Determining the effects on f(x) = x when replaced by f(x) + d or f(x - c) | Khan Academy

We're told here is a graph of a segment of f of x is equal to x. That's this graph right over here. And they say that g of x is equal to f of x minus 4. Graph g, and we can graph g with this little widget here. Now I would normally ask you to pause this video and try to do it on your own, but you don't have access to this widget. So you can think about what I should do, and then I will do it.

All right, so let's see how I think about it. This is the segment of f of x, x is equal to x. Now g of x is defined by whatever f of x is; we now subtract four from that. So, for example, f of three, if you put three into f, you get f of three is actually equal to three. And that makes sense; f of x is equal to x, f of three is equal to three. But now g of three is going to be equal to f of three minus 4. So g of three is going to be equal to f of three, but then we're going to subtract four from it. So it's going to be down by four, or we're going to subtract four to get to negative one.

Let's go to the other endpoint here, and actually let me scroll down a little bit so that we can see it. So if we go to this other endpoint, f of negative 3 is of course equal to negative 3, but now g of three is going to be equal to that minus four. So we're going to shift down again, so we're going to go to negative seven, just like that. So we have now graphed g, and as you can see, if you take f of x and you subtract four from it, you have just shifted the whole graph down by four.

Let's do another example. So this one here, here's a graph of a segment of f of x equals x. And now they're saying h of x is equal to f of x plus 6. Graph h, so we have to be careful here. You might say, "Oh, maybe this just shifts up by six," but that's not exactly what's happening here. If this was f of x and then that whole thing plus 6, then yes, we would shift up by six, but something different is happening here.

So, let's think what might happen at eight, and I'll show you in a second why I care about negative eight. f of x is not defined at g of eight, at least the way it is here. We don't see the graph going all the way down to negative eight, at least if you view it as being defined by this graph. But if you say x equal eight, if you said h of eight, that's the same thing as f of eight plus 6. So h of eight is the same thing as f of negative two. We know f of negative two is negative two, so h of eight is going to be negative two.

And then now let's move forward, and I'm picking these points intentionally to h of negative four. So if we say h of negative four, that is the same thing as f of negative four plus 6, or it's the same thing as f of two. f of two is two, so h of negative four is the same thing as f of two. Now what just happened here? Well, it looks like when we replaced our x with x plus 6, it shifted the graph six to the left.