Hyperbolas are typically the last conic section studied. The equation of a hyperbola appears very similar to that of the ellipse, the difference being a minus sign. Here is the equation of the standard from of a hyperbola.

or

 

 

There are a few things that are important to notice about these equations. First, notice that the x term is notice always first. When the x term is negative it is put after the y term. It is also important to notice that the a value does not always go underneath the x value as it has in other conic sections. The a value will always go underneath the positive x or y whichever the case may be.

 

Following are the two different kinds of graphs of hyperbolas. One opens back and forth, when the x is positive, and the other opens up and down, when the y is positive.

 

 

 

 

 

 

 

 

(0, c)

(0, a)

 

Asymptotes

 

 

 

(0, -a)

(0, -c)

 

 

 

 

 

 

 

 

Asymptotes

 

(c,0)

(c,0)

 

 

(a, 0)

(-a, 0)

 

 

 

 

 

is easy to tell from these graphs that a hyperbola looks like two parabolas opening away from each other. The equation used to find the c value is . The dotted lines are called asymptotes and the equations to find them are as follows.

 

If the parabola opens up and down (y² is positive), Then the asymptotes are

and

 

 

 

 

If the parabola opens back and forth (x² is positive), Then the asymptotes are

 

and

 

.

 

It is important to note that the hyperbola will never cross the asymptotes. They serve as boundaries.

 

Steps to follow when graphing hyperbolas:

1)      See which direction it opens

2)      Graph the vertices

3)      Graph the foci

4)      Draw the asymptotes

5)      Graph

 

 

 

 

Example

 

Graph

 

 

Since x is the positive term, the hyperbola will open back and forth. Looking at the equation, a² is 25, which means a = 5. Since the hyperbola opens back and forth and a = 5, the vertices are at ( -5, 0) and (5, 0). Graph these two points.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Next, find the foci. Use the equation to find c.

 

 

Simplify.

 

 

 

 

The foci are at (7.81, 0) and (–7.81, 0). To find the asymptotes use the equations for the asymptotes of a hyperbola that opens back and forth.

The equations are and .

 

 

These are the asymptotes. Graph them along with the foci.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

All that is left now is to graph the hyperbola. The vertices are known as well as the foci and the asymptotes. Just draw two parabolas with that information.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

 

 

 

Example

 

Graph 25y² = x² + 100

 

The first thing to do is to get it into standard form. Subtract x² from both sides.

 

25y² - x² = 100

 

Divide everything by 25.

 

Divide everything by 4.

 

Now it is in standard from. Since the y term is positive, it opens up and down. The a value is 2, and the b value is 10. Plug this into the equation, , to find c.

 

 

 

 

The foci are at (0, 10.20) and (0, -10.20). The vertices are at (0, 2) and (0, -2). Find the asymptotes.

 

Graph all the information along with the hyperbola.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!