Reflecting & compressing functions | Mathematics III | High School Math | Khan Academy

• [Voiceover] So we're told g of x is a transformation of f of x. The graph below shows f of x as a solid blue line. So this is the graph of y is equal to f of x, and the g of x is a dotted red line. So that's the graph of y is equal to g of x. What is g of x in terms of f of x? And they gave us some choices here, and I encourage you to pause the video and see if you can figure this out.

Well, there's a couple of ways that you can think about it. One is if you just eyeball it, it looks like if you flipped f of x over the x-axis, it looks a little bit like g(x), but g(x) looks like a version of that that's diminished a little bit.

So for example, if you were to flip it perfectly over the x-axis, you would get something—you would get something that looks like—and I'm just going to sketch it. So if you just flipped perfectly over the x-axis, you would get something that looks like this, and trying my best to eyeball it.

It would look something like this. If you perfectly just flipped it over, it would look something like that. I might not get it perfectly right. But that would be—this right over here. So let me be clear. If this is y is equal to f of x, then this line right over here that I just drew, that would be y is equal to negative f of x.

Because whatever f of x would give you, just take the negative of it you're flipping over the x-axis and now g of x looks like a diminished version of that, and if I were to just eyeball it, it looks like it's roughly 1/3 of this.

So my initial guess—and we can verify this little bit—is that this right over here is 1/3 the value of this. So I would say it is... So g of x is equal to 1/3 of negative f of x or negative 1/3 of f of x.

And that is not one of the choices, which makes me extra cautious, but let me just emphasize why I like this choice. Because we said this is negative f of x right there, and it looks like for any x value, what g of x is is 1/3 of that.

So instead of getting to four, we're getting to a little bit over one. Instead of getting to three right over here, we're getting to one. Instead of getting to one right over here, we are only getting to 1/3.

So it looks like it is 1/3 of this line that I just hand drew, which is negative f of x. So it would be 1/3 of negative f of x, which would be negative 1/3 f of x.

Once again, not a choice here, but let's actually look at some values. So if we said, "Let's see some values where it looks like we're hitting integer values." So for example, right at this point, right over here, it looks like f of negative seven is equal to negative one.

It looks like g of negative seven... So it looks like g of negative seven is equal to positive 1/3, positive 1/3, if I am just eyeballing it. So that seems consistent with this, because if f of negative seven is negative one, you take the negative 1/3 times that, and you get positive 1/3.

We can do that in a couple of other places. If we look right over... If we look right over here, it looks like f of negative one. f of negative one is equal to negative three, and it looks like g of one is negative 1/3 times that.

Negative 1/3 times that is that point right over there. Negative 1/3 times negative three is positive one. So g of... g is colored in that red color.

So we get for that point right over there, we get g of negative one is equal to positive one. So once again, this is negative 1/3 times this right over here.

So I feel pretty confident in my answer. g of x is equal to negative 1/3 times f of x. Once again, negative f of x would just flipped over, and then multiplying it by the 1/3 diminishes that, which we see right over there.

So I feel pretty confident with my none of the above.