Lesson 1 | Quadratic Functions and Solutions | 9th Grade Mathematics

Objective

Compare quadratic, exponential, and linear functions represented as graphs, tables, and equations.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Interpreting Functions

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.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Linear, Quadratic, and Exponential Models

F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Foundational Standards

The foundational standards covered in this lesson

F.LE.A.1

Linear, Quadratic, and Exponential Models

F.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Distinguish between linear, exponential, and quadratic functions in tables, equations, and graphs.
Describe features of quadratic functions represented in graphs.
Describe features of quadratic functions represented in equations.
Describe features of quadratic functions represented in tables, including constant second differences.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The first three lessons in this unit introduce students to features of quadratic functions, as seen in their graphs, tables, and equations. Specifically, they compare and contrast quadratic functions to linear and exponential functions they have studied in previous units.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Identify each graph below as an example of a linear function, exponential function, or quadratic function. Use features of the graphs to support your classification.

Guiding Questions

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Problem 2

The three tables below represent a linear function, an exponential function, and a quadratic function. Determine which type of function each table represents and explain your reasoning.

Guiding Questions

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Problem 3

The first three steps of three visual patterns are shown below.

The functions that define the number of tiles in step $$n$$ of each pattern are shown below. Decide which function defines which pattern, and explain your reasoning using the structures seen in the pattern.

$$f(n)=3n^2$$

$$g(n)=n^2+4$$

$$h(n)=n^2-1$$

Guiding Questions

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References

Illustrative Mathematics Quadratic Sequence 1 — Part a

Quadratic Sequence 1, accessed on July 2, 2018, 9:52 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

A set of suggested resources or problem types that teachers can turn into a 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 task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Categorize each representation of a function as linear, exponential, or quadratic. Explain your reasoning for each one.

$$x$$	$$y$$
1	-2
2	1
3	6
4	13
5	22

$$x$$	$$y$$
1	48
2	24
3	12
4	6
5	3

c. $$f(x)=x(x-4)$$

d. $$f(x)=4-{x\over2}$$

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 where students analyze, describe, and categorize tables, equations, and graphs as linear, exponential, or quadratic

Approximately Normal Experiences in Math Perplexed in Patterns — (This is a great activity to follow Anchor Problem #3 and to engage students in investigating quadratic patterns)
Illustrative Mathematics Identifying Functions — (Students may have seen an adapted version of this task in Unit 6)
Mathematics Vision Project: Secondary Mathematics Two Module 1: Quadratic Functions — Lesson 1.1 "Go"

Lesson 2

Lesson Map

Topic A: Features of Quadratic Functions

Compare quadratic, exponential, and linear functions represented as graphs, tables, and equations.

F.IF.B.4 F.LE.A.2

Identify key features of a quadratic function represented graphically. Graph a quadratic function from a table of values.

F.IF.B.4 F.IF.C.7.A

Calculate and compare the average rate of change for linear, exponential, and quadratic functions.

F.IF.B.4 F.IF.B.6 F.LE.A.3

Topic B: Factoring and Solutions of Quadratic Equations

Factor quadratic expressions using the greatest common factor. Demonstrate equivalence between expressions by multiplying polynomials.

A.APR.A.1 A.SSE.A.2 A.SSE.B.3.A

Identify solutions to quadratic equations using the zero product property (equations written in intercept form).

A.APR.B.3 F.IF.C.8.A

Factor quadratic equations and identify solutions (when leading coefficient is equal to 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor quadratic equations and identify solutions (when leading coefficient does not equal 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor special cases of quadratic equations—difference of two squares.

A.SSE.A.1.A A.SSE.A.2 A.SSE.B.3.A

Factor special cases of quadratic equations—perfect square trinomials.

A.SSE.A.2 A.SSE.B.3.A

Solve quadratic equations by factoring. Compare solutions in different representations (graph, equation, and table).

A.SSE.B.3.A F.IF.C.8.A F.IF.C.9

Solve quadratic equations by taking square roots.

A.REI.B.4.B

Graph quadratic functions using $${x-}$$intercepts and vertex.

A.APR.B.3 F.IF.B.4 F.IF.C.7.A F.IF.C.8.A

Topic C: Interpreting Solutions of Quadratic Functions in Context

Interpret quadratic solutions in context.

A.CED.A.1 F.IF.B.4 F.IF.B.5 F.IF.C.8.A

Quadratic Functions and Solutions

Objective

Common Core Standards

Core Standards

Interpreting Functions

Linear, Quadratic, and Exponential Models

Foundational Standards

Linear, Quadratic, and Exponential Models

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

Additional Practice

Lesson Map

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