Graphs of Rational Functions

Chapter 2.7 - Graphing Rational Functions

this chapter contains:
-how to analyze and sketch the graphs of rational functions
-how to decide if graphs have slant asymptotes and what they are

Guidelines for Graphing Rational Functions

1. Evaluate f(0) to find the y-intercept and then plot it on the graph.
2. Set the numerator equal to zero -> N(x) = 0. The real solutions represent the x-intercepts and should also be plotted on the graph. 
3. Set the denominator equal to zero -> D(x) = 0. The real solutions represent the vertical asymptotes. They can be plotted on the graph as dashed vertical lines.
4. Find the horizontal asymptote and sketch it on the graph as a dashed horizontal line.
5. Using the end-behavior, fill in the graph with smooth, curved lines.

Example: 

1. f(0) = -3/2 
2. 3 = 0 -> No real solutions, so no x-intercepts
3. 0 = x - 2 -> x = 2 is a vertical asymptote.
4. y = 0 is the horizontal asymptote because the degree of x in the numerator is less than the degree of x in the denominator.
5. Since no asymptotes are multiplicities, the parts of the graph are opposites. The left piece has to be below y = 0 in order to touch the y-intercept. The opposite in the right would the be above      y = 0. 

                         

Slant Asymptotes

If the degree of the numerator is exactly one more than the degree of the denominator, the graph has a slant (or oblique) asymptote.

For example: has a slant asymptote.

To find the asymptote, use long division.

The slant asymptote portion of the answer is everything but the remainder over the divider. This leaves the line  as the slant asymptote