Example scaling parabola

Function G can be thought of as a scaled version of f of x equal to x^2. Write the equation for G of x. So like always, pause this video and see if you can do it on your own.

All right, now let's work through this together. So the first thing that we might appreciate is that G seems not only to be flipped over the x-axis but then flipped over and then stretched wider. So let's do these in steps.

First, let's flip over the x-axis. So if we were to do this visually, it would look like this: instead, when x is equal to 0, y is still going to be equal to 0. But when x is equal to -1, instead of y being equal to 1, it now will be equal to -1. When x is equal to 1, instead of squaring one and getting one, you then take the negative of that to get -1.

So when you flip it, it looks like this: when x is equal to -2, instead of y being equal to 4, it would now be equal to -4. As we just talked through, as we're trying to draw this flipped-over version, whatever y value we were getting before for a given x, we would now get the opposite of it or the negative of it.

So this green function right over here is going to be y is equal to the negative of f of x, or we could say y is equal to -x^2. Whatever x is, you square it, and then you take the negative of it. You see that this will flip it over the x-axis. But that by itself does not get us to G of x.

G of x also seems to be stretched in the horizontal direction. So let's think about can we multiply this by some scaling factor so that it does that stretching so that we can match up to G of x? The best way to do this is to pick a point that we know sits on G of x. They, in fact, give us one: they show us right over here that the point (2, -1) sits on G of x. When x is equal to 2, y is equal to -1 on G of x, or you could say G of 2 = -1.

Now, on our green function, when x is equal to 2, y is equal to -4. So let's see, maybe we can just multiply this by 1/4 to get our G. Let's see if we were to scale by 1/4; does that do the trick? Scale by 1/4.

So in that case, we're going to have y is equal to not just x^2, but (1/4) x^2. If you're saying, “Hey, how did you get 1/4?” well, I looked at when x is equal to 2 on our green function. When x is equal to 2, I get (2, -4), but we want that when x is equal to 2 to be equal to -1.

Well, -1 is 1/4 of -4, so that's why I said, “Okay, let’s see if we can take our green function. If I multiply it by 1/4, that seems like it'll match up with G of x.” So let's verify that: when x is equal to 0, well, this is still all going to be equal to 0, so that makes sense.

When x is equal to 1... let me do this in another color. When x is equal to 1, then (1^2) * (1/4) well, that does indeed look like -1/4 right there. When x is equal to 2, (2^2) is 4 * (-1/4) is indeed equal to -1.

When... let's try this point here because it looks like this is sitting on our graph as well. When x is equal to 4, (4^2) is 16; 16 * (-1/4) is indeed equal to -4, and it does work also for the negative values of x as well.

So I'm feeling really good that this is the equation of G of x. G of x is equal to (-1/4) * x^2. In general, when we say we're scaling it, we're scaling it by a negative value. This is what flips it over the x-axis. Then multiplying it by this fraction that has an absolute value less than one is actually stretching it wider. If the absolute value of this value right over here was greater than one, then it would stretch it vertically or make it thinner in the horizontal direction.