Lesson 19

End Behavior of Rational Functions

19.1: Different Divisions, Revisited (5 minutes)

Warm-up

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.

Student Facing

Complete all three representations of the polynomial division following the forms of the integer division.

Diagram of integer long division of 2,775 by 11 and the equivalent expression 252 + 3/11.

\( \require{enclose} \begin{array}{r}  2x^2 \phantom{+7x+55} \\ x+1 \enclose{longdiv}{2x^3+7x^2+7x+5} \\    \end{array}\)

\(2775=11(252)+3\)

\(2x^3+7x^2+7x+5=\)

\(\dfrac{2775}{11}=252+\dfrac{3}{11}\)

\(\dfrac{2x^3+7x^2+7x+5}{x+1}=\)

Student Response

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Activity Synthesis

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.

19.2: Combined Fuel Economy (15 minutes)

Activity

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.

Launch

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.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret the idea of fuel efficiency, and to increase awareness of language used to talk about rational expressions in context. Display only the task statement of this problem (“In 2000, the Environmental Protection Agency (EPA) reported a combined fuel efficiency for cars that assumes 55% city driving and 45% highway driving. The expression for the combined fuel efficiency of a car that gets \(x\) mpg in the city and \(h\) mpg on the highway can be written as \(\frac{100xh}{55x+45h}\).”), and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions that connect rational expressions to the context.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

In 2000, the Environmental Protection Agency (EPA) reported a combined fuel efficiency for cars that assumes 55% city driving and 45% highway driving. The expression for the combined fuel efficiency of a car that gets \(x\) mpg in the city and \(h\) mpg on the highway can be written as \(\frac{100xh}{55x+45h}\).

  1. Several conventional cars have a fuel economy for highway driving is that is about 10 mpg higher than for city driving. That is, \(h=x+10\). Write a function \(f\) that represents the combined fuel efficiency for cars like these in terms of \(x\).
  2. Rewrite \(f\) in the form \(q(x)+\frac{r(x)}{b(x)}\) where \(q(x)\), \(r(x)\), and \(b(x)\) are polynomials.

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. Show the algebraic process of substituting \((x + 10)\) for \(h\) and distributing the terms to get \(\frac{100x^2+1000x}{100x+450}\). Continue with the long division that yields the expression as a sum of polynomials. They can use the annotated calculations as a reference in the next activity.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Memory

19.3: Exploring End Behavior (15 minutes)

Activity

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).

Launch

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.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. Using the division from the previous activity as a model, highlight the corresponding elements in the five functions they are working with. If students struggle with rewriting a particular function longer than is productive, encourage them to move to another, and to make generalizations that they can apply to ones they struggle with.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Memory

Student Facing

function degree
of num.
degree
of den.
rewritten in the form of
\(q(x)+\frac{r(x)}{b(x)}\)
end behavior
\(g(x)=\text-\frac{5}{x+2}\)
\(h(x)=\frac{7x-5}{x+2}\)
\(j(x)=\frac{3x^2+7x-5}{x+2}\)
\(k(x)=\frac{2x^3+3x^2+7x-5}{x+2}\)
\(m(x)=\frac{x+2}{2x^3+3x^2+7x-5}\)
  1. Complete the table to explore the end behavior for rational functions.
  2. What do you notice about the end behavior of different types of rational functions?

Student Response

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Student Facing

Are you ready for more?

  1. Graph \(y=j(x)\) and the line it approaches.
  2. Under what conditions would the end behavior of the graph of a rational function approach a line that is not horizontal?
  3. Create a rational function that approaches the line \(y=2x-3\) as \(x\) gets larger and larger in either the positive or negative direction.

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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.

Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain how their group completed their row and determined the end behavior of the function. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is for students to return to the surface area discussion from an earlier lesson and use their new knowledge for identifying end behavior.

Display the following to remind students of the problem that started their investigation into rational functions: There are many cylinders with volume 452 cm3. Let \(r\) represent the radius of these cylinders, \(h\) represent the height, and \(S\) represent the surface area. Here are some questions for discussion.

  • “The height of these cylinders is defined by the function \(h(r)=\frac{452}{\pi r^2}\). What is the end behavior of this function, and what does it tell us about the situation?” (Since the degree of the denominator is larger than the degree of the numerator, the function has a horizontal asymptote at 0, so as \(r\) gets larger and larger, \(h(r)\) gets closer and closer to zero. This means that for a cylinder with a volume of 452 cm3, as the radius increases, the height decreases toward zero.)
  • “The surface area of these cylinders is defined by the function \(S(r)=\frac{2\pi r^3 + 904}{r}\). What is the end behavior of this function, and what does it tell us about the situation?” (We can rewrite this as \(S=2\pi r^2+\frac{904}{r}\), which makes it clearer that as \(r\) gets larger and larger, the value of \(\frac{904}{r}\) approaches zero, and the value of \(S(r)\) gets larger and larger, approaching the value of \(2\pi r^2\).)

19.4: Cool-down - Finding End Behavior (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

In earlier lessons, we saw rational functions whose end behavior could be described by a horizontal asymptote. For example, we can rewrite functions like \(d(x)=\frac{x+4}{x}\) as \(d(x)=1+\frac{4}{x}\) to see more clearly that as \(x\) gets larger and larger in either the positive or negative direction, the value of \(\frac{4}{x}\) gets closer and closer to 0, which means the value of \(d(x)\) gets closer to 1. We can use similar thinking to understand rational functions that do not have horizontal asymptotes.

For example, consider \(f(x)=\frac{x^2+4x+5}{x-3}\). Using division, the expression can be rewritten as \(f(x)=x+7+\frac{26}{x-3}\). As \(x\) gets larger and larger in either the positive or negative direction, the value of the term \(\frac{26}{x-3}\) gets closer and closer to 0, which means the value of \(d(x)\) gets closer to the value of \(x+7\). This means that the end behavior of \(f\) can be described by the line \(y=x+7\). Here is a graph of \(y=f(x)\), the line \(y=x+7\), and the vertical asymptote of the function at \(x=3\):

Graph of a rational function f(x) with no horizontal asymptotes, two dashed intersecting lines on coordinate plane.