Determining the effects on f(x) = x when replaced by af(x) or f(bx) | Khan Academy

We're told here is a graph of a segment of f of x is equal to x, and so they've graphed that segment right over here. Then they tell us that g of x is equal to -2 times f of x, and they want us to graph g. So think about how you would approach that now. Let's work on this together.

So g of x, whatever x is, I would evaluate f of x, and then I would multiply that by -2. So let's pick, say, when we are at x = 3. f of 3, it looks like that is 3. In fact, we know that's 3 because f of x is equal to x. So f of 3 is 3, but if we want to figure out g of 3, that's going to be f of 3 * -2, which is 3 * -2, which is -6. So f of 3 is -6.

Now let's think about when x is equal to -5. We see that f of 5 is 5, which makes sense. But if we were to take g of 5, that's going to be -2 * f of 5, so it's going to be -2 * 5, which is -10. So it would get us down here.

So really, what you see happening when we multiply f of x by -2, well, if we multiply it by two, we would just be scaling everything up by a factor of two, but then that negative flips it over the x-axis to get what we see right over here.

Let's do another example, but it's going to be a little different. Here, instead of multiplying times our f of x, we're multiplying the x by a number. Here’s a graph of the segment f of x is equal to x; we see that again. Now they've defined h of x as being equal to f of (1/3)x.

So let's graph h. One way to think about it is I know what f of 2 is; f of 2 is equal to 2. Now for h, I could actually input 6 in here. I could figure out what h of 6 is. How do I know what that is? How do I know I can do that? Because h of 6 is going to be f of (1/3) * 6. Another way of saying it, h of 6 is going to be the same thing as f of 2.

So h of 6 is the same thing as f of 2, which is 2. Then we could do that on the negative side. For example, we know that f is defined at -3; f of -3 is -3. Now, if we were to go three times that value and we would say what is h of 9? h of 9, we could go over here. h of 9 is equal to f of (1/3) * 9, or it's going to be the same thing as f of -3. f of -3 is -3, so h of 9 is -3.

So notice now we are scaling; we're making it wider when we multiplied inside of our function. As we multiply x times a fraction, if we multiplied this times a value greater than one, then we would be squeezing it in the horizontal direction.