Determining the effects on f(x) = x when replaced by f(x) + d or f(x - c)
About this video
Starting from a segment of f(x) = x, the video graphs g(x) = f(x) - 4 by moving every point down 4. It then graphs h(x) = f(x + 6) by evaluating points such as h(-8) = f(-2) and h(-4) = f(2), showing that adding 6 inside the function shifts the graph 6 to the left.
You'll be able to
- graph f(x) + d by shifting the graph of f up or down by d.
- graph f(x + c) by evaluating points and recognise it as a horizontal shift.
- explain why f(x + 6) shifts left rather than up.
Key words
- translation
- Sliding a graph up, down, left or right without changing its shape.
- segment
- A piece of a line with two endpoints.
- horizontal shift
- Moving a graph left or right, caused by adding or subtracting inside the function, like f(x + 6).
Transcript
- [Tutor] 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 four, 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. Alright, so let's see how I think about it. This is the segment of F of 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, if 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 four. So G of three is going to be equal to F of three, but then we're gonna subtract four from it. So it's going to be down by four or we're gonna subtract four to get to negative one.
And let's go to the other endpoint here. And actually let me scroll down a little bit so that we can see it. So we go to this other endpoint. F of negative three is of course equal to negative three, but now G of negative three is gonna be equal to that minus four.
So we're gonna shift down again. So we're going to go to minus seven, just like that. So we have now graft G, and as you can see, if you kick 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 six 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 six, then yes, we would shift up by six.
But something different is happening here. So let's think what might happen at negative eight, and I'll show you in a second why I care about negative eight. F of X is not defined at negative 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 equals negative eight, if you said H of negative eight, that's the same thing as F of negative eight plus six. So H of negative eight is the same thing of F of negative two. We know F of negative two is negative two, so H of negative 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 six or so 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 six, it shifted the graph six to the left.
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