Lesson 18

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.