While end behavior of rational functions has been examined in a previous lesson, the focus has been on those functions whose end behavior is a result of a horizontal asymptote. In this lesson, students look at rational functions with other types of end behavior. In order to determine the exact end behavior, students learn how to rewrite rational expressions using long division. Students generalize their work to see how the structure of the expression, specifically the relationship between the degrees of the numerator and denominator, affects the type of end behavior the function has (MP8).
The lesson begins with students learning how to rewrite a rational equation using long division, building on what they already know about dividing polynomials. Students then consider a rational expression used to model fuel economy in cars. Rewriting this expression from the form \(\frac{a(x)}{b(x)}\) to \(q(x) + \frac{r(x)}{b(x)}\), where \(a(x)\), \(b(x)\), \(q(x)\), and \(r(x)\) are polynomials, and then considering the structure of the equivalent expression, allows students to make connections to their work from the previous lesson in order to identify end behavior. However, students need to recognize that the end behavior is not particularly relevant due to the domain the function has in this context (MP2). The last activity gives students an opportunity to summarize what they have learned about how to identify the end behavior of different types of rational functions.
Lesson overview
 19.1 Warmup: Different Divisions, Revisited (5 minutes)
 19.2 Activity: Combined Fuel Economy (15 minutes)

19.3 Activity: Exploring End Behavior (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 19.4 Cooldown: Finding End Behavior (5 minutes)
Learning goals:
 Calculate the end behavior of a rational function by rewriting it in the form \(f(x)=q(x)+\frac{r(x)}{b(x)}\).
 Generalize from specific rational functions to state relationships between the end behavior and the degrees of the numerator and denominator.
Learning goals (student facing):
 Let’s explore the end behavior of rational functions.
Learning targets (student facing):
 I can find the end behavior of a rational function by rewriting it as \(f(x)=q(x)+\frac{r(x)}{b(x)}\).
Standards:
 This lesson builds on the standards: CCSS.HSASSE.A.1.aMS.ASSE.1aCCSS.HSFIF.C.7MS.FIF.7MO.A2.NQ.A.2MO.A2.IF.A.1
 This lesson builds towards the standard: CCSS.HSFIF.C.7MS.FIF.7MO.A2.IF.A.1
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