End Behavior (Part 2)
9.1: It’s a Cover Up (5 minutes)
In this warm-up, students use the structure of expressions to match equations and graphs of polynomials (MP7). While three of the graphs show end behavior students have encountered in previous lessons, one of the graphs has end behavior the opposite of what they have seen, and this graph matches an equation with a negative leading coefficient. The connection between a negative leading coefficient and “flipped” end behavior is explored further in the following activity, so only an informal understanding is expected at this time.
Monitor for students using different strategies to identify a matching equation for the third graph. For example, some students may test different inputs while others may rewrite \(y=x(x+3)(2-x)\) as \(y=\text-x^3-x^2+6x\) and reason about the effect of the -1 coefficient on the leading term.
Since the focus of this warm-up is on structure, students should reason about the matches without using technology to graph the given equations.
Match each of the graphs to the polynomial equation it represents. For the graph without a matching equation, write down what must be true about the polynomial equation.
Some questions to start the discussion:
- “Which graphs must match even degree polynomials? Odd degree?” (Graphs A and B are even degree since the end behavior on each side matches. Graphs C and D must be odd degree polynomials, since the end behavior on each side is opposite.)
- “How did you identify the match for Graph B?” (I rewrote each equation in standard form and saw that \(y=5(x+3)-5x\) is the same as \(y=15\), so that must be the match to Graph B.)
Select 2–3 previously identified students to share how they identified what equation matched the third graph, starting with students who used specific input-output pairs. Highlight any students who rewrote \(y=x(x+3)(2-x)\) in standard form to see that it is a third degree polynomial. Tell students that in the next activity, they will investigate which parts of an equation do and do not affect the end behavior.
9.2: The Case of Unexpected End Behavior (15 minutes)
The purpose of this activity is to address how the leading coefficient being positive or negative changes the end behavior of a polynomial. Working in small groups is meant to give students more equations to compare as they learn which features of an equation match with different output behaviors.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Arrange students in groups of 4. Give individual work time for the first question followed by small-group sharing of the different equations written. Invite groups to share 1–2 equations and record these for all to see. Tell groups to discuss what similarities and differences they see between the equations before starting work on the last question.
Supports accessibility for: Memory; Organization
Write an equation for a polynomial with the following properties: it has even degree, it has at least 2 terms, and, as the inputs get larger and larger in either the negative or positive directions, the outputs get larger and larger in the negative direction.
Pause here so your teacher can review your work.
- Write an equation for a polynomial with the following properties: it has odd degree, it has at least 2 terms, as the inputs get larger and larger in the negative direction the outputs get larger and larger in the positive direction, and as the inputs get larger and larger in the positive direction, the outputs get larger and larger in the negative direction.
Are you ready for more?
In the given graph all of the horizontal intercepts are shown. Find a function with this general shape and the same horizontal intercepts.
If some students have trouble getting started because the end behavior is different from most of the functions they've worked with, remind them of the odd-degree function with a negative leading coefficient that they saw in the warm-up, which also had flipped end behavior. Encourage them to think about how they could make a similar function that has the end behavior they’re looking for.
The goal of this discussion is to make sure students understand that if the leading term has a negative coefficient, then the end behavior of the graph will be “flipped” when compared to an equation of the same degree with a positive leading coefficient. This also means that even polynomials still have matching end behavior and odd polynomials do not.
Select 2–3 groups to share some of the odd degree equations they wrote and record these for all to see alongside the list of the even degree equations. After some quiet think time, ask groups to write a brief summary of which parts of an equation can be used to identify end behavior (the leading term) and which parts do not affect the end behavior (all other terms). Invite groups to share their summaries and help the class reach agreement.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
9.3: Which is Greater? (10 minutes)
While we describe end behavior of different functions in similar ways, not all end behavior is the same. In this activity, students investigate two functions with different degrees but the same end behavior when \(x>0\) and they are asked to decide which function they believe is greater. Students can use graphs, expressions, or tables to analyze the functions. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
The goal of this activity is that students understand what the degree of a polynomial can tell us about the end behavior of its graph. A higher-degree function will always have output values that eventually exceed in magnitude the outputs of a lower-degree function, but in this case, both functions increase in the negative direction as \(x\) increases in the positive direction, so the lower-degree function will have greater values. Students will need to be clear about their understanding of “greater” when explaining their thinking to the class (MP3).
Monitor for students using different strategies to build their case for why they think one function is greater than another. For example, some students may graph each function on the same axes, while others make a table of various input-output pairs.
Supports accessibility for: Language; Organization
\(M\) and \(N\) are each functions of \(x\) defined by \(M(x)=\text-x^3 - 2x + 8\) and \(N(x)=\text-20x^2+3x+8\).
- Describe the end behavior of \(M\) and \(N\).
- For \(x>0\), which function do you think has greater values? Be prepared to share your reasoning with the class.
If students have trouble making connections between these specific functions and the general point that the output of any function will eventually exceed the output of a function of lower degree, it may help if they graph simpler functions like \(x^4\) and \(x^3\) and compare their outputs for large positive values of \(x\). Even if the lower-degree function has a large coefficient, the higher-degree function will exceed it eventually. For example, \(x^4\) will exceed \(300x^3\) when \(x\) is larger than 300.
Invite previously identified students to share their reasoning for which function has greater values when \(x>0\), starting with any students who tested specific points or made tables of input-output pairs and then moving to students who used graphs. While graphing with an appropriate window size may have helped students decide on an answer faster, focus the discussion on how a table of input-output values can help us understand why the output of a polynomial with higher degree will always exceed in magnitude the output of a polynomial with lower degree when moving away from the horizontal axis, connecting back to their work from the previous lesson.
Some students may have concluded that \(M\) is greater than \(N\) because it has more extreme values as \(x\) increases. Remind these students that “greater” is a term with a precise meaning, and that negative numbers with large magnitude are less than negative numbers with small magnitude. So they are correct that the values of \(M\) are more extreme, and it’s important to recognize the pattern that higher-degree functions have more extreme values, but in this case, that makes \(M\)’s values less than \(N\)’s.
Design Principle(s): Support sense-making
Tell students that they are going to play the same game today as in the previous lesson, only this time, they will supply the equations. Give each student a slip of paper and ask them to write a polynomial equation for the game. Collect the slips and then remind students how the game works.
- 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!”, 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.
After playing several rounds of the game, if time allows, invite students who do particularly well to share any strategies they have with the class.
9.4: Cool-down - Describe the End Behavior (5 minutes)
Student Lesson Summary
What happens when we multiply a number by a negative number? If the original number was positive, the product is negative. But if the original number was negative, the product is positive. The sign of the new number is the opposite of the original number.
Now let’s consider the polynomial functions \(f(x)=x^2\) and \(g(x)=\text-x^2\). For any non-zero real number \(x\), the output of \(f\) is positive while the output of \(g\) is negative. The signs of all the output values for \(g\) are the opposite of those of \(f\). The difference between these two functions is also easy to see when we look at their graphs.
This is the effect of a negative leading coefficient: the end behavior of the polynomial is the opposite of what it would be if the leading coefficient were positive. For polynomials of odd degree, we can see that a negative leading coefficient has the same effect on the end behavior.
Here are the graphs of \(y=\text-(x-1)(2x+3)(x+4)\), which has a leading term of \(\text-2x^3\), and \(y=(x-1)(2x+3)(x+4)\), which has a leading term of \(2x^3\). They have the same zeros, but opposite end behavior, because they have opposite signs on their leading coefficients.