In this final section we need to discuss graphing rational
functions. It’s is probably best to
start off with a fairly simple one that we can do without all that much
knowledge on how these work.
Let’s sketch the graph of . First, since this is a rational function we
are going to have to be careful with division by zero issues. So, we can see from this equation that we’ll
have to avoid since that will give division by zero.
Now, let’s just plug in some values of x and see what we get.
x

f(x)

4

0.25

2

0.5

1

1

0.1

10

0.01

100

0.01

100

0.1

10

1

1

2

0.5

4

0.25

So, as x get large
(positively and negatively) the function keeps the sign of x and gets smaller and smaller.
Likewise as we approach the function again keeps the same sign as x but starts getting quite large. Here is a sketch of this graph.
First, notice that the graph is in two pieces. Almost all rational functions will have
graphs in multiple pieces like this.
Next, notice that this graph does not have any intercepts of
any kind. That’s easy enough to check
for ourselves.
Recall that a graph will have a yintercept at the point . However, in this case we have to avoid and so this graph will never cross the yaxis.
It does get very close to the yaxis,
but it will never cross or touch it and so no yintercept.
Next, recall that we can determine where a graph will have xintercepts by solving . For rational functions this may seem like a
mess to deal with. However, there is a
nice fact about rational functions that we can use here. A rational function will be zero at a
particular value of x only if the
numerator is zero at that x and the
denominator isn’t zero at that x. In other words, to determine if a rational
function is ever zero all that we need to do is set the numerator equal to
zero and solve. Once we have these
solutions we just need to check that none of them make the denominator zero as
well.
In our case the numerator is one and will never be zero and
so this function will have no xintercepts. Again, the graph will get very close to the xaxis but it will never touch or cross
it.
Finally, we need to address the fact that graph gets very
close to the x and yaxis but never crosses. Since there isn’t anything special about the
axis themselves we’ll use the fact that the xaxis
is really the line given by and the yaxis
is really the line given by .
In our graph as the value of x approaches the graph starts gets very large on both sides
of the line given by . This line is called a vertical asymptote.
Also, as x get
very large, both positive and negative, the graph approaches the line given by . This line is called a horizontal asymptote.
Here are the general definitions of the two asymptotes.
Determining asymptotes is actually a fairly simple
process. First, let’s start with the
rational function,
where n is the
largest exponent in the numerator and m
is the largest exponent in the denominator.
We then have the following facts about asymptotes.
 The
graph will have a vertical asymptote at if the denominator is zero at and the numerator isn’t zero at .
 If then the xaxis is the horizontal asymptote.
 If then the line is the horizontal asymptote.
 If there will be no horizontal asymptotes.

The process for graphing a rational function is fairly
simple. Here it is.
Process for Graphing
a Rational Function
 Find
the intercepts, if there are any.
Remember that the yintercept
is given by and we find the xintercepts by setting the numerator equal to zero and solving.
 Find
the vertical asymptotes by setting the denominator equal to zero and
solving.
 Find
the horizontal asymptote, if it exits, using the fact above.
 The
vertical asymptotes will divide the number line into regions. In each region graph at least one
point in each region. This point
will tell us whether the graph will be above or below the horizontal
asymptote and if we need to we should get several points to determine
the general shape of the graph.
 Sketch
the graph.

Note that the sketch that we’ll get from the process is
going to be a fairly rough sketch but that is okay. That’s all that we’re really after is a basic
idea of what the graph will look at.
Let’s take a look at a couple of examples.
Example 1 Sketch
the graph of the following function.
Solution
So, we’ll start off with the intercepts. The yintercept
is,
The xintercepts
will be,
Now, we need to determine the asymptotes. Let’s first find the vertical asymptotes.
So, we’ve got one vertical asymptote. This means that there are now two regions
of x’s. They are and .
Now, the largest exponent in the numerator and denominator
is 1 and so by the fact there will be a horizontal asymptote at the line.
Now, we just need points in each region of x’s.
Since the yintercept and xintercept are already in the left
region we won’t need to get any points there.
That means that we’ll just need to get a point in the right
region. It doesn’t really matter what
value of x we pick here we just
need to keep it fairly small so it will fit onto our graph.
Okay, putting all this together gives the following graph.
Note that the asymptotes are shown as dotted lines.

Example 2 Sketch
the graph of the following function.
Solution
Okay, we’ll start with the intercepts. The yintercept
is,
The numerator is a constant and so there won’t be any xintercepts since the function can
never be zero.
Next, we’ll have vertical asymptotes at,
So, in this case we’ll have three regions to our graph : ,
,
.
Also, the largest exponent in the denominator is 2 and
since there are no x’s in the
numerator the largest exponent is 0, so by the fact the xaxis will be the horizontal asymptote.
Finally, we need some points. We’ll use the following points here.
Notice that along with the yintercept we actually have three points in the middle
region. This is because there are a
couple of possible behaviors in this region and we’ll need to determine the
actual behavior. We’ll see the other
main behaviors in the next examples and so this will make more sense at that
point.
Here is the sketch of the graph.

Example 3 Sketch
the graph of the following function.
Solution
This time notice that if we were to plug in into the denominator we would get
division by zero. This means there
will not be a yintercept for this
graph. We have however, managed to
find a vertical asymptote already.
Now, let’s see if we’ve got xintercepts.
So, we’ve got two of them.
We’ve got one vertical asymptote, but there may be more so
let’s go through the process and see.
So, we’ve got two again and the three regions that we’ve
got are ,
and .
Next, the largest exponent in both the numerator and
denominator is 2 so by the fact there will be a horizontal asymptote at the
line,
Now, one of the xintercepts
is in the far left region so we don’t need any points there. The other xintercept is in the middle region. So, we’ll need a point in the far right
region and as noted in the previous example we will want to get a couple more
points in the middle region to completely determine its behavior.
Here is the sketch for this function.
Notice that this time the middle region doesn’t have the
same behavior at the asymptotes as we saw in the previous example. This can and will happen fairly often. Sometimes the behavior at the two
asymptotes will be the same as in the previous example and sometimes it will
have the opposite behavior at each asymptote as we see in this example. Because of this we will always need to get
a couple of points in these types of regions to determine just what the
behavior will be.
