Shifting functions | Mathematics III | High School Math | Khan Academy

So we have these two graphs that look pretty similar: Y is equal to F of x and Y is equal to G of x. What they ask us to do is write a formula for the function G in terms of F.

Let's think about how to do it, and like always, pause the video and see if you can work through it on your own.

All right, well, what I like to do is focus on this minimum point because I think that's a very easy thing to look at. Both of them have that minimum point right over there. We could think about how do we shift F, at least especially this minimum point? How do we shift it to get it overlapping with G?

Well, the first thing that might jump out at us is that we would want to shift to the left, and we'd want to shift to the left four. So let me do this in a new color. I would want to shift to the left by four. So we have shifted to the left by four, or you could say we shifted by 4, either way you could think about it.

Then we need to shift down. We need to go from Y = 2 to Y = -5, so let me do that. Let's shift down. So we shift down by seven, or you could say we have a -7 shift.

So how do you express G of x if it's a version of F of x that's shifted to the left by four and shifted down by seven? Or you could say it had a -4 horizontal shift and had a -7 vertical shift.

Well, one fun way to think about it is: G of x is going to be equal to F of (x - horizontal shift) + vertical shift.

Well, what is our horizontal shift here? Well, we're shifting to the left, so it was a negative shift, so our horizontal shift is -4.

And what's our vertical shift? Well, we went down, so our vertical shift is -7. So it's -7.

So there you have it: G of x is equal to F of (x + 4) - 7.

And when I look at things like this, the 7 is somewhat more intuitive to me, as I shifted it down. It makes sense to have a -7, but at first, when you work on these, you say, "Hey wait, I shifted to the left. Why is it a plus? Why is it a +4?"

The way I think about it is, in order to get the same value out of the function, instead of inputting x = 0, so if you want to get the value of F of 0, you now have to put x = -4 in, and then you get that same value. You still get to zero.

So that's—I don't know if that helps or hurts in terms of your understanding, but it often helps to try out some different values for x and see how it actually does shift the function. If you're just trying to get your head around this piece, the horizontal shift, I recommend, you know, not even using this example. Use an example that only has a horizontal shift, and it'll become a little bit more intuitive.

We have many videos that go into much more depth to explain that. Let's do another example of this.

So here we have Y = G of x in purple and Y = F of x in blue. They say, given that F of x is equal to √(x + 4) - 2, write an expression for G of x in terms of x.

First, let me just write an expression for G of x in terms of F of x. We can see, once again, it's just a shifted version of F of x. Remember, I'll just write in general: G of x is going to be equal to F of x - horizontal shift + vertical shift.

So to go from F to G, what is your horizontal shift? Well, your horizontal shift, if you take this point right over here, which should map to that point once we shift everything, your horizontal shift is two to the left. So, you could say it's a -2 horizontal shift.

So that should be -2. Then what is our vertical shift? Well, our vertical shift is we go from Y = -2 to Y = 3, so we're shifting five up. This is a vertical shift of positive five, so your vertical shift is five.

If we just wanted to write G of x in terms of F of x, like we just did in the previous example, we could say G of x is going to be equal to F of (x + 2) + 5. But that's not what they asked us to do. They asked us to write an expression for G of x in terms of x.

So here we're actually going to use the definition of F of x. Let me make it clear: We know that F of x is going to be equal to √(x + 4) - 2.

So given that, what is F of (x + 2)? Well, F of (x + 2) is going to be equal to everywhere we see an x, we're going to replace it with (x + 2): √((x + 2) + 4) - 2, which is equal to √(x + 6) - 2.

Well, that's fair enough; that's just F of (x + 2).

Now, what is F of (x + 2) + 5? So F of (x + 2) + 5 is going to be this thing right over here plus 5, so it's going to be equal to √(x + 6) - 2 + 5.

What we end up with is going to be √(x + 6) - 2 + 5, which is +3.

So that is equal to G of x. Just as a reminder, what did we do here? First, I expressed G of x in terms of F of x. I said, "Hey, to get from F of x to G of x, I shift two to the left."

Two to the left is a little counterintuitive: that's +2, which makes it a shift of two to the left. This was -2, which would be a shift of two to the right.

But like I just said in the previous example, it's good to try out some x's to see why that makes sense. Then we shifted five up.

This was G of x in terms of F of x, but then they told us what F of x actually is in terms of x.

So I said, "Okay, well, what is F of (x + 2)?" F of (x + 2) we substituted (x + 2) for x and we got this. But G of x is F of (x + 2) + 5, so we took what we figured out: F of (x + 2) is, and then we added five. That's what G of x is, and then we are all done.