If students forget that a single number, like 49.5, qualifies as a polynomial when they are rewriting the expression, consider asking:

• “Can you explain how you rewrote your function.”
• “What is the same and different about 49.5 and ?”

The goal of this discussion is to support students in recognizing how using long division to rewrite the original expression helps us identify the end behavior of the function.

Select 2–3 students to share what they calculated for in terms of and how they rewrote in the form , recording these for all to see and refer to throughout the Activity Synthesis.

Then, ask students, “What can you say about the end behavior of ?” After a brief quiet think time, display a graph of for all to see, showing the function for positive values of . Invite students to share their explanation for what the end behavior of the function is. (As gets larger and larger, the combined fuel efficiency also gets larger and larger. For large values of , the relationship between and is close to , since the value of is less than 1 for any -value greater than 20.25.)

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

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

Students might not be sure whether the remainder in the last row approaches 0 as gets larger, since appears in both the numerator and denominator. Encourage these students to plug in some large values of 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 , where the degree of polynomial is less than the degree of polynomial , is the same as the end behavior of . Students should also recognize that the degree of 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, encourage students to connect back to the original prediction for the function and why it was correct or incorrect. If any groups graphed their functions, display these so students can 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. It is important for students to notice that there are three cases that define the end behavior of the rational function:

• 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

In each case, using division to rewrite the equation so that the polynomial is less than the degree of polynomial , meaning will approach 0 for large values of , is a key insight. 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 need to be only three cases. When appropriate, revoice student explanations to demonstrate and amplify mathematical language use.