Scaling & reflecting parabolas

- [Instructor] Function G can be thought of as a scaled version of F of X is equal to X squared. Write the equation for G of X. So like always, pause this video and see if you can do it on your own. Alright 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. So first let's flip over, flip over the x-axis. So if we were to do this visually it would look like this. Instead when X is equal to zero, Y is still gonna be equal to zero. But when X is equal to negative one, instead of Y being equal to one, it'd now be equal to negative one. When X is equal to one, instead of squaring one and getting one, you then take the negative of that to get to negative one. So when you flip it, it looks like this. Y when is X is equal to negative two instead of Y being equal to four, it would now be equal to negative four. So it would look like this. So as we just talk 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 negative X squared. Whatever the X is, you square it, and then you take the negative of it. Whatever X is, you square it, and then you take the negative of it, and you see that that 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. And so let's think about, can we multiply this times some scaling factor so that it does that stretching so that we can match up to G of X? And the best way to do this is to pick a point that we know sits on G of X, and they in fact give us one. They show us right over here that at the point two comma negative one, sits on G of X. When X is equal to two, Y is equal to negative one on G of X. So you could say G of two is negative one. Now on our green function, when X is equal to two Y is equal to negative four. So let's see. Maybe we can just multiply this by 1/4 to get our G. So let's see. If we were to, let's see if we scale by 1/4, does that do the trick? Scale by 1/4. So in that case, we're gonna have Y is equal to not just negative X squared, but negative 1/4 X squared. And if you're saying hey, so how did you get 1/4? Well I looked at when X is equal to two. On our green function, when X is equal to two I get to negative four. Well we want that when X is equal to two to be equal to negative one. Well negative one is 1/4 of negative four, so that's why I said okay, well let's up take to see if we could take our green function, and if I multiply it by 1/4, that seems like it will match up with G of X. And so let's verify that. When X is equal to zero, well this is still all gonna be equal to zero so that makes sense. When X is equal to one, let me do this in another color, when X is equal to one, then one squared times negative 1/4, well that does indeed look like negative 1/4 right there. When X is equal to two, two squared is four, times negative 1/4 is indeed equal to negative one. Let's try this point here 'cause it looks like this is sitting on our graph as well. When X is equal to four, four squared is 16. 16 times negative 1/4 is indeed equal to negative four. 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 negative 1/4 times X squared. And so in general, that when we were saying we were scaling it, we're scaling it by negative value. This is what flips it over the x-axis, and then multiplying it by this fraction that has an absolute value less than one, this is actually stretching it wider. If this value right over here, its absolute value was greater than one, then it would stretch it vertically, or would make it thinner in the horizontal direction.