Unit 2, Lesson 9
End Behavior (Part 2)
Algebra 2
9.1
Warm-up
It’s a Cover Up
Instructional Routines
Materials
To Gather
Math Community Chart
Activity Narrative
Students use the structure of expressions to match equations and graphs of polynomials (MP7). The details of what each graph looks like close to an input of 0 is concealed so that students must rely on their knowledge of end behavior to make the matches. While three of the graphs show end behavior students have encountered in previous lessons, one of the graphs has end behavior opposite of what they have seen. 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
In this activity, students are using structure to reason about the matches, so technology is not an appropriate tool.
Launch
Since the focus of this Warm-up is on structure, this activity was designed to be completed without technology. Ask students to put away any devices.
Activity
NoneMatch 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.
ABlank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 2 upper left, descends to the square box, comes out in Quadrant 1 and ascends to upper right.
BBlank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 3, horizontally to the square box and comes out in Quadrant 4 and horizontally to the right.
CBlank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 2 upper left, descending to the square box and comes out in Quadrant 4 and descends to lower right.
DBlank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 3 lower left, ascending to the square box and comes out in Quadrant 1 and ascends to upper right.
Activity Synthesis
The purpose of this discussion is to consider end behavior that has not been seen before and how it might be expressed in an equation. Some questions to start the discussion:
- “Which graphs must match even-degree polynomials? Odd-degree polynomials?” (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
Then select 2–3 previously identified students to share how they identified which equation matched the third graph, starting with students who used specific input-output pairs. Highlight any students who rewrote
Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, plan to discuss the remaining suggestions over the next few lessons.
Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves as math learners.
Student Lesson Summary
Consider the polynomial functions
Graph of polynomial function y = -x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 3, passes through about (-63 point 3 comma -4,000), ascends in a smooth curve to the origin, into Quadrant 4 and descends in a smooth curve to the lower right through about (63 point 3 comma -4,000).
Graph of polynomial function y = x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes through about (-63 point 3 comma 4,000), descends in a smooth curve to the origin, into Quadrant 4 and ascends in a smooth curve to the upper right through about (63 point 3 comma 4,000).
Now consider the graphs of
Graph of a polynomial, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2 passes through about (-14 point 3 comma 4,000) descends to the origin, into Quadrant 4 and descends to lower right through about (11 point 3 comma -4,000).
Graph of a polynomial, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 3 passes through about (-14 point 3 comma -4,000) ascends to the origin, into Quadrant 1 and ascends to upper right through about (11 point 3 comma 4,000).
For polynomials of both even and odd degree, a negative leading coefficient results in the end behavior of the polynomial being the opposite of what it would be if the leading coefficient were positive.
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