Graphs of Rational Functions

How to Graph a Rational Function:

• Find the y-intercept by solving for f(0)
• Set the numerator equal to zero and solve to find the x-intercepts
• Set the denominator equal to zero and solve to find the vertical asymptotes
• Find and sketch any horizontal asymptotes
• Use smooth curves to complete the graph

Example:

First, find the y-intercept of the graph by moving for f(0), since f(0)= 2, the y-intercept is (0,2).
Then set the numerator equal to zero to find the x-intercepts:

         -1(x-2)=0

               x-2=0

                  x=2

The x-intercept is (2,0)

Next set the denominator equal to zero to find the vertical asymptotes. Since the denominator is in factored form we know that the vertical asymptote will be x= -1 because that is what makes it equal zero. We also know that the vertical asymptote will have a multiplicity of 2 because it is squared in the denominator. 

From the information previously gathered, we are able to come up with the above graph for the function. Note that the graph never crosses over negative one because it is the vertical asymptote and that towards the end of the graph, it curves up to meet the horizontal asymptote because the end behavior of the graph is restricted by the horizontal asymptote. 

Slant Asymptotes:

If the numerator of a rational function has a degree one more than its denominator, the graph of the function has a slant/oblique asymptote. 

Example:



To find the slant asymptote, use long division and divide  into , which results in .
y=x-2 would be the slant asymptote. Then graph the function like a normal rational function.

Notice that the graph approaches, but does not touch the line y=x-2

Works Cited

"Desmos Graphing Calculator." Desmos Graphing Calculator. N.p., n.d. Web. 19 Jan. 2015. 

Hostetler, and Edwards. "Graphs of Rational Functions." Precalculus with Limits a Graphing Approach. By Larson. Third ed. N.p.: Houghton Mifflin, n.d. N. pag. Print.