# 12.2: Graphs of Rational Functions

## Learning Objectives

At the end of this lesson, students will be able to:

- Compare graphs of inverse variation equations.
- Graph rational functions.
- Solve real-world problems using rational functions.

## Vocabulary

Terms introduced in this lesson:

- rational function
- horizontal asymptote, vertical asymptote
- oblique (slant) asymptote

## eaching Strategies and Tips

Reconstruct the tables in Examples 2-4 to remind students of the inverse relationship.

Explore several rational functions side-by-side.

- Have students make note of the degrees of the numerator and denominator and any horizontal and vertical asymptotes.
- Point out that what sets rational functions apart from other functions in this course is
*division*. - Division creates the asymptotes and branches.
- Remind students that dividing by zero is undefined and is denoted on the graph by a vertical dashed line.

Asymptotes are denoted by dashed lines. Remind students that asymptotes are not part of the function and only serve to show how the graph approaches certain values.

- Point out that graphing calculators may display asymptotes using a solid line.

Have students rewrite the steps for finding asymptotes preceding Example 5 for themselves.

Encourage graphing rational functions by hand. Use a graphing calculator only as a way to check.

- Sketching graphs can solidify student understanding of intercepts, intercepts, factoring, and domains.

## Error Troubleshooting

In Examples 2-4, have students choose enough values for their tables to determine the behavior of the function accurately. Remind them to pick values close to the vertical asymptotes.