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# 12.2: Graphs of Rational Functions

Difficulty Level: At Grade Created by: CK-12

## 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 x\begin{align*}x-\end{align*}intercepts, y\begin{align*}y-\end{align*}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.

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