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.
Terms introduced in this lesson:
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−intercepts, y−intercepts, factoring, and domains.
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.