Lesson 8

End Behavior (Part 1)

8.1: Notice and Wonder: A Different View (5 minutes)

Warm-up

The purpose of this warm-up is to elicit student language about the general shape of graphs of polynomials of even and odd degree with a focus on the end behavior of the graphs. 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.

Launch

Display the graphs 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 the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

$y=x^3+4x^2-x-4$

Graph of polynomial function, xy-plane.

$y=x^4-10x^2+9$

Graph of polynomial function, xy-plane.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, point out contradicting information, etc.

At the end of the discussion, tell students that today, they are going to investigate the end behavior of polynomial functions. The end behavior of a polynomial 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 \(x=0\), pairs that are often outside of the graphing windows we use.

8.2: Polynomial End Behavior (20 minutes)

Activity

The goal of this activity is for students to understand why the leading term of a polynomial determines the end behavior.

Launch

Arrange students in groups of 2–3 and assign each group one of the polynomials:

\(y=x^2+1\)

\(y=x^3+1\)

\(y=x^4+1\)

\(y=x^5+1\)

Display a blank version of the table for all to see. As groups finish completing their assigned column of the table, select 1 group for each polynomial to fill in their values on the displayed table before the discussion. Provide access to devices that can run Desmos or other graphing technology.

Conversing: MLR2 Collect and Display. As groups discuss characteristics of their polynomial, listen for and collect the language students use to describe the end behavior. Call students’ attention to language such as “output values,” “negative/positive direction,” or “larger and larger.” Write the students’ words and phrases on a visual display. Be sure to add language to the display during the synthesis discussion, especially when discussing the difference between even and odd degrees of the leading term. Remind students to borrow language from the display as needed. This will help students use appropriate mathematical language for describing the end behavior of a polynomial function.
Design Principle(s): Maximize meta-awareness

Student Facing

  1. For your assigned polynomial, complete the column for the different values of \(x\). Discuss with your group what you notice.

    \(x\) \(y = x^2 + 1\) \(y = x^3 + 1\) \(y = x^4 + 1\) \(y = x^5 + 1\)
    -1000        
    -100        
    -10        
    -1        
    1        
    10        
    100        
    1000        
  2. Sketch what you think the end behavior of your polynomial looks like, then check your work using graphing technology.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Mai is studying the function \(p(x) = \text-\frac{1}{100}x^3 + 25,\!422x^2 + 8x + 26\). She makes a table of values for \(p\) with \(x = \pm 1, \pm 5, \pm 10, \pm 20\) and thinks that this function has large positive output values in both directions on the \(x\)-axis. Do you agree with Mai? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may use parentheses incorrectly for the exponents when calculating negative values in the table. When this happens, encourage students to consider what happens when a negative number is raised to an even power.

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 \(y\) for large negative values of \(x\). The equations with an odd exponent have very large negative values of \(y\) for large negative values of \(x\).)
  • “How would the end behavior of the polynomials change if there was another term in the equations, like \(100x\)?” (For values of \(x\) 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.)

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 \(y= x^3 + 1\), we can say that as \(x\) gets larger and larger in the negative direction, \(y\) gets larger and larger in the negative direction. As \(x\) gets larger and larger in the positive direction, \(y\) gets larger and larger in the positive direction. For \(y=x^4+1\), as \(x\) gets larger and larger in the negative direction, \(y\) gets larger and larger in the positive direction. As \(x\) gets larger and larger in the positive direction, \(y\) 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.

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

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings to summarize the discussion. Include a variety of examples of equations and diagrams that include a set of axes and the overall features of the graph. Label the end behavior on each example with the phrasing used in the activity. For example, “As \(x\) gets larger and larger in the (positive/negative) direction, \(y\) gets larger and larger in the (positive/negative) direction . . . .”
Supports accessibility for: Conceptual processing; Memory

8.3: Two Polynomial Equations (10 minutes)

Activity

This activity is meant to continue the work begun in the previous activity, investigating the end behavior of polynomials and how it 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 \(x\). At each \(x\)-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.

Launch

Tell students to close their books or devices and then display the the equation \(y=(x+1)(x^2-4)(x-5)(2x+3)\) for all to see. Ask students to consider what features of the polynomial they can identify from the equation. After 1–2 minutes, select students to share their observations. (Sample responses: the polynomial has 5 zeros at -1, -2, 2, 5, and -1.5. The graph intercepts the vertical axis at 60. The degree of the polynomial is 5.)

If time allows, ask students to rewrite the equation in standard form before starting the rest of the activity. While students work on the question about which term is greatest, if they wonder where the negative signs went, let them know that the purpose is to consider the magnitude of the term at different inputs, so they were all written as positive terms.

Student Facing

Consider the polynomial \(y=2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60\).

  1. Identify the degree of the polynomial.
  2. Which of the 6 terms, \(2x^5\), \(5x^4\), \(30x^3\), \(5x^2\), \(88x\), or \(60\), is greatest when:
    1. \(x=0\)
    2. \(x=1\)
    3. \(x=3\)
    4. \(x=5\)
  3. Describe the end behavior of the polynomial.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students have trouble deciding what the end behavior will be, remind them of the previous activity, in which the leading term determined the end behavior. Encourage students to connect this to the question in this activity about which term is greatest, in which the leading term has greater magnitude than all the other terms when \(x\) is not near 0.

Activity Synthesis

The purpose of this discussion is for students to use mathematically correct language about the end behavior of a polynomial and to consider what changes to an equation will, and will not, affect the end behavior of the polynomial. Select 2–3 students to share their descriptions about the end behavior of the polynomial. Discuss:

  • “How would the end behavior change if the leading term was \(20x^5\) instead of \(2x^5\)?” (Since the exponent on the \(x\) did not change, the end behavior wouldn’t change.)
  • “How can we change the term \(5x^4\) in order to change the end behavior of the polynomial?” (Changing the exponent to 6 would change the end behavior.)

Consider displaying a graph of the equation for all to see and using graphing technology to change the equation in ways students suggest and verifying the end behavior.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their verbal and written responses to the question “Describe the end behavior of the polynomial.” Give students time to meet with 2–3 partners to share and get feedback on their responses. The listener should press for detail and precision in language by asking, “What will that look like on the graph?”, “How do you know?”, or “Can you say that a different way?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use mathematically correct language when describing the end behavior of a polynomial.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings to summarize the discussion. If a chart was used in an earlier activity, then conclusions can be added onto the same chart. Ask students, “What matters when identifying end behavior?” and write the question on the chart. Add student-driven answers such as “leading term” and “highest exponent.” Non-examples can also be added for reference, such as “coefficient doesn’t matter” and “output when \(x = 0\) does not change end behavior.”
Supports accessibility for: Conceptual processing; Memory

Lesson Synthesis

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.

  1. A series of polynomial equations will be displayed one at a time.
  2. 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.
  3. 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 \(y=x^2\), you put both hands up in the air. For something like \(y=x^3\), 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.

  • \(y=x\)
  • \(y=x^4\)
  • \(y=3x^2+2x-2000000\)
  • \(y=3x-7+x^2\)
  • \(y=x^5+6x^4-x^5-2x+8\)
  • \(y=(x-3)(x+4)(2x+1)\)
  • \(y=x^{10}\)
  • \(y=x^{25}\)
  • \(y=x^{50}\)
  • \(y=x^{101}\)

8.4: Cool-down - Identifying End Behavior (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We know that if the expression for a polynomial function \(f\) written in factored form has the factor \((x-a)\), then \(a\) is a zero of \(f\) (that is,\(f(a)=0\)) and the point \((a, 0)\) is on the graph of the function. But what about other values of \(x\)? In particular, as we consider values of \(x\) that get larger and larger in either the negative or positive direction, what happens to the values of \(f(x)\)?

The answer to this question depends on the degree of the polynomial, because any negative real number raised to an even power results in a positive number. For example, if we graph \(y=x^2\), \(y=x^3\) and \(y=x^4\) and zoom out, we see the following:

$y=x^2$

Graph of polynomial function y = x squared, xy-plane, origin O.

$y=x^3$

Graph of polynomial function y = x cubed, xy-plane, origin O.

$y=x^4$

Graph of polynomial function, xy-plane, origin O.

For both \(y=x^2\) and \(y=x^4\), large positive values of \(x\) or large negative values of \(x\) each result in large positive values of \(y\). But for \(y=x^3\), large positive values of \(x\) result in large positive values of \(y\), while large negative values of \(x\) result in large negative values of \(y\).

Consider the polynomial \(P(x)=x^4-30x^3-20x^2+1000\). The leading term, \(x^4\), almost seems smaller than the other 3 terms. For certain values of \(x\), this is even true. But, for values of \(x\) far away from zero, the leading term will always have the greatest value. Can you see why?

\(x\) \(x^4\) \(\text-30x^3\) \(\text-20x^2\) \(1000\) \(P(x)\)
-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

The value of the leading term \(x^4\) 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. In the case of \(P(x)\), as \(x\) gets larger and larger in the positive and negative directions, the output of the function gets larger and larger in the positive direction.