# Functions and Transformations

Students take a deeper look at piecewise functions and absolute value functions, and study how transformations of functions can be identified graphically and represented algebraically.

## Unit Summary

In Unit 5, Functions and Transformations, students take a deeper look at piecewise functions and absolute value functions, and then study how transformations of functions can be identified graphically and represented algebraically. Throughout the unit, students will recall and utilize skills from eighth grade as they manipulate equations and inequalities, write equations from graphs, and solve systems of equations.

In Topic A, students recall from Unit 1 how a graph can depict different behaviors over different domains. They define these functions as piecewise functions and look at various real-world applications that can be modeled using piecewise and step functions.

In Topic B, students identify the graph of an absolute value function as a piecewise function. They solve absolute value equations and inequalities algebraically, and students also leverage their understanding of systems of equations to conceptualize their solutions graphically in the coordinate plane.

In Topic C, students use the absolute value function as a vehicle to understand, identify, and represent transformations to function graphs. Along the way, they also apply transformations to other parent functions and learn how the graph of any function can be manipulated in certain ways using algebraic rules.

## Assessment

This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Essential Understandings

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• Piecewise functions model different behaviors over various domains. They can be defined using multiple function rules or equations with specified domain intervals.
• Piecewise functions are often used to model real-world situations such as varying price rates, income tax brackets, or overtime pay. Step functions are an example of piecewise functions and are often used to model situations such as postage cost by weight.
• Absolute value equations, ${|x|=a}$, can be thought of as two functions, ${ f(x)=|x|}$ and ${g(x)=a}$, that are equal to one another, ${f(x)=g(x)}$. The solution(s) are the values for $x$ where the two functions intersect in the coordinate plane.
• The graphs of functions can be transformed in the coordinate plane and represented algebraically. For a function $f(x)$, $f(x)+k$ and $f(x+h)$ represent vertical and horizontal translations, respectively; $af(x)$ and $f(bx)$ represent vertical and horizontal scaling, respectively; and specifically, $-f(x)$ and $f(-x)$ represent reflections over the $x$- and $y$-axes, respectively.

### Vocabulary

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 Domain/range Absolute value inequality Piecewise function Parent function Step function Transformations of functions Absolute value functions Translation (vertical and horizontal) Extraneous solution Scaling (vertical and horizontal)

### Unit Materials, Representations and Tools

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• Graphing calculators or other graphing technology
• Computers for Desmos activities

### Intellectual Prep

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Internalization of Standards via the Unit Assessment

• Take unit assessment. Annotate for:
• Standards that each question aligns to
• Purpose of each question: spiral, foundational, mastery, developing
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that assessment points to

Internalization of Trajectory of Unit

• Read and annotate “Unit Summary.”
• Notice the progression of concepts through the unit using “Unit at a Glance.”
• Essential understandings
• Connection to assessment questions

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### Building Functions
• F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

##### Creating Equations
• A.CED.A.1 — Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

• A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

• 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.

##### 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.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

• 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.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 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.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 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.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

• F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

##### 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.B.3 — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

• 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.

##### Seeing Structure in Expressions
• A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context 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.

• A.SSE.A.1.A — Interpret parts of an expression, such as terms, factors, and coefficients.

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• 8.EE.B.5

• 8.EE.B.6

• 8.EE.C.7

• 8.EE.C.8

• 8.F.B.4

• 8.G.A.3

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• F.BF.B.3

• F.IF.B.6

• A.REI.D.11

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.