Transforming quadratic functions
Here I've drawn the most classic parabola, y is equal to x squared. And what I want to do is think about what happens-- or how can I go about shifting this parabola. And so let's think about a couple of examples. So let's think about the graph of the curve. This is y is equal to x squared. Let's think about what the curve of y minus k is equal to x squared. What would this look like? Well, right over here, we see when x is equal to 0, x squared is equal to 0. That's this yellow curve. So x squared is equal to y, or y is equal to x squared. But for this one, x squared isn't equal to y. It's equal to y minus k. So when x equals a 0, and we square it, 0 squared doesn't get us to y. It gets us to y minus k. So this is going to be k less than y. Or another way of thinking about it, this is 0. If it's k less than y, y must be at k, wherever k might be. So y must be at k, right over there. So at least for this point, it had the effect of shifting up the y value by k. And that's actually true for any of these values. So let's think about x being right over here. For this yellow curve, you square this x value, and you get it there. And it's clearly not drawn to scale the way that I've done it right over here. But now for this curve right over here, x squared doesn't cut it. It only gets you to y minus k. So y must be k higher than this. So this is y minus k. y must be k higher than this. So y must be right over here. So this curve is essentially this blue curve shifted up by k. So making it y minus k is equal to x squared shifted it up by k. Whatever value this is, shift it up by k. This distance is a constant k, the vertical distance between these two parabolas. And I'll try to draw it as cleanly as I can. This vertical distance is a constant k. Now let's think about shifting in the horizontal direction. Let's think about what happens if I were to say y is equal to, not x squared, but x minus h squared. So let's think about it. This is the value you would get for y when you just square 0. You get y is equal to 0. How do we get y equals 0 over here? Well, this quantity right over here has to be 0. So x minus h has to be 0, or x has to be equal to h. So let's say that h is right over here. So x has to be equal to h. So one way to think about it is, whatever value you were squaring here to get your y, you now have to have an h higher value to square that same thing. Because you're going to subtract h from it. Just to get to 0, x has to equal h. Here, if you wanted to square 1, x just had to be equal to 1. So here, let's just say, for the sake of argument, that this is x is equal to 1. And this is 1 squared, clearly not drawn to scale. So that would be 1, as well. But now to square 1, we don't have to just get x equals 1. x has to be h plus 1. It has to be 1 higher than h. It has to be h plus 1 to get to that same point. So you see the net effect is that instead of squaring just x, but squaring x minus h, we shifted the curve to the right. So the curve-- let me do this in this purple color, this magenta color-- will look like this. We shifted it to the right. And we shifted it to the right by h. Now let's think of another thought experiment. Let's imagine that-- let's think about the curve y is equal to negative x squared. Well, now whatever the value of x squared is, we're going to take the negative of it. So here, no matter what x we took, we squared it. We get a positive value. Now we're always going to get a negative value once we multiply it times a negative 1. So it's going to look like this. It's going to be a mirror image of y equals x squared reflected over the horizontal axis. So it's going to look something like that. So that's y is equal to negative x squared. And now let's just imagine scaling it even more. What would y equal negative 2x squared? Well, actually, let me do two things. So what would y equals 2x squared look like? So let's just take the positive version, so y equals 2x squared. Well, now as we square things, we're going to multiply them by 2. So it's going to increase faster. So it's going to look something like this. It's going to be narrower and steeper. So it might look something like this. And once again, I'm just giving you the idea. I haven't really drawn this to scale. So increasing it by a factor will make it increase faster. If we did y equals negative 2x squared, well, then it's going to get negative faster on either side. So it's going to look something like this. It's going to be the mirror image of what I just drew. So it's going to be a narrower parabola just like that. And similarly-- and I know that my diagram is getting really messy right now-- but just remember we started with y equals x squared, which is this curve right over here. What happens if we did y equals 1/2 x squared? I'm running out of colors, as well. If we did y equals 1/2 x squared, well, then the thing's going to increase slower. It's going to look the same, but it's going to open up wider. It's going to increase slower. It's going to look something like this. So this hopefully gives you a sense of how we can shift parabolas around. So for example, if I have-- and I'm doing a very rough drawing here to give you the general idea of what we're talking about. So if this is y equals x squared, so that's the graph of y equals x squared. Let me do this in a color that I haven't used yet-- the graph of y minus k is equal to A times x minus h squared will look something like this. Instead of the vertex being at 0, 0, the vertex-- or the lowest, or I guess you could say the minimum or the maximum point, the extreme point in the parabola, this point right over here, would be the maximum point for a downward opening parabola, a minimum point for an upward opening parabola-- that's going to be shifted. It's going to be shifted by h to the right and k up. So its vertex is going to be right over here. And it's going to be scaled by A. So if A is equal to 1, it's going to look the same. It's going to have the same opening. So that's A equals 1. If A is greater than 1, it's going to be steeper, like this. If A is less than 1 but greater than 0, it's just going to be wider opening, like that. Actually, if A is 0, then it just turns into a flat line. And then if A is negative but less than negative 1, it's kind of a broad-opening thing like that. Or I should say greater than negative 1. If it's between 0 and negative 1, it will be a broad-opening thing like that. At negative 1, it'll look like a reflection of our original curve. And then if A is less than negative 1-- so it's even more negative-- then it's going to be even a steeper parabola that might look like that. So hopefully that gives you a good way of how to shift and scale parabolas.