Lesson | Loading...

Unit 2, Lesson 8

End Behavior (Part 1)

Algebra 2

8.1

Warm-up

5 mins

Notice and Wonder: A Different View

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

The purpose of this Warm-up is to elicit student language about the general shape, especially the end behavior, of graphs of polynomials of even and odd degrees, which will be useful when students look for patterns between end behavior and the degree of the polynomial in a later activity. While students may notice and wonder many things about these equations and graphs, the end behavior of the graphs is the important discussion point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). This activity is meant only to introduce the idea of end behavior. Throughout this lesson and the next, students will continue to refine their language use. They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Launch

Arrange students in groups of 2. Display the image and equations for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

Student Task Statement

What do you notice? What do you wonder?

<p>Graph of polynomial function, xy-plane.</p>

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

At the end of the discussion, tell students that they are going to investigate the end behavior of polynomial functions, which is what happens to the output as the input gets larger and larger in either the positive or negative directions. For end behavior, we are using “larger” to refer to magnitude, that is, distance from 0. End behavior does not describe the zeros of a function or the intercepts of the graph, but rather, the trend of the input-output pairs of the function as we move farther from . These input-output pairs are often outside of the graphing windows we use.

8.2

Activity

20 mins

Polynomial End Behavior

Instructional Routines

Graph It MLR2: Collect and Display

Materials

To Gather

Graphing technology

Activity Narrative

In this activity, students understand why the leading term of a polynomial determines the end behavior.

This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.

Launch

Graphing technology is needed for every student. Arrange students in groups of 2–3, and assign each group one of the polynomials:

Display a blank version of the table for all to see. As groups complete their assigned column of the table, select 1 group to fill in their values on the displayed table for each polynomial.

Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe the end behavior of their polynomial. Display words and phrases, such as “output values,” “negative/positive direction,” or “larger and larger.”

8.3

Activity

10 mins

Two Polynomial Equations

Instructional Routines

None

Materials

None

Activity Narrative

This activity continues to investigate how the end behavior of polynomials can be determined from the structure of an expression. Starting with a 5th-degree polynomial that has 6 terms, students consider the equation for the polynomial in both factored and standard forms before considering the value of the individual terms at specific values of . At each -value, a different term has the greatest magnitude. This is meant to reinforce that, as the name implies, end behavior is about the input-output pairs of the function at the ends of the graph away from any intercepts or other unique features.

Students use precise language as they refine their descriptions of a polynomial's end behavior (MP6).

Lesson Synthesis

Ask students to stand up to play a game. Suggest that students wiggle their arms and do some stretches while explaining the rules of the game.

A series of polynomial equations will be displayed one at a time.
After an equation is displayed, there will be a brief quiet think time to identify the end behavior. Give a hand signal when you are ready.
When you hear “Pose!” (or a different word chosen by the class), use your arms to show the end behavior of the function. For example, for , you put both hands up in the air. For something like , you have your left arm down and your right arm up.

Here are some polynomial equations to use for the game. Have a graph of each ready to show for students to check their pose against.

to access Cool-down.

Student Lesson Summary

The value of the leading term determines the end behavior of the function, that is, how the outputs of the function change as we look at input values farther and farther from 0.

Consider the polynomial . The leading term, , almost seems smaller than the other three terms, and for certain values of , this is even true. But, for values of far away from 0, the leading term will always have the greatest value. In the case of , as gets larger and larger in the positive and negative directions, the output of the function gets larger and larger in the positive direction.


-500	62,500,000,000	3,750,000,000	-5,000,000	1,000	66,245,001,000
-100	100,000,000	30,000,000	-200,000	1,000	129,801,000
-10	10,000	30,000	-2,000	1,000	39,000
0	0	0	0	1,000	1000
10	10,000	-30,000	-2,000	1,000	-21,000
100	100,000,000	-30,000,000	-200,000	1,000	69,801,000
500	62,500,000,000	-3,750,000,000	-5,000,000	1,000	58,745,001,000

If we graph , and and zoom out, we see the following:

<p>Graph of polynomial function y = x squared, xy-plane, origin O.</p>

<p>Graph of polynomial function y = x cubed, xy-plane, origin O.</p>

<p>Graph of polynomial function, xy-plane, origin O.</p>

For both and , large positive values of or large negative values of each result in large positive values of .

But for , large positive values of result in large positive values of , while large negative values of result in large negative values of .

Student Task Statement

In the column for your assigned polynomial, evaluate for the different values of . Discuss what you notice with your group.


-1000
-100
-10
-1
1
10
100
1000

Sketch what you think the end behavior of your polynomial looks like, then check your work using graphing technology.

Activity Synthesis

The purpose of this discussion is for students to understand why the end behavior of polynomials with a leading term of odd degree differs from polynomials with a leading term of even degree. Discuss:

“Describe the similarities and differences between the polynomials listed in the table.” (The equations with an even exponent have very large positive values of for large negative values of . The equations with an odd exponent have very large negative values of for large negative values of .)
“How would the end behavior of the polynomials change if there was another term in the equations, like ?” (For values of near 0, there would be a change, but the end behavior would not change, since 100 is a small number compared to 1 million or -999,999,999.)

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they would describe the end behavior they see in the graph of the polynomial. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Tell students that one way to describe the end behavior is by stating what happens to the output values as the input values move away from 0. For example, for , we can say that as gets larger and larger in the negative direction, gets larger and larger in the negative direction. As gets larger and larger in the positive direction, gets larger and larger in the positive direction. For , as gets larger and larger in the negative direction, gets larger and larger in the positive direction. As gets larger and larger in the positive direction, gets larger and larger in the positive direction. The “larger and larger” language is used to be clear that no matter how large you go (since you can never zoom out far enough to see the entire polynomial), you know what the output values are doing. Students will have more practice using this language in the future, and do not need to be fluent with the exact words at this time.

Add the language "As gets larger and larger in the positive/negative direction, gets " to the reference created using Collect and Display.

If time allows, ask students to consider the end behavior of linear functions, that is, polynomials with degree 1.

Lesson | Loading...

Settings

Audience

Assessment Access

Lesson 8

End Behavior (Part 1)

Warm-up

Notice and Wonder: A Different View

Standards Alignment

Building On

Addressing

Building Toward

HSF-IF.C.7.c

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Polynomial End Behavior

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Two Polynomial Equations

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Are You Ready for More?

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-SSE.A

Building Toward

HSA-APR.B.3

HSF-IF.C

HSF-IF.C.7.c

Standards Alignment

Building On

Addressing

HSA-SSE.A

Building Toward

HSF-IF.C.7.c