Lesson 18

Graphs of Rational Functions (Part 2)

Lesson Narrative

In this lesson, students continue their investigation connecting features of graphs and equations of rational functions with the focus now on horizontal asymptotes as a type of end behavior. The line \(y=c\) is a horizontal asymptote for \(f\) if \(f(x)\) gets closer and closer to \(c\) as the magnitude of \(x\) increases.

Students begin the lesson considering a common error when rewriting fractions, in preparation for rewriting fractions themselves. Next, they reason about the average cost to produce \(x\) number of books, first making sense of a non-zero horizontal asymptote in context, and then by rewriting the original expression (MP2). In the following activity, students match equations to graphs of rational functions with an emphasis on rewriting as needed to make the match.

In the following lesson, students will more formally address how to use polynomial long division to rewrite rational functions of the form \(\frac{a(x)}{b(x)}\) as \(q(x) + \frac{r(x)}{b(x)}\), where \(a(x)\), \(b(x)\), \(q(x)\), and \(r(x)\) are all polynomials and \(b(x) \neq 0\). For this lesson, students are expected to take a more informal approach, rewriting by inspection and applying their knowledge of fractions.

Learning Goals

Teacher Facing

  • Identify horizontal asymptotes of simple rational functions from their equations.
  • Interpret a graph of a rational function and explain the meaning of the horizontal asymptote in context.

Student Facing

  • Let’s learn about horizontal asymptotes.

Learning Targets

Student Facing

  • I can identify a horizontal asymptote from a graph or an equation of a rational function.

CCSS Standards

Addressing

Building Towards

Glossary Entries

  • horizontal asymptote

    The line \(y =c\) is a horizontal asymptote of a function if the outputs of the function get closer and closer to \(c\) as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the \(x\)-axis.

  • rational function

    A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.

  • vertical asymptote

    The line \(x=a\) is a vertical asymptote for a function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. This means the graph goes off in the vertical direction on either side of the line.