Unit 2, Lesson 8
End Behavior (Part 1)
Algebra 2
8.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity 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
NoneWhat do you notice? What do you wonder?
Graph of polynomial function, xy-plane. Horizontal axis, scale -20 to 20, by 10’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through about (-20 comma -8,000), ascending in a smooth curve flattened near and through the origin, flattened up into Quadrant 1 and ascends in a smooth curve to the upper right through about (20 comma 8,000).
Graph of polynomial function, xy-plane. Horizontal axis, scale -20 to 20, by 10’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 2, passes through about (-10 comma 8,000), descending in a smooth curve flattened near and through the origin, flattened up into Quadrant 1 and ascends in a smooth curve to the upper right through about (10 comma 8,000).
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.
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:
Graph of polynomial function y = x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes down through (-50 comma 2,500), descending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (50 comma 2,500).
Graph of polynomial function y = x cubed, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes up through (-20 comma 8,000), ascending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (20 comma 8,000).
Graph of polynomial function, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through (-10 comma 10,000), descending in a smooth curve through the origin, up into Quadrant 1 and ascends in a smooth curve to the upper right through (10 comma 10,000).
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 .