Curriculum  /  Math  /  9th Grade  /  Unit 8: Quadratic Equations and Applications  /  Lesson 4

Quadratic Equations and Applications

Lesson 4

Math

Unit 8

9th Grade

Lesson 4 of 15

Objective

Solve and interpret quadratic applications using the vertex form of the equation.

Common Core Standards

Core Standards

  • F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  • A.SSE.B.3.B — Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Foundational Standards

  • F.IF.A.2
  • F.IF.B.5

Criteria for Success

  1. Interpret features of quadratic functions in context of the situation (for example, roots represent “break-even” values in profit models).
  2. Use the process of completing the square to determine the vertex and roots of a quadratic function.
  3. Understand that quadratic equations can be used to represent profit functions (the amount of money a business makes on the sale of a product).

Tips for Teachers

This lesson touches upon some applications of quadratic functions in order to have students interpret features in context. Later in the unit in Lessons 11–13, students will engage more deeply with applications by writing and then analyzing equations that model quadratic situations.

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Anchor Problems

25-30 minutes

Problem 1

A business is marketing its product and has collected data on sales and prices for the past few years. The company, through careful modeling, has determined that their profit is dependent upon the selling price of the item and that it follows a quadratic model. 

The function below describes the profit, $$P$$, dependent upon the selling price, $$s$$, of its product. 

$$P(s)=-20s^2+1,400s-12,000$$

a.  Write the function in intercept form and identify the roots. 

b.  Explain what the roots mean in context of the situation.

c.  Write the function in vertex form and identify the vertex. 

d.  Explain what the vertex means in context of the situation. 

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 4 > Topic B > Lesson 12 — Example 2

Algebra I > Module 4 > Topic B > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

The height, $$h$$, in feet, of a projectile launched upward from a bridge is given by $$ h=-16t^2+16t+96$$, where $$t$$ is time in seconds. 

  1. Write the equation in vertex form.
  2. How long does it take for the projectile to reach maximum height?
  3. What is the maximum height of the projectile? 

Guiding Questions

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Problem Set

15-20 minutes

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

Target Task

5-10 minutes

The profit that a company makes selling an item (in thousands of dollars) depends on the price of the item (in dollars). If $$p$$ is the price of the item, then three equivalent forms for the profit are:

Standard form: $$-2p^2+24p-54$$

Factored form: $$-2(p-3)(p-9)$$

Vertex form: $$-2(p-6)^2+18$$

 

  1. Which form is most useful for finding the prices that give a profit of zero dollars? Explain your reasoning and find the values. 
  2. Which form is most useful for finding the profit when the price is $$0$$? Explain your reasoning and find the value. 
  3. Which form is most useful for finding the price that gives the maximum profit? Explain your reasoning and find the value.

References

Illustrative Mathematics Profit of a Company

Profit of a Company, accessed on July 12, 2018, noon, 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.

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.

  • Include other application problems with the equation provided and have students determine and interpret the roots and vertex
  • Include problems that review completing the square for students to continue to build fluency with this skill

Next

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

Lesson 5
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Lesson Map

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

Topic A: Deriving the Quadratic Formula

Topic B: Transformations and Applications

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