Graphing Rational Functions and Vertical Asymptotes

Date: 03/28/2008 at 00:36:38
From: Hans
Subject: Trying to find the shape of a rational function.

When working with rational functions such as y=(x-5)/(x-3), how do you
know if at the vertical asymptote, does the graph curve up or curve down?

What is the the general pattern with all rational function regardless
of the degree?  And also, how can we quickly interpret the function
and determine the shape of the graph? 

We only started graphing with graphing calculators, however I would
like to know the reason behind such behavior, and how to determine it
simply on paper.

Date: 03/28/2008 at 22:57:02
From: Doctor Peterson
Subject: Re: Trying to find the shape of a rational function.

Hi, Hans.

One way to examine the behavior of the function near the vertical
asymptote is to look at the signs of factors in the expression on
either side of the asymptote.  If x is just a little GREATER than 3,
then x-3 > 0 (a very small positive), and x-5 is just a little greater
than -2 (and therefore negative).  Thus, (x-5)/(x-3) will be something
near -2 divided by a very small positive number, and will therefore be
a very large NEGATIVE number (that is, going in the direction of
-infinity) just to the right of the asymptote at x = 3.

On the other hand, when x is just a little LESS than 3, then x-3 < 0
(a very small negative), and x-5 is just a little less than -2 (and
therefore still negative).  Thus, (x-5)/(x-3) will be something near
-2 divided by a very small negative number, and will therefore be a
very large POSITIVE number (that is, going in the direction of
+infinity) on the left side of the asymptote.

In general, because the factor that causes the asymptote (the (x-3) in
this case) changes sign as you evaluate it on each side of the
asymptote, often the function rises to +infinity on one side of the
asymptote and falls to -infinity on the other side.  Can you think of
a case where that would not happen? 

Another approach is to use polynomial division to rewrite the function
as a "mixed rational function" rather than an "improper rational
function".  Here, division gives

  x - 5        -2
  ----- = 1 + -----
  x - 3       x - 3

You can see that this is the sum of y = 1 (a horizontal line) and
-2/(x-3), which falls as it approaches x=3 from the right and rises as
it approaches from the left.

This division is also very useful in understanding behavior for large
x, such as horizontal asymptotes.  Can you see the horizontal
asymptote standing out very visibly in the "mixed" form of our function?

I applaud your desire to be able to analyze a function algebraically
rather than depending on a tool.  Seeing a graph gives you a good idea
of the possibilities, but to really understand what is going on you
have to use your own mind to look behind the graph and see the reasons
for the behavior.  Tools like those I've mentioned are really 
important for this process.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/