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Quadratic Functions

Unit Plan

Quadratic Functions

Objectives

Students will gain a greater understanding of quadratic functions. They will:

  • learn advanced mathematics associated with quadratic functions.

  • learn the complete-the-square technique, and understand how to use it as appropriate.

  • understand how to graph quadratics using transformations.

  • learn how to transform quadratic functions into vertex form.

Essential Questions

  • How can we determine if a real-world situation should be represented by a quadratic, polynomial, or exponential function?

  • How do quadratic equations and their graphs and/or tables help us interpret events that occur in the world around us?

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.

Formative Assessment

  • View

    Multiple Choice Items:

    1. How does the graph of equation y = −2(x + 1)2 + 3 compare to the graph of y = x2?

    A

    wider, upside down, left 1, up 3

    B

    skinnier, upside down, right 1, up 3

    C

    skinnier, upside down, left 1, up 3

    D

    wider, upside down, right 1, up 3

    2. What is the vertex of the graph of y = (x − 3)2 − 4?

    A

    (3, −4)

    B

    (−3, −4)

    C

    (3, 4)

    D

    (−4, 4)

    3. What is the stretch factor in the equation y = −(x + 4)2 − 1?

    A

    −

    B

    −

    C

    4

    D

    −1

    4. Which equation is the standard form of the equation y = (x − 2)2 + 3?

    A

    y = x2 – 1

    B

    y = x2 − 4x + 7

    C

    y = x2 − 4x − 1

    D

    y = x2 + 4x + 7

    5. What are the solutions to x2 = −64?

    A

    		  

    B

    		  

    C

    		  

    D

    		  

    6. What are the solutions to x2 + 3x + 5 = 0? (Use the quadratic formula.)

    A

    		  

    B

    		  

    C

    		  

    D

    		  

    7. Simplify: 4 + 5i − (6 + 2i)

    A

    −2 + 7i

    B

    10 + 7i

    C

    10 + 3i

    D

    −2 + 3i

    8. Simplify: (4 − 2i) ÷ (3 + i)

    A

    1 − i

    B

    1 + i

    C

    		  

    D

    10 − 10i

    9. The total impedance in a series alternating-current (AC) circuit is ZT = Z1 + Z2, where the impedances of the individual circuits, Z1 and Z2, are expressed as complex numbers. What is the total impedance, ZT, of a series circuit where Z1 = 6 + 2i ohms and Z2 = 5 − 3i ohms?

    A

    8 + 2i ohms

    B

    8 + 8i ohms

    C

    11 − i ohms

    D

    11 + 5i ohms

    Multiple Choice Answer Key:

    1. C

    2. A

    3. B

    4. B

    5. B

    6. C

    7. D

    8. A

    9. C

     

    Short Answer Items:

    10. Samantha kicked a soccer ball 20 feet in the air to her friend who was 50 feet away. Find the equation of the soccer ball’s path.





    11. Convert the following equation from standard form to vertex form by completing-the-square:

    y = x2 – 8x + 22





    12. The voltage (V) is 25 + 5i volts and the impedance (Z) is 2 + i ohms in an AC circuit. What is the current (I) in amps in the circuit (V = ZI)?





    13. What are the x-intercepts of x2 − 6x + 13?

    Short-Answer Key and Scoring Rubrics:

    10. Samantha kicked a soccer ball 20 feet in the air to her friend who was 50 feet away. Find the equation of the soccer ball’s path.

    y = −.032(x – 25)2 + 20

    11. Convert the following equation from standard form to vertex form by completing-the-square:

    y = x2 – 8x + 22

    y = (x − 4)2 + 6

    12. The voltage (V) is 25 + 5i volts and the impedance (Z) is 2 + i ohms in an AC circuit. What is the current (I) in amps in the circuit (V = ZI)?

    11 − 3i amps

    13. What are the x-intercepts of x2 − 6x + 13?

    3 + 2i and 3 − 2i

    Points

    Description

    2

    • Response is complete, correct, and detailed.

    • Student demonstrates thorough understanding of quadratic functions.

    1

    • Response is partially correct or true but does not answer the specific question, or is correct but lacking detail.

    • Student demonstrates partial understanding of quadratic functions.

    0

    • Response is incorrect.

    • Student demonstrates no understanding of quadratic functions.

    Performance Assessment:

    Compare and contrast two methods of finding the solution to a quadratic equation.

    1. Graphing

    2. Another method of your choice

    When is it most appropriate to use one or the other method and why? Give specific examples.

    Performance Assessment Scoring Rubric:

    Points

    Description

    4

    • Graph is drawn correctly.

    • Rectangles are drawn correctly with a width of 1.

    • The area is correct.

    • Work is shown.

    3

    • Graph is drawn correctly.

    • Rectangles are drawn but not as specified.

    • The area is correct.

    • Work is shown.

    2

    • Graph is drawn but has two or fewer errors.

    • Rectangles are drawn but incorrectly or not as specified.

    • The area is incorrect.

    • Work is unclear or not shown.

    1

    • Graph is drawn but has three or four errors.

    • Rectangles are not drawn.

    • The area is incorrect.

    • Work is unclear or not shown.

    0

    • Graph is drawn but has many errors or was not completed.

    • Rectangles are not drawn.

    • The area is incorrect.

    • Work is unclear or not shown.

     

DRAFT 08/31/2011
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