Example translating parabola

Function G can be thought of as a translated or shifted version of f.

Of x is equal to x^2. Write the equation for G of x. Now pause this video and see if you can work this out on your own.

All right, so whenever I think about shifting a function, and in this case, we're shifting a parabola, I like to look for a distinctive point. On a parabola, the vertex is going to be our most distinctive point.

If I focus on the vertex of f, it looks like if I shift that to the right by three, and then if I were to shift that down by four, at least our vertices would overlap. I would be able to shift the vertex to where the vertex of G is, and it does look, and we'll validate this at least visually in a little bit.

So, I'm going to go minus 4 in the vertical direction. That not only would it make the vertices overlap, but it would make the entire curve overlap. So we're going to make, we're going to first shift to the right by three.

We're going to think about how we would change our equation so it shifts F to the right by three, and then we're going to shift down by four. Shift down by four.

Now, some of you might already be familiar with this, and I go into the intuition in a lot more depth in other videos. But in general, when you shift to the right by some value, in this case, we're shifting to the right by three, you would replace x with x minus 3.

So, one way to think about this would be Y is equal to f of x - 3, or Y is equal to instead of it being x^2, you would replace x with x - 3, so it'd be (x - 3)^2.

Now, when I first learned this, this was counterintuitive. I'm shifting to the right by three; the x-coordinate of my vertex is increasing by 3, but I'm replacing x with x + 3. Why does this make sense?

Well, let's graph the shifted version just to get a little bit more intuition here, and once again, I go into much more depth in other videos. Here, this is more of a worked example.

So, this is what the shifted curve is going to look like. Think about the behavior that we want right over here at x = 3. We want the same value that we used to have when x equals 0. When x equals 0 for the original F, 0 squared was 0, y equals 0. We still want y equals 0.

Well, the way that we can do that is if we square zero. And the way that we're going to square zero is if we subtract 3 from x. And you can validate that at other points.

Think about what happens now when x = 4. 4 - 3 is 1 squared; it does indeed equal 1, the same behavior that you used to get at x equal to 1. So it does look like we have indeed shifted to the right by three when we replace x with x - 3.

If you replaced x with x + 3, it would have had the opposite effect; you would have shifted to the left by three. I encourage you to think about why that actually makes sense.

So now that we've shifted to the right by three, the next step is to shift down by four, and this one is a little bit more intuitive.

So, let's start with our shifted to the right, so that's y equal to (x - 3)^2. But now whatever y value we were getting, we want to get four less than that.

So when x = 3, instead of getting y = 0, we want to get y = 4 less, or -4. When x = 4, instead of getting 1, we want to get y = -3.

So whatever y value we were getting, we want to now get four less than that. So the shifting in the vertical direction is a little bit more intuitive. If we shift down, we subtract that amount; if we shift up, we add that amount.

So this right over here is the equation for G of x. G of x is going to be equal to (x - 3)^2 - 4.

And once again, just to review, replacing the x with x - 3 on F of x, that's what shifted right by three, and then subtracting the four, that shifted us down by four to give us this next graph.

You can visualize or you can verify visually that if you shift each of these points exactly down by four, we are indeed going to overlap on top of G of x.