Curriculum  /  Math  /  9th Grade  /  Unit 8: Quadratic Equations and Applications  /  Lesson 5

Quadratic Equations and Applications

Lesson 5

Math

Unit 8

9th Grade

Lesson 5 of 15

Objective

Convert and compare quadratic functions in standard form, vertex form, and intercept form.

Common Core Standards

Core Standards

  • F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Foundational Standards

  • A.SSE.B.3.A

Criteria for Success

  1. Understand the features of a quadratic function that are revealed by different forms of the equation. 
  2. Show algebraically and through analysis of features that different quadratic forms represent the same parabola. 
  3. Describe efficient forms and methods to identify features of a parabola in mathematical and contextual situations.

Tips for Teachers

  • In regard to pacing, this lesson may be extended over more than one day to ensure students are skillful at working with all three forms. There are several great resources in the Problem Set Guidance that can be split over two days or incorporated into later lessons for review. 
  • Students will also get more opportunities to work with all three forms of quadratic equations in Lesson 8, where they will focus on graphing.
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Anchor Problems

Problem 1

Which of the following could be the function of a real variable $$x$$ whose graph is shown below? Explain your reasoning for each equation.

 

$$f_1(x)=(x+12)^2+4$$ $$f_5(x)=-4(x+2)(x+3)$$
$$f_2(x)=-(x-2)^2-1$$ $$f_6(x)=(x+4)(x-6)$$
$$f_3(x)=(x+18)^2-40$$ $$f_7(x)=(x-12)(-x+18)$$
$$f_4(x)=(x-12)^2-9$$ $$f_8(x)=(24-x)(40-x)$$

 

Guiding Questions

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References

Illustrative Mathematics Which Function?

Which Function?, accessed on July 12, 2018, 4:57 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

Suppose Andre, Brett, and Carlo each throw a baseball into the air. 

  • The height of Andre’s baseball is given by $${h(t)=-16t^2+80t+6}$$, where $$h$$ is in feet and $$t$$ is in seconds. 
  • The height of Brett’s baseball is given by $$h(t)=-16(t-1.5)^2+65$$, where $$h$$ is in feet and $$t$$ is in seconds.
  • The height of Carlo’s baseball is given by the graph below. 

  1. Whose baseball went the highest?
  2. How long is each baseball airborne?

Guiding Questions

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References

Illustrative Mathematics Throwing Baseballs

Throwing Baseballs, accessed on Aug. 18, 2017, 2:52 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 3

All three equations shown below represent the same function. Prove this algebraically. 

$$f_1  (x)=2(x-3)(x+5) $$

$${f_2 (x)=2x^2+4x-30}$$
$${f_3 (x)=2(x+1)^2-32 }$$

Guiding Questions

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Problem Set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

Here are 4 equations of quadratic functions and 4 sketches of the graphs of the quadratic functions.

A.   $${y=x^2-6x+8}$$

B.   $${y=(x-6)(x+8)}$$

C.   $${y=(x-6)^2+8}$$

D.   $${y=-(x+8)(x-6)}$$

  1. Match each equation to its graph and explain your decision.
  2. Write the coordinates of the points: $${P(\space\space,\space\space)}$$, $${Q(\space\space,\space\space)}$$, $${R(\space\space,\space\space)}$$, $${S(\space\space,\space\space)}$$

References

MARS Formative Assessment Lessons for High School Representing Quadratic Functions Graphically — Quadratic Functions, #1

Representing Quadratic Functions Graphically from the Classroom Challenges by the MARS Shell Center team at the University of Nottingham is made available by the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed July 12, 2018, 12:06 p.m..

Modified by Fishtank Learning, Inc.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

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Lesson 4

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Lesson 6

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Deriving the Quadratic Formula

Topic B: Transformations and Applications

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