10.3 Hyperbolas

Here are the equations for a hyperbola:

The first equation is used for horizontal hyperbolas, while the second is used for vertical hyperbolas.

Below is an example of a problem that asks you to sketch a hyperbola. Unless the coefficients of x^2 and y^2 are already one, the first step to these types of problems is completing the square. This is done on the second line of this problem. (Don't forget to add equal amounts to the other side of the equation!) to sketch the graph, mark the center (h,k), then mark points that are "a" distance away in the direction "a" corresponds to (in this case, x). Do the same for "b" (in this case, in the y direction) and make a box; the asymptotes, or lines the graph will never touch but approach infinitely, will run diagonally out of the corners of the box. Then draw the graph as shown, leaving from the vertex, or the "a" points you marked, and approaching the asymptotes.