The next graph
that we need to look at is the hyperbola. There are two basic forms of a hyperbola.
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Hyperbolas
consist of two vaguely parabola shaped pieces that open either up and down or
right and left. Also, just like parabolas each of the pieces has a vertex. Note
that they aren’t really parabolas, they just resemble parabolas.
There are also
two lines on each graph. These lines are called asymptotes and as the graphs
show as we make x large (in both the positive and negative sense) the
graph of the hyperbola gets closer and closer to the asymptotes. The asymptotes
are not officially part of the graph of the hyperbola. However, they are
usually included so that we can make sure and get the sketch correct. The point
where the two asymptotes cross is called the center of the hyperbola.
There are two standard
forms of the hyperbola, one for each type shown above. Here is a table giving
each form as well as the information we can get from each one.
Form 
Opens
Opens left and right Opens up and down
Vertices
(h + a, k) and (h − a, k) (h, k
+ b) and (h, k
− b)
Slope
of Asymptotes 
Equations
of Asymptotes 
Note that the
difference between the two forms is which term has the minus sign. If the y term
has the minus sign then the hyperbola will open left and right. If the x term
has the minus sign then the hyperbola will open up and down.
We got the
equations of the asymptotes by using the point-slope form of the line and the
fact that we know that the asymptotes will go through the center of the
hyperbola.
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