What do quadratic approximations look like

In the last couple of videos, I talked about the local linearization of a function. In terms of graphs, there's a nice interpretation here. If you imagine the graph of a function and you want to approximate it near a specific point, you picture that point somewhere on the graph. It doesn't have to be there; it could choose to be kind of anywhere else along the graph.

But if you have some sort of point and you want to approximate the function near there, you can have another function whose graph is just a flat plane. Specifically, it's a plane that is tangent to your original graph at that point. That's kind of visually how you think about the local linearization.

What I'm going to start doing here in this next video, and in the ones following, is talking about quadratic approximations. Quadratic approximations take this to the next level. First, I'll show what they look like graphically, and then I'll show you what that actually means in formulas.

Graphically, instead of having a flat plane, you have a few more parameters to deal with. You can give yourself some kind of surface that hugs the graph a little bit more closely. It's still going to be simpler in terms of formulas; it can still be notably simpler than the original function. But this actually hugs it closely.

As we move around the point that it's approximating near, the way that it hugs can look pretty different. If you want to think graphically about what a quadratic approximation is, you can basically say if you slice this surface—this kind of ghostly white surface—in any direction, it'll look like a parabola of some kind.

Notice that, given that we're dealing in multiple dimensions, that can make things look pretty complicated. For example, if you slice it kind of in this direction, moving things about, if you look at it from this angle, it kind of looks like a concave up parabola. But from another direction, it kind of looks concave down.

All in all, you get a surface that actually has quite a bit of information carried within it. By hugging the graph very closely, this approximation is going to be even closer because, near the point where you're approximating, you can take a couple of steps away, and the approximation is still going to be very close to what the graph is. It's only when you step really far away from the original point that the approximation starts to deviate away from the graph itself.

This is going to be something that, although it takes more information to describe than a local linearization, gives us a much closer approximation. So a linear function, which you know, one that just draws a plane like this, in terms of actual functions, what this means is: I'll kind of write linear.

This is going to be some kind of function of X and Y. What it looks like is some kind of constant, which I'll say a, plus another constant times the variable X, plus another constant times the variable Y. This is sort of the basic form of linear functions.

Technically, this isn't linear. If one were going to be really pedantic, they would say that that's actually a line, because, strictly speaking, linear functions shouldn't have this constant term; it should be purely X's and Y's. But in the context of approximations, people usually call this the linear term.

So what does a quadratic term look like? A quadratic term is allowed to have all the same terms as that linear one. You can have a constant, you can have these two linear terms, BX and CY, and then you're allowed to have anything that has two variables multiplied into it.

So maybe I'll have D * X², and then you can also have something times XY. This is considered a quadratic term, which is a little bit weird at first because usually we think of quadratics as associated with that exponent 2. But really, it's just saying anytime you have two variables multiplied in.

Then we can add some other constant, say F * Y², where now we're multiplying two Y's into it. All of these guys are what you would call your quadratic terms—things that have either Xs, Y², or XY—anything that has two variables in it.

You can see this gives us a lot more control. Because previously, as we tweaked the constants A, B, and C, you're able to get yourself some control over all sorts of planes in space. If you choose the most optimal one, you'll get one that's tangent to your curve at the specific point. It kind of depends on where that point is; you'll get different planes, but they're all tangent.

What we're going to do in the next couple of videos is talk about how you tweak all of these six different constants so that you can get functions that really closely hug the curve. They're all going to depend on the original point because, as you move that point around, what it takes to hug the curve is going to be different.

It's going to have to do with partial differential information about your original function—the function whose graph this is. It’s going to look pretty similar to the local linearization case, just with added complexity. So we have to add a few more steps in there, and I'll see you next video talking about that.