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Nonlinear Functions

Unit Plan

Nonlinear Functions

Objectives

In this unit, students will model nonlinear functions, including absolute value functions and quadratic functions. Students will:

  1. create a table of values and graphs of absolute value functions and quadratic functions.
  2. develop an understanding of transformations of functions.
  3. model real-world situations with absolute value functions and quadratic functions and interpret components of the function, including vertex, domain, range, and rate of change.
  4. develop the ability to imagine real-world scenarios best modeled by each type of function.

Essential Questions

  1. How can a given function be represented? What connections can be made between the various representations?
  2. Which function best models a given real-world scenario? What does the function look like in the real world?
  3. How can an equation, table, and graph be used to analyze the rate of change and other applicable information, related to a real-world problem and the representative function?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

  1. http://enlvm.usu.edu/ma/nav/activity.jsp?sid=nlvm&cid=4_2&lid=109
  2. http://www.mathwarehouse.com/geometry/parabola/real-world-application.php

Formative Assessment

  • View

    Multiple Choice Items


    1. Which of the following is the graph of  y = |x – 1| ?

    A

    01a.PNG

    B

    01b.PNG

    C

    01c.PNG

    D

    01d.PNG

    2. Which is the vertex of y = –|x + 3|?

    A

    (3, 0)

    B

    (0, 3)

    C

    (0, −3)

    D

    (−3, 0)


    3. What is the domain and range of the function y = |x| – 11?

    A

    D = [−11, ∞); R = (−∞, ∞)

    B

    D = (−∞, ∞); R = [−11, ∞)

    C

    D = [11, ∞); R = (−∞, ∞)

    D

    D = (−∞, ∞); R = (11, ∞)

    4. Which of the following is the graph of y = −(x − 2)2 + 5

    A04a.PNG

    B04b.PNG

    C04c.PNG

    D04d.PNG


    5. Which is the vertex of y=-3x2-4?

    A

    (4, 0)

    B

    (0, 4)

    C

    (0, −4)

    D

    (0, −3)

    6. What is the domain and range of the function y = (x + 5)² – 9?

    A

    D = [9, ∞); R = (−∞, ∞)

    B

    D = [−9, ∞); R = (−∞, ∞)

    C

    D = (−∞, ∞); R = [−9, ∞)

    D

    D = (−∞, ∞); R = [9, ∞)

    7. A science student wishes to model the path of a projectile from launch to landing. Which of the following functions is the most accurate model?

    A

    y=x-6

    B

    y=-x+6

    C

    y=x²+6

    D

    y=-x² + 6


    8. A baseball pitcher throws the ball to the batter. Which of the following functions is a the most representative model?

    A

    y=14x²+16x+3

    B

    y=-14x²+16x + 3

    C

    y=14x+3

    D

    y=-14x + 3

    9. Which statement is true about the graphs of f(x) = |1 – x| and f(x) = (1 – x)²?

    A

    Both ranges are positive.

    B

    Both domains are positive.

    C

    They have two common zeros.

    D

    They intersect at the origin.

    Multiple-Choice Answer Key

    1. A

    2. D

    3. B

    4. B

    5. C

    6. C

    7. D

    8. B

    9. A



    Short Answer Items:


    10. Write the vertex of y = –2|x + 6|. Does the graph open up or down? Is the vertex a minimum or maximum?
    11. Provide a transformed function of the parent quadratic function y=x². Write the vertex and domain and range.
    12. Describe a quadratic function found in the business world. Explain the meaning of the orientation, domain, and range within the context of the problem.

    Short-Answer Key and Scoring Rubrics:


    10. Write the vertex of y = –2|x + 6|. Does the graph open up or down? Is the vertex a minimum or maximum?

    Scoring Rubric for Item 10:

    Points

    Description

    3

    • The student writes the following:

    • (−6, 0) AND

    • opens down AND

    • maximum.

    2

    • The student writes two of the three answers.

    1

    • The student writes one of the three answers.

    0

    • The student does not provide any answer.


    11. Provide a transformed function of the parent quadratic function y=x². Write the vertex and domain and range.

    Scoring Rubric for Item 11:

    Points

    Description

    3

    • The student writes the following:

    • a possible transformed quadratic function AND

    • the correct vertex AND

    • the correct domain and range.

    2

    • The student writes two of the three answers.

    1

    • The student writes one of the three answers.

    0

    • The student does not provide any answer.
    12. Describe a quadratic function found in the business world. Explain the meaning of the orientation, domain, and range within the context of the problem.

    Scoring Rubric for Item 12:

    Points

    Description

    3

    • The student writes the following:

    • a possible business-related quadratic function AND

    • gives orientation and relates orientation to context AND

    • gives domain and range and relates domain and range to context.

    2

    • The student writes two of the three answers.

    1

    • The student writes one of the three answers.

    0

    • The student does not provide any answer.

    Performance Assessment:

    Determine one specific area of confusion related to either absolute value functions or quadratic functions. Write an article for submission to a scholarly journal in the mathematics field. Explain the area of confusion. Include examples (at least one) and provide possible solutions for clearing up such confusion, including possible explorations, investigations, etc. The article should be two to three pages in length.

    Performance Assessment Scoring Rubric:

    Points

    Description

    5

    • The student:

    • identifies a specific area of confusion related to either absolute value functions or quadratic functions.

    • explains the area of confusion.

    • includes at least one example.

    • provides possible solutions.

    • adheres to the article length requirement.

    4

    • The student includes 4 of the 5 requirements.

    3

    • The student includes 3 of the 5 requirements.

    2

    • The student includes 2 of the 5 requirements.

    1

    • The student includes 1 of the 5 requirements.

    0

    • The student does not write anything.

DRAFT 11/08/2010
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