Lesson 19

End Behavior of Rational Functions

Lesson Narrative

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.

Learning Goals

Teacher Facing

• 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.

Student Facing

• Let’s explore the end behavior of rational functions.

Student Facing

• I can find the end behavior of a rational function by rewriting it as $f(x)=q(x)+\frac{r(x)}{b(x)}$.

Building On

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.

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