 A conic section in standard form will have equation after a suitable translation, as follows. A parabola has the form for p y equals x squared, or for p x equals y squared. An ellipse has the form x squared divided by a squared plus y squared divided by b squared equal to 1. And a hyperbola will have one of these two forms. So a useful general strategy. Identify the conic, identify the extreme points, sketch, and, as necessary, translate. So let's sketch the graph of this conic section. And remember the secret to graphing is, graph first, then label. So the first thing you might notice here is the x squared and y squared terms have opposite signs, and so this will be some sort of hyperbola. It's also useful to note that the graph of this equation will be symmetric about the x and y axes. And something that's useful to do is to look for extremes. The least a squared can be is zero. And so the least x squared can be is zero. But in that case, our equation becomes, but this is unsolvable. Well, it's not that bad, but it does mean that there is no point for which x is equal to zero. On the other hand, the least y squared can be is zero. And so in that case, our equation becomes, and we can solve this. So we find the points three zero and negative three zero are on the curve. Now, if we rearrange our equation a little bit, we note that if y squared is larger, x squared will also be larger. And so there are going to be points to the right of this point that we graphed. And since the graph is symmetric about the y axis, there will be a corresponding point on the other side of the y axis. And since the graph is symmetric about the x axis, these points will also be duplicated below the x axis. And so this hyperbola opens horizontally. Another secret to graphing is transformations. So notice that this equation is actually the ellipse shifted vertically and horizontally. So let's consider what happens with this ellipse. The least x squared can be is zero, in which case. We find two points on the ellipse, and we can graph these points. Meanwhile, the least y squared can be is zero, in which case. We find another two points on the ellipse, and we can graph the ellipse. Now, we'll take this graph of x squared divided by five squared plus y squared divided by three squared equals one, and then we'll translate it horizontally to the right by four units, then vertically upward by seven units. Or we could take something like this, and if we ignore the plus two and minus five, we see that this is a translation of the parabola y squared equals four x. And this time we see that the graph will be symmetric about the x axis. And going to extreme, see least y squared can be is zero, in which case. So the graph passes through zero, zero. And if y squared is larger, then x will also be larger, so there are points to the right. And since the graph is symmetric about the x axis, the points must be above and below the axis. And so the parabola opens to the right. And so we'll sketch y squared equals four x, and then we'll translate this five units to the right, and two units downward. And this gives us the graph of y plus two squared equals four times x minus five.