Lesson 11 | Functions and Transformations | 9th Grade Mathematics

Objective

Identify solutions to a system of absolute value and linear functions graphically and algebraically.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Reasoning with Equations and Inequalities

A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 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.

Reasoning with Equations and Inequalities

A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 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

The foundational standards covered in this lesson

8.EE.C.7

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.
8.EE.C.8

Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Criteria for Success

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

Describe that two sides of the equals or inequality sign can be represented as two functions, equal or compared to one another, as in $${ f(x)=g(x)}$$ or $${ f(x)>g(x)}$$.
Describe how the process of finding the solutions to an absolute value equation or inequality is the same, even when there are variables in the non-absolute value function.
Find the solution(s) to a system of absolute value and linear functions, and identify extraneous solutions.

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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

Functions $${f(x) }$$ and $${g(x)}$$ form a system of equations. Let $${f(x)=|x+2|-3}$$ and $${g(x)}=0.5x+1$$.

When is $$f(x)>{g(x)}$$? Find the solution(s) to the system algebraically and graphically (may use technology).

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 3 > Topic C > Lesson 16 — Opening Exercise 1-2

Algebra I > Module 3 > Topic C > Lesson 16 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

Function $${ f(x)}$$ is shown below.

$${f(x)=|x-1|}$$

Find a function $${g(x)}$$ such that the system has:

two solutions
one solution
no solutions

Guiding Questions

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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

Find the solution to this system algebraically and graphically.

$${f(x)=|x+2|-1}$$
$${g(x)={1\over2}x+3 }$$

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 a mix of problems, similar to the Anchor Problems and Target Task, that are equations and inequalities. Ensure students check for extraneous solutions by either substituting or graphing.

EngageNY Mathematics Algebra I > Module 3 > Topic C > Lesson 16 — Example 2, Exercises 1, 4, 5, and Problem Set #3-6
Illustrative Mathematics Find a System — Change students to write a system that involves an absolute value

Lesson 10

Lesson 12

Lesson Map

Topic A: Piecewise Functions

Write equations and graph linear relationships with different domains.

A.CED.A.1 A.CED.A.2 A.CED.A.3 F.IF.B.5 F.IF.C.7

Graph piecewise functions presented algebraically and write piecewise functions from graphs.

F.IF.A.2 F.IF.B.4 F.IF.C.7.B

Evaluate a piecewise function written algebraically, with and without context.

A.SSE.A.1.A F.IF.A.1 F.IF.A.2

Write and graph piecewise functions from contextual situations.

A.CED.A.2 A.CED.A.3 F.IF.B.5

Graph, evaluate, and write step functions, with and without context.

A.CED.A.3 F.IF.A.1 F.IF.C.7.B

Topic B: Absolute Value Functions

Graph absolute value functions in the coordinate plane and identify domain and range.

F.BF.B.3 F.IF.B.4 F.IF.B.5 F.IF.C.7.B F.IF.C.9

Identify the solutions to an absolute value equation.

A.REI.A.1 A.REI.D.11

Solve absolute value equations algebraically and identify extraneous solutions.

A.REI.B.3

Identify the solution(s) to an absolute value inequality.

A.REI.A.1 A.REI.B.3 A.REI.D.11 F.IF.B.4 F.IF.C.7.B

Solve absolute value inequalities algebraically and verify graphically.

A.REI.B.3

Identify solutions to a system of absolute value and linear functions graphically and algebraically.

A.REI.A.1 A.REI.C.6 A.REI.D.11

Topic C: Function Transformations

Identify and describe vertical translations of functions.

F.BF.B.3 F.IF.B.4 F.IF.B.5

Identify and describe horizontal translations of functions.

F.BF.B.3 F.IF.B.5

Identify and describe vertical scaling of functions, including reflections over the $$x$$-axis.

F.BF.B.3 F.IF.B.5

Identify and describe horizontal scaling of functions, including reflections over the $$y$$-axis.

F.BF.B.3 F.IF.B.5

Apply transformations to functions.

F.BF.B.3

Functions and Transformations

Objective

Common Core Standards

Core Standards

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Expressions and Equations

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem Set

Target Task

Additional Practice

Lesson Map

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