The next graph that we need to look at is the
hyperbola. There are two basic forms of
a hyperbola. Here are examples of each.
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
|
|
|
|
|
|
|
|
Center
|
|
|
|
|
|
|
|
Opens
|
Opens left and right
|
Opens up and down
|
|
|
|
|
|
Vertices
|
and
|
and
|
|
|
|
|
|
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.
Let’s take a look at a couple of these.
|
Example 1 Sketch
the graph of each of the following hyperbolas.
(a) [Solution]
(b) [Solution]
Solution
(a) Now, notice that the y
term has the minus sign and so we know that we’re in the first column of the
table above and that the hyperbola will be opening left and right.
The first thing that we should get is the center since
pretty much everything else is built around that. The center in this case is and as always watch the signs! Once we have the center we can get the
vertices. These are and .
Next we should get the slopes of the asymptotes. These are always the square root of the
number under the y term divided by
the square root of the number under the x
term and there will always be a positive and a negative slope. The slopes are then .
Now that we’ve got the center and the slopes of the
asymptotes we can get the equations for the asymptotes. They are,
We can now start the sketching. We start by sketching the asymptotes and
the vertices. Once these are done we
know what the basic shape should look like so we sketch it in making sure
that as x gets large we move in
closer and closer to the asymptotes.
Here is the sketch for this hyperbola.
[Return to Problems]
(b) In this case the hyperbola will open up and down since the x term has the minus sign. Now, the center of this hyperbola is . Remember that since there is a y2 term by itself we had to
have . At this point we also know that the
vertices are and .
In order to see the slopes of the asymptotes let’s rewrite
the equation a little.
So, the slopes of the asymptotes are . The equations of the asymptotes are then,
Here is the sketch of this hyperbola.
[Return to Problems]
|
