Lesson 5 | Functions and Transformations | 9th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Creating Equations

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

F.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Interpreting Functions

F.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F.IF.C.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Interpreting Functions

F.IF.C.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Criteria for Success

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

Identify the equations in a step function as constant functions.
Understand the appropriate contexts that can be modeled with step functions.
Interpret the graphs and equations of step functions.
Draw a graph of a step function to best represent a contextual situation.

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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

A step function is shown below.

When does the function have a value of 4? Select all that apply.
1. $${f(4)}$$
2. $${f(5)}$$
3. $${f(7.99)}$$
4. $${f(8)}$$
5. $${f(8.01)}$$
Describe a situation that could be represented by this step function.

Guiding Questions

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Problem 2

At the beginning of the week, Jessie had $500 in her bank account. She deposited a check for $50 on Monday and then paid $250 in rent on Wednesday. On Friday, Jessie deposited $200 in the account and then on Saturday she paid $100 for groceries from her bank account. Jessie made the following graph for the balance in her bank account during this week:

Is the depiction of how the account balance varies over the week accurate? Explain.
How can Jessie graphically represent the bank account balance in a way that better shows how it changes?
Draw a new graph to represent Jessie’s balance, and represent it algebraically as a step function.

Guiding Questions

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References

Illustrative Mathematics Bank Account Balance

Bank Account Balance, accessed on May 10, 2018, 11:05 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

The graph below shows the postage cost for mailing a letter in 2013.

Use the graph to complete the table below.

Weight in ounces, $$x$$	2	2.2	3	3.5	4
Cost of postage, $$C(x)$$

Write a formula involving step functions that represents the cost of postage based on the graph shown above.
If it costs Trina $0.54 to mail her letter, how many ounces could it have weighed?

References

EngageNY Mathematics Algebra I > Module 3 > Topic D > Lesson 24 — Exit Ticket

Algebra I > Module 3 > Topic D > Lesson 24 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.

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 involving context where students write a step function, graph it, and then answer questions. Some contexts to include are:
- postage weights and costs
- bulk purchases (with $${f(x) }$$ representing unit price rather than total)
- greatest integer or least integer
- shipping costs for online orders for various totals

EngageNY Mathematics Algebra I > Module 3 > Topic C > Lesson 15 — Problem Set, Question #4
EngageNY Mathematics Algebra I > Module 3 > Topic D > Lesson 24 — Opening Exercise and Problem Set Question #1 and #4

Lesson 4

Lesson 6

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

Creating Equations

Interpreting Functions

Interpreting Functions

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Problem Set

Target Task

References

Additional Practice

Lesson Map

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