Horizontal Asymptotes of Rational Functions

Author:
Jason Grebe, Mäkiö, Mathguru
In this activity, we will examine how to determine whether a given rational function has a horizontal asymptote. We will also determine some basic rules for how to determine the equation of the horizontal asymptote by examining the degree and leading coefficient of the numerator and denominator. Make sure you have the show asymptotes as x approaches positive and negative infinity check box selected. A reminder that a rational function is an algebraic fraction where the numerator and denominator are both polynomials. For this activity, we will examine the function h(x) = f(x)/g(x). For the questions below, you may need to zoom in and out on the graph or move the entire graph left/right to effectively analyze the behavior of the graph. To reset the graph, use the reset button in the upper right-hand corner of the graph.

1. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

2. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

3. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

4. Examine your results from questions 1-3. Do these graphs have a horizontal asymptote? Explain your reasoning.

5. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

6. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

7. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

8. Examine your results from questions 5-7. Do these graphs have a horizontal asymptote? Explain your reasoning.

9. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

10. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

11. Graph the following function: a) What is the degree of the numerator? b) What is the degree of the denominator? c) Describe the end behavior of the graph.

12. Examine your results from questions 9-11. Do these graphs have a horizontal asymptote? Explain your reasoning.

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