Precalculus: Crossing the Asymptote

Objectives

• Students will test whether the graph of a given rational function crosses its horizontal asymptote.
• Students will examine the relationship among the coefficients of the polynomials in the numerator and denominator of various rational functions whose graph does or does not cross its asymptote.

Vocabulary

• rational function
• horizontal asymptote
• vertex

About the Lesson

This lesson involves the graph of a rational function of the form r(x) = p(x) / q(x).
As a result, students will:

• Discover conditions under which the graph of y = r(x) does or does not cross its horizontal asymptote. The functions p(x) and q(x) are assumed to be linear or quadratic polynomials.
• Manipulate graphs of rational functions and their asymptotes to determine whether they intersect.
• Make conjectures about the relationship between the coefficients of the polynomials in the numerator and denominator of a rational function whose graph does or does not cross its horizontal asymptote.