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

So we're told this is the graph of function f right over here, and then they tell us that function g is defined as g of x is equal to one third f of x. What is the graph of g? If we were doing this on Khan Academy, this is a screenshot from our mobile app; it has multiple choices, but I thought we would just try to sketch it.

So pause this video; maybe in your mind, imagine what you think the graph of g is going to look like or at least how you would tackle it.

Alright, so g of x is equal to one third f of x. So for example, we can see here that f of three is equal to negative three, so g of three should be one-third that, so it should be negative one, likewise. So g of three would be right over there. Likewise, g of negative three, what would that be? Well, f of negative three is three, so g of negative three is going to be one third that or it's going to be equal to one.

F of zero is zero; one third of that is still zero, so g of zero is still going to be right over there. We know that's going to happen there and there as well. We already have a sense of what this graph is going to look like.

The function g is going to look something like this. I'm just connecting the dots, and they did give us some dots that we can use as reference points. So the graph of g is going to look something like this. It gets a little bit flattened out or a little bit squished or smushed a little bit to look something like that, and you would pick the choice that looks like that.

Let's do another example. So here we are told this is the graph of f of x and it's defined by this expression. What is the graph of g of x? g of x is this, so pause this video and think about it again.

Alright, now the key realization is it looks like g of x is if you were to take all the terms of f of x and multiply it by 2, or at least if you were to multiply the absolute value by 2, and then if you were to multiply this negative 2 by 2. So it looks like g of x is equal to 2 times 2 times f of x.

We could even set up a little table here; this is another way that we can think about it. We can think about x, we can think about f of x, and now we could think about g of x, which should be two times that. We can see that when x is equal to zero, f of x is equal to one, so g of x should be equal to two because it's two times f of x—so g of zero, I should say, is going to be equal to two.

What about when at x equals, we'll say, when x equals three? When x equals three, f of x is negative two, g of x is going to be two times that because it's two times f of x, so it's going to be negative four. So g of three is going to be negative four; it's going to be right over there.

Then maybe let's think about one more point. So f of five is equal to zero; g of five is going to be two times that, which is still going to be equal to zero, so it's going to be right over there. So the graph is going to look something like this. I'm just connecting the dots, trying to draw some straight lines; it's going to look something like this. You can see it's kind of stretched in the vertical direction, so if you were doing this on Khan Academy, it'd be multiple choice; you look for the graph that looks like that.

Let's do a few more examples. So here we're given a function g is a vertically scaled version of f. We can see that g is a vertically scaled version of f; the functions are graphed where f is solid and g is dashed. Yeah, we see that. What is the equation of g in terms of f?

So pause this video and try to think about it. Well, the way that I would tackle this is once again let's do it with a table. Let's see the relationship between f and g. So this column is x, this column is f of x, and then this column is g of x. I'll make another column right over here, and so let's see some interesting points.

So when actually I could pick zero, but zero is maybe less interesting than this point over here. So this is when x is equal to negative three; f of negative three is negative three. What is g of negative three? It looks like it is negative nine. When f is—when x is zero, f of zero is negative two. What is g of zero? It is equal to negative six.

And so we already see a pattern forming: whatever f is, g is three times that. Whatever f is, g is three times that. So we don't even actually need these big columns, but we can see that g of x is equal to three times f of x. So that is the equation of g in terms of f.