What is the hyperbola?

The hyperbola belongs to a family known as the “conic sections” and is defined as all the points in a plane for whom the difference of their distance from two fixed points (foci) is constant.  Its general equation looks like:

OR

When graphed, it looks like two symmetrical curves facing away from each other.

How can we use the concept:

First, it is important to know some of the vocabulary surrounding hyperbolas!

Foci: There are two foci for a hyperbola and they are the fixed points from which the difference in distance is constant for every point on the parabola.

Vertices: A line drawn through the foci intersects the parabola at two points called the vertices.  The vertices are the point at which the curves seem to “turn”.

Transverse Axis: The line segment that joins the vertices.

Center: The midpoint of the transverse axis.

Looking at the general form of the equation of a hyperbola in which the transverse axis is horizontal

we can pick out some of these features!  The point (h,k) will be the center of the hyperbola.  a is the distance the vertices are from the center.  See below for a worked example of how to find the vertices given the equation of a hyperbola.

Note that

is the general form for a hyperbola with a vertical axis.  However, all the features just mentioned - the center and vertices - can be deduced from this equation in the same way.

Another feature we are interested in finding is the foci! The foci are c units away from the center, where

Last but certainly not least, to graph a hyperbola, it is best to find and graph its vertices.  Then we need to find its asymptotes!  These are lines that the hyperbola will curve towards, but never touch.  The asymptotes for a horizontal hyperbola are found by using the formula

y =

y =