Rational Functions

For graphing rational functions we need to follow these steps: 

1)  Find any intercepts, if there are any.  (Remember to find the y-intercept with f(0) and x intercepts by setting the numerator equal to zero).

2)  Find the vertical asymptotes by setting the denominator equal to zero and solving.

3)  Find the horizontal asymptote.

4)  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.

5)   Sketch the graph.

Example:

Sketch the graph of the following function.

                                                            

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 y-intercept for this graph.  We have however, managed to find a vertical asymptote already.

Now, let’s see if we’ve got x-intercepts.

                                     

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 x-intercepts is in the far left region so we don’t need any points there.  The other x-intercept 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.