plot these points, and we get a general sense of the shape of the graph.

So let's see: g of 4 is equal to 0, g of 8 is equal to negative 4, right over there. And then g of 12 is equal to 0 again. So everything has been stretched out. So there you go; it's been stretched out in at least in the horizontal direction is one way to think about it—in the horizontal direction.

You can see that this point in f corresponds to this point in g; it's gotten twice as far from the origin because everything is growing half as fast. You input an x, you take a half of it, and then you input it into f. And then this point right over here corresponds to this point; instead of happening at 4, this vertex point is now happening in 8.

And last but not least, this point right over here corresponds to this point; instead of happening at 6, it's happening at 12. Everything is getting stretched out. Let's do one more example. f of x is equal to all of this. We have to be careful; there's a cube root over here, and g is a horizontally scaled version of f.

The functions are graphed, where f is solid and g is dashed. What is the equation of g? So pause this video and see if you can figure that out.

All right, let's do this together. It looks like they've given us some points that seem to correspond with each other. To go from f to g, it looks like these corresponding points have been squeezed in closer to the origin.

What we can see is that f of negative 3 seems to be equal to g of negative 1. And f of 6 over here seems to be equal to g of 2. Or another way to think about it: whatever x you input in g, it looks like that's going to be equivalent to 3 times that x inputted into f.

So g of x is equal to f of 3x. And so if you want to know the equation of g, we just evaluate f of 3x. So f of 3x is going to be equal to—and I could just actually put an equal sign like this—f of 3x is going to be equal to negative 3 times the cube root of, instead of an x, I'll put a 3x right over there, 3x plus 2, and then we have plus 1.

And that's it; that's what g of x is equal to. It's equal to f of 3x, which is that we substituted this x with a 3x, and we are done.