2.7 Graphs of Rational Functions

How to graph a rational function:

1. Find and plot the y-intercept by evaluating f(0).
2. Set the numerator equal to zero and solve to find the x-intercepts.
3. Set the denominator equal to zero to find the vertical asymptotes.
4. Find and sketch the horizontal asymptote if there is one.
5. Use curves to complete the graph between and after the vertical asymptotes.

Example:

Find the y-intercept by plugging in 0 for x.  You then find f(0)=5, therefore making the y-intercept (0,5).
Now to find the x-intercepts set the numerator equal to 0.

0=5+2x
x= -5/2

This tells you that the x-intercept in (-5/2,0)
To then find the vertical asymptote you must set the denominator equal to 0.  From here we can see that the vertical asymptote is line x= -1.
From the information above we can also see that the horizontal asymptote is line y=2

The facts from above allowed us to create this graph of a rational function.

Most of the time with vertical asymptotes, the two parts are on different ends of the asymptote, for example the graph above. Although when you have a vertical asymptote with a multiplicity of two, they are on the same end.

Example:





This has a multiplicity of 2 at the asymptote x=1.
See how in this graph the two parts are at the same end of the asymptote, that is due to the multiplicity of two.

Some graphs can also have holes. When there are the same factors located in the numerator and the denominator of the function it cause a hole at that x value in the graph.

Example:




With this on the graph we should see a hole when x=1 due to the same factor in the numerator and denominator.

Slant Asymptotes:
If the degree of the numerator is one more than the degree of the denominator then the function has a slant asymptote. To find the line of the asymptote you must use long division and divide the numerator by the denominator.

Example:


This divides out to x-2 with a remainder of 2.  The slant asymptote is the line f(x)=x-2.
You can see this in the graph.