 So we graphed linear equations, can we graph nonlinear equations? So for nonlinear equations, we might begin as follows. In a linear equation, all terms are constant or have degree 1. So it would seem that in the simplest possible nonlinear equation, all terms would be constant or have degree 1 or 2. While we need at least one term of degree 2, we have a choice. We can have an x squared term, we can have a y squared term, or we can have an xy term. As it turns out, having an xy term makes the problem much harder. So let's not choose that option. And intuitively, it would seem that having both an x squared and a y squared term also makes the problem harder. So let's not have both options. Let's just have an x squared term. Putting all these together, this suggests that the easiest nonlinear equation to graph will have the form y equals ax squared plus bx plus c, where a, b, and c are real numbers, and a is not equal to 0. Actually, this is still too hard, so let's make it even easier. a can't be equal to 0, but if b equals 0 and c equals 0, we have y equals ax squared, which is a second degree equation in two variables. Actually, y equals ax squared is still too hard, so let's make it simpler. Now at this point, we do have to be a little bit careful. We could take a specific value for a. y equals 389x squared. The problem is that anything we found would only be true for that particular value of a. So we might learn a lot about the graph of y equals 389x squared, but it would tell us nothing about the graph of y equals 388x squared. So this is maybe a little too simple. Instead, we'll assume that a has a specific property. Since a is a real number and a is not equal to 0, then either a is greater than 0 or a is less than 0. Suppose a is greater than 0. If we wanted to graph y equals ax squared, we could try to find ordered pairs x, y that make the equation true. And since the equation is already solved for y, we'll choose values of x and compute y. So if x equals 0, then y is equal 2. If x is equal to 1, then y is equal 2. If x is equal to 2, then y is. And if x equals 3, y is. While we could find y when x equals minus 3, minus 2, minus 1, remember that b squared is the same as minus b squared. So the values for x equals 3, 2, and 1 will be the same as the values for x equals minus 3, minus 2, and minus 1. So now we have a bunch of x and y coordinates. Now let's graph. What's that? We don't know what a is? Ah, well that could be a problem. But remember, the secret to graphing. Graph first, then label. So let's consider this first ordered pair minus 3, 9a. While we don't know where minus 3, 9a is, we know that a is greater than 0, and so 9a must also be greater than 0. And so that means that this point minus 3, 9a, is to the left of the origin and above the axis somewhere. How about here? By the same reasoning, minus 2, 4a is to the left of the origin and above the axis somewhere, but since 4a is less than 9a, it's not as high up. So maybe we'll put it here. Similarly, minus 1a is to the left of the origin, but not nearly as high as minus 2, 4a. So maybe it goes about here. Well, 0, 0, we know exactly where that is. For 1a, we know we're to the right of the origin, and since we've already placed the point minus 1a at height a, we need to be consistent. This point has to be at the same height as this point, which means we'll want to put the point about here. And similarly for 2, 4a, and 3, 9a. And this gives us the general shape of the graph, and if we graph a few million more points, we'll fill out the shape. What if a is less than 0? We'll find some points on the graph of y equals ax squared. And the thing to realize here is that the only thing that has changed is the property of a. The algebraic formulas are still the same, so the algebraic results are still the same. But when we go to graph, again, while we don't know where minus 3, 9a is, we do know that a is less than 0, so 9a is less than 0, and so this means our point is to the left of the origin and below the axis. Similarly for minus 2, 4a, a is less than 0, so 4a is less than 0, so minus 2, 4a is to the left of the origin and below the axis. And just like before, if this is 9a below the axis, 4a is not as far below the axis, so maybe it's around here. And as before, we can fill out our graph, and now we have the following result. Because we didn't choose a specific value of a, we have a result that holds for all values of a. For the graph of y equals ax squared, if a is greater than 0, our graph looks like this, and we saw that the graph opens upward. And finally, we need a name for this shape, and so we name the shape a parabola.