Scaling functions vertically: examples

- [Instructor] So, we're told that 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? And if we were doing this on Khan Academy, this is a screenshot from our mobile app, it has multiple choices, but I thought we could 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. All right, 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, and 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. And we know that's going to happen there and there as well, and so we already have a sense of what this graph is going to look like. The function g is going to look something like, 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 squooshed or smooshed 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 the graph of f of x and it's defined by this expression. What is the graph of g of x, and g of x is this. So pause this video and think about it again. All right, now the key realization is, is it looks like g of x is, if you were to take all the terms of f of x and multiply it by two, or at least if you were to multiply the absolute value by two, and then if you were to multiply this negative two by two. So it looks like g of x is equal to two times, two times f of x. And we could even set up a little table here, this is another of the way that we can think about it. We can think about x, we can think about f of x, and now we can think about g of x, which should be two times that. So 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 x is going to be equal to-- Or 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, 'cause it's two times f of x, so it's going to be negative four. So, g of x, or I should say g of three is going to be negative four. It's going to be right over there. And 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. And so the graph is going to look something like this, I'm just really just connecting, 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'd 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. So we can see that g is a vertically scaled version of f. The functions are graphed where f is a 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 and 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 two 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. And 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.