Lesson 19

The purpose of this activity is for students to use long division to rewrite a rational expression. While students have done polynomial long division previously, the result of dividing polynomials \(a(x)\) and \(b(x)\) were written as \(a(x)=b(x)q(x)+r(x)\), where \(q(x)\) and \(r(x)\) are also polynomials, since the previous goal was identifying factors of the polynomial \(a(x)\). Now, students are asked to write an equivalent expression for the rational function \(\frac{a(x)}{b(x)}\), and so they will write the equivalent form as \(q(x)+\frac{r(x)}{b(x)}\). In the following activities, students will use this technique to rewrite a variety of different rational expressions in order to identify the end behavior of the function.

Select students to share how they completed the long division and found an equivalent expression for \(\frac{2x^3+7x^2+7x+5}{x+1}\). Once students are in agreement that \(2x^2+5x+2 +\frac{3}{x+1}\) is the equivalent form, here are some questions for discussion:

• “If \(f(x)=\frac{2x^3+7x^2+7x+5}{x+1}=2x^2+5x+2 +\frac{3}{x+1}\), what happens to the output of the function as \(x\) gets larger and larger in the positive direction? The negative direction?” (The value of \(\frac{3}{x+1}\) approaches 0, so the output of the function is close to the value of \(2x^2+5x+2\), which gets larger and larger in the positive direction as \(x\) gets larger and larger in either the positive or negative direction.)
• “Does \(f\) have any vertical asymptotes? If yes, where?” (\(f\) has one vertical asymptote at \(x=\text-1\).)

Conclude the discussion by displaying the graphs of \(y=f(x)\) and \(y=2x^2+5x+2\) on the same axes for all to see, giving students time to notice the end behavior and how the graph of \(f\) is “split” due to the vertical asymptote. Tell students that previously, they studied rational functions with end behavior described by horizontal asymptotes. Today, they are going to look at rational functions with other types of end behavior that can be identified by rewriting the expression for the function.

The goal of this activity is for students to work with a more complicated rational expression with two input variables that they then rewrite as a function of only one of the variables. From there, students use long division to rewrite the expression as \(q(x)+\frac{r(x)}{b(x)}\), from which they can reason to find the end behavior of the function (MP2). In this instance, the end behavior has little meaning since the domain of this function doesn’t go much above \(x=40\) for modern gas-powered vehicles. It is important for students to develop the habit of recognizing a reasonable domain for models. While in this instance, the end behavior does not have meaning, it is still true that \(q(x)\) defines the end behavior of the rational function. In the following activity, students will investigate in more depth how to rewrite functions in order to reveal end behavior.

A note about the expression used in this activity: to help students focus on the mathematics of rewriting a rational expression using division and then reasoning about the end behavior in context, the details of where the expression for fuel efficiency is from have been omitted. Additionally, the expression used by the EPA in 2000 to calculate combined fuel efficiency for a conventional car has been rewritten to make the weighted percents clearer for students to see. This was done because the actual expression is the result of a harmonic mean and is often written as \(\dfrac{1}{\frac{0.55}{x}+\frac{0.45}{h}}\), and making sense of this type of rational expression is not a goal of this course.

Display the statement from the activity for all to see. If students are unfamiliar with the idea of fuel efficiency, invite students to share what they think it means. Highlight that:

• car advertisements often feature fuel efficiency by stating how many miles a car can travel per gallon of gasoline
• fuel efficiency is measured separately for highway driving and for city driving, and vehicles typically have better fuel efficiency on the highway than in a city
• smaller vehicles often have better fuel efficiency than larger vehicles

Ask “When \(x=25\) and \(h=35\), the expression is about 29.7. What does that mean in this context?” (The combined fuel efficiency of a car that gets 25 mpg in the city and 35 mpg on the highway is about 29.7 mpg.) Select students to share what the value means in the context before students continue with the questions in the activity.

Students may forget that a single number, like 49.5, qualifies as a polynomial when they are rewriting the expression, making them believe they have done something incorrectly. Remind these students of the variety of forms polynomials can take.

Select students to share their functions and the rewritten forms. Ask, “What can you say about the end behavior of \(f\)?” After a brief quiet think time, display a graph of \(y=f(x)\) for all to see, showing the function for positive values of \(x\). Tell students to share their thinking with a partner and then consider whether the graph agrees with their thinking. Invite students to share their explanation for what the end behavior of the function is. (As \(x\) gets larger and larger, the combined fuel efficiency also gets larger and larger. For large values of \(x\), the relationship between \(x\) and \(f(x)\) is close to \(x+5.5\), since the value of \(\frac{49.5}{2x+9}\) is less than 1 for any \(x\) value greater than 20.25.)

If the connection between the graph of \(f\) and the graph of \(y=x+5.5\) is not brought up, add the graph of \(y=x+5.5\) to the same axes as \(y=f(x)\). Ask students to consider why the two graphs look more and more alike as the values of \(x\) increase. While previously, students learned to phrase end behavior using “as \(x\) gets larger and larger in the positive direction, \(y\) gets larger and larger in the positive direction,” we can now be more specific and state that as \(x\) gets larger and larger in the positive direction, \(y\) approaches the value of \(x+5.5\).

Conclude the discussion by asking students what the end behavior of \(f\) tells us about the situation. An important takeaway here is that the end behavior of \(f\) doesn’t mean much, since conventional cars don’t have city gas mileage over 40 miles per gallon.

The goal of this activity is for students to summarize what they have learned about rational equations. Specifically, students state how the relationship between the degrees of the numerator and denominator of a rational function written in the form \(\tfrac{a(x)}{b(x)}\) affects the end behavior. They rewrite each function in the form \(q(x)+\tfrac{r(x)}{b(x)}\) to see the end behavior more easily. Asking students to complete the same types of calculations several times and look for patterns encourages them to generalize the patterns they see (MP8).

Arrange students in groups of 2–4. Display the table from the activity for all to see. Assign each group one of the functions from the table to consider, and ask them to make a prediction about the end behavior. After a brief think time, ask each group for their prediction and a brief explanation of their reasoning. Record the predictions for all to see during work time. While each group should fill out the entire table, encourage them to start with their assigned row so they can check their prediction using technology before moving on to consider the other functions. Note that “numerator” and “denominator” have been abbreviated in the table, so students may need clarification about what the abbreviations mean.

Students may choose to use graphing technology during this activity. Encourage these students to limit their technology use to checking their work after they have completely filled out a row.

Students might not be sure whether the remainder in the last row approaches 0 as \(x\) gets larger, since \(x\) appears in both the numerator and denominator. Encourage these students to plug in some large values of \(x\) and see what happens to the value of the rational expression. They can also graph it to confirm that it approaches 0.

The goal of this discussion is for students to recognize that the end behavior of a rational function written in the form \(y=q(x)+\frac{r(x)}{b(x)}\), where the degree of polynomial \(r(x)\) is less than the degree of polynomial \(b(x)\), is the same as the end behavior of \(q\). Students should also recognize that the degree of \(q\) can be determined by looking at the difference between the degrees of the numerator and denominator of the original expression for the function. The focus of this discussion should stay on reading the structure of the original function to determine features of the function and not on memorizing different cases.

Select 1–2 groups for each function to share how they filled out the row and determined the end behavior of the function. During this discussion, help students connect back to the original prediction for the function and why it was correct or incorrect. If any groups graphed their functions, display these to help students make connections between the graph and equation.

If the relationship between the degrees of the numerator and denominator of the rational function does not come up during the discussion, ask students what they notice about the end behavior and the relationship between the degrees. Help students precisely define the three cases (degree of the denominator is greater than the degree of the numerator, degree of the denominator is equal to the degree of the numerator, degree of the denominator is less than the degree of the numerator), and how these cases determine the end behavior of the rational function. Some students may have been more precise in describing the relationship, such as by writing the numerical difference between the degrees, and part of the discussion should be on why there only need to be three cases. When appropriate, help summarize student explanations as clearer mathematical explanations.