HyperBola


Example 7  Graph

Solution
This is a hyperbola.  There are actually two standard forms for a hyperbola.  Here are the basics for each form.

Form



Center
(h, k)
(h, k)



Opens
Opens right and left
Opens up and down



Vertices
a units right and left
of center.
b units up and down
from center.



Slope of Asymptotes

So, what does all this mean?  First, notice that one of the terms is positive and the other is negative.  This will determine which direction the two parts of the hyperbola open.  If the x term is positive the hyperbola opens left and right.  Likewise, if the y term is positive the parabola opens up and down.

Both have the same “center”.  Note that hyperbolas don’t really have a center in the sense that circles and ellipses have centers.  The center is the starting point in graphing a hyperbola.  It tells us how to get to the vertices and how to get the asymptotes set up.

The asymptotes of a hyperbola are two lines that intersect at the center and have the slopes listed above.  As you move farther out from the center the graph will get closer and closer to the asymptotes.

For the equation listed here the hyperbola will open left and right.  Its center is
(-1, 2).  The two vertices are (-4, 2) and (2, 2).  The asymptotes will have slopes .

Here is a sketch of this hyperbola.  Note that the asymptotes are denoted by the two dashed lines.

CommonGraphs_G7

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