Compare and Contrast Polynomial Graphs Generalization.docx - Section 2: Investigation and New Learning
Compare and Contrast Graphs of Polynomial Functions
Lesson 6 of 9
Objective: Students will be able to describe the similarities and differences between graphs of similar but not identical polynomial functions. Students will be able to describe accurately the end behavior, degree, x-intercepts and general shape of a polynomial function and explain how slight changes to a function rule can affect these aspects of the graph.
Closing
Ask students to examine the graphs they created while working on the classwork today. Ask them to choose a few questions that they can answer from the Polynomial Discussion Questions. These questions are very abstract and challenging for students. This is a time to explicitly model for students how to use their work to create examples. The whole point of creating all these graphs is to use them as examples to make generalizations about the behavior of polynomial functions. Once they have some ideas, facilitate a brief discussion of selected questions, asking students to share their answers. For their check-out, then can produce 2 thorough answers using complete sentences and examples. If students struggle to organize their thinking, discuss on sample question while taking notes on the projector to illustrate how to use examples to support a generalization.
Here is a model response to one of the more challenging questions in the discussion. Students will not generate this full response, but should be able to address some of these ideas.
How does changing the exponent in a polynomial equation written in factored form affect the end behavior, degree and x-intercepts of a polynomial?
If you square a factor (changing the exponent from 1 to 2), the behavior of the function changes at the root that corresponds to the factor. If the exponent that the factor is raised to is 1 (or any other odd number), then the graph will cross the x-axis at that root. If the exponent is even, then the graph will “bounce off” the x-axis at that root.
This definitely does affect the degree, because the highest exponent in the equation will be affected by the different exponent. It does not affect the x-intercepts, however, because the equation still has the same factors. It will change the end behavior because it could switch the degree from even to odd, or odd to even, so this will affect the end behavior.
Refer to problem #2 or problem #6 to see examples of this. |
Resources (1)
Resources (1)
Resources
- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions
- LESSON 1: The Painted Cube Problem
- LESSON 2: The Painted Cube Part 2 and End Behavior
- LESSON 3: Surface Area and Volume Functions
- LESSON 4: Writing Rules for Polynomial Functions using Data Tables
- LESSON 5: Sketching Graphs of Polynomial Functions
- LESSON 6: Compare and Contrast Graphs of Polynomial Functions
- LESSON 7: Relationship between the Degree and the Number of X-intercepts of a Polynomial
- LESSON 8: Writing Equations for Polynomial Graphs
- LESSON 9: Graphing Polynomial Transformations