Curriculum  /  Math  /  9th Grade  /  Unit 8: Quadratic Equations and Applications  /  Lesson 1

Quadratic Equations and Applications

Lesson 1

Math

Unit 8

9th Grade

Lesson 1 of 15

Objective

Describe features of the vertex form of a quadratic function and write quadratic equations in vertex form from graphs.

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.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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.

Foundational Standards

  • A.SSE.A.1

Criteria for Success

  1. Identify the vertex from an equation written in vertex form, $${ f(x)=a(x-h)^2+k}$$, where the vertex is $${(h,k)}$$.
  2. Describe the features that different forms of quadratic equations reveal about the graph of the function. 
  3. Write the equation for a quadratic function given as a graph or a function described verbally; use the most appropriate form of the equation.

Tips for Teachers

With this lesson, students will have seen the three common forms of quadratic equations: vertex form, intercept form, and standard form. Make this point explicit for students. 

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Anchor Problems

Problem 1

A quadratic function is shown in the graph below, along with its equation in three different forms: intercept form, standard form, and vertex form.

 

$${f(x)=-(x-5)(x+1)}$$

$${f(x)=-x^2+4x+5}$$

$${f(x)=-(x-2)^2+9}$$

 

 

 

 

 

 

 

Determine which form each equation is written in and describe the features seen in the graph that each equation form reveals.

Guiding Questions

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Problem 2

The graph of a quadratic function is shown below.

Write the equation that models this graph in vertex form, $${f(x)=a(x-h)^2+k}$$.

Guiding Questions

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Problem 3

Make up an equation for a quadratic function whose graph satisfies each given condition. Use whatever form is most convenient for each condition.

  1. Has a vertex at $${(-2,-5)}$$
  2. Has a $${ y-}$$intercept at $${ (0,6)}$$
  3. Has $${x-}$$intercepts at $${(3,0)}$$ and $${(5,0)}$$
  4. Has $$ {x-}$$intercepts at the origin and $${(4,0)}$$

Guiding Questions

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References

Illustrative Mathematics Graphs of Quadratic Functions — Part c

Graphs of Quadratic Functions, accessed on Aug. 18, 2017, 10:57 a.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 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

A quatratic equation is shown in three different forms: intercept form, standard form, and vertex form.

$${f(x)=2(x-1)(x-3)}$$

$${f(x)=2x^2-8x+6}$$

$${f(x)=2(x-2)^2-2}$$

 

  1. Identify 4 different coordinate points on the graph of the equation.
  2. Sketch a graph of the parabola.

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 problems in the format $${(x-2)^2=6}$$ and/or include error analysis problems where students incorrectly identify the vertex from this form.
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Lesson 2

Lesson Map

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

Topic A: Deriving the Quadratic Formula

Topic B: Transformations and Applications

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