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Hyperbola | Calculus Math Help

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:

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1(horizontal-axis)

\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1(vertical-axis)

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

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

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

\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1

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

c^2 = a^2+b^2

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 =