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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 02: Absolute Value Functions, Equations, and Inequalities SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address transformations, characteristics, and applications of absolute value functions. Absolute value equations and inequalities are solved in mathematical problems. Absolute value equations are formulated for real-world problem situations, solved, and solutions justified in terms of the situation. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Algebra I Units 01 – 04 and Unit 11 and Algebra II Unit 01, students transformed and analyzed linear functions and solved linear equations and inequalities.

During this Unit
Students graph the absolute value parent function, f(x) = |x|, and analyze the key attributes such as domain, range, intercepts, symmetries, and maximum and minimum. Students analyze the effect on the graph of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d, and investigate parameter changes and key attributes in terms of mathematical and real-world problem situations. Students solve absolute value linear equations and inequalities in mathematical problems. Students also formulate absolute value linear equations that model real-world situations, solve the equations, and justify the solutions in terms of the problem situation.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Algebra II

After this Unit
In Unit 12, and in subsequent mathematics courses, students will continue to apply these concepts when absolute value functions, equations, and inequalities arise in problem situations.

In Algebra II, analyzing and graphing absolute value functions are identified in STAAR Readiness Standards 2A.2A and STAAR Supporting Standard 2A.6C; and subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses and STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. Solving and applying absolute value equations and inequalities are identified in STAAR Readiness Standards 2A.6E, and STAAR Supporting Standards 2A.6D, 2A.6F; and subsumed under STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, C2, C3, D1, D2; V. Statistical Reasoning A1, C2; VI. Functions A2, B1, B2, C1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to research from the National Council of Teachers of Mathematics (NCTM), “Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone” (2009, p. 41). This unit places particular emphasis on multiple representations. State and national mathematics standards support such an approach. The Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions, such as algebraic, tabular, graphical, and verbal descriptions. This skill is mirrored in the Principles and Standards for School Mathematics (NCTM, 1989). Specifically, this work calls for instructional programs that enable all students to understand relations and functions and select, convert flexibly among, and use various representations for them. More recently, the importance of multiple representations has been highlighted in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (NCTM, 2007). According to this resource, students should be able to translate among verbal, tabular, graphical, and algebraic representations of functions and describe how aspects of a function appear in different representations as early as Grade 8. Also, in research summaries such as Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement (2001), such concept development is even cited among strategies that increase student achievement. Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences has been statistically shown to improve student performance on standardized measures of progress (Marzano, Pickering & Pollock, 2001).

Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2007). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Absolute value functions are characterized by a pair of corresponding positive and negative constant rates of change (over restricted domains) and can be used to describe, model, and make predictions about situations.
• How can the rates of change of an absolute value function be determined?
• What kinds of mathematical and real-world situations can absolute value functions model?
• What graphs, key attributes, and characteristics are unique to absolute value functions?
• What pattern of covariation is associated with absolute value functions?
• How can the key attributes of absolute value functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of an absolute value function?
• What are the real-world meanings of the key attributes of an absolute value function model?
• How can the key attributes of an absolute value function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How are functions …
• shifted?
• scaled?
• reflected?
• How do transformations affect the …
• representations
• key attributes
… of a function?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Rate of change/Slope
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Vertex
• Symmetries
• Functions
• Absolute value
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Equations and inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation or inequality.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can equations and inequalities be used to represent relationships between quantities?
• How do solutions to inequalities differ from solutions to equations?
• Why must solutions be justified in terms of problem situations?
• What types of absolute value equations produce extraneous solutions, and why are the solutions considered extraneous?
• What methods can be used to write absolute value equations and absolute value inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing absolute value equations and absolute value inequalities?
• What methods can be used to solve absolute value equations and absolute value inequalities?
• How does the structure of the equation influence the selection of an efficient method for solving absolute value equations?
• How can the solutions to absolute value equations and absolute value inequalities be determined and represented?
• How are properties and operational understandings used to transform absolute value equations and absolute value inequalities?
• Functions, Equations, and Inequalities
• Functions, Equations, and Inequalities
• Absolute value
• Patterns, Operations, and Properties
• Relations and Generalizations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that |x| can equal a negative number such as |x| = –1 rather than the fact that it is the x inside the absolute that can equal –1.
• Some students may think that |x| = 2 only gives x = 2 rather than x = 2 and x = –2.

Underdeveloped Concepts:

• Some students may not understand why they must switch the second inequality when separating an absolute value inequality; |x| < 2 x < 2 and x > –2.

#### Unit Vocabulary

• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Domain – set of input values for the independent variable over which the function is defined
• Extraneous solution – solution derived by solving the equation algebraically that is not a true solution of the equation and will not be valid when substituted back into the original equation
• Inequality notation – notation in which the solution is represented by an inequality statement
• Interval notation – notation in which the solution is represented by a continuous interval(s)
• Range – set of output values for the dependent variable over which the function is defined
• Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
• Relative maximum – largest y-coordinate, or y-value, a function takes over a given interval of the curve
• Relative minimum – smallest y-coordinate, or y-value, a function takes over a given interval of the curve
• Set notation – notation in which the solution is represented by a set of values
• x-intercept(s)x-coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y-coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

 Absolute value Function notation Representations of solutions Restricted domain Simplify Stretch/compress Transformation Translate Vertex Vertical/horizontal shift
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Algebra II Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
2A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
2A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
2A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
2A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
2A.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
2A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
2A.1G Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
2A.2 Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:
2A.2A

Graph the functions f(x)=, f(x)=1/x, f(x)=x3, f(x)= , f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.

Graph

THE FUNCTIONS f(x) = |x|

Including, but not limited to:

• Representations of functions, including graphs, tables, and algebraic generalizations
• Absolute value, f(x) = |x|
• Connections between representations of families of functions
• Comparison of similarities and differences of families of functions

Analyze

THE KEY ATTRIBUTES OF THE FUNCTIONS SUCH AS DOMAIN, RANGE, INTERCEPTS, SYMMETRIES, AND MAXIMUM AND MINIMUM GIVEN AN INTERVAL, WHEN APPLICABLE

Including, but not limited to:

• Domain and range of the function
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Representation for domain and range
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, y ∈ Ζ
• Set notation – notation in which the solution is represented by a set of values
• Braces are used to enclose the set.
• Solution is read as “The set of x such that x is an element of …”
• Ex: {x|x ∈ ℜx < 5}
• Ex: {x|x ∈ ℜ}
• Ex: {y|y ∈ ℜ, –3 < y ≤ 6}
• Ex: {y|y ∈ Ζy ≥ 0}
• Interval notation – notation in which the solution is represented by a continuous interval
• Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
• Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
• Ex: (–, 5)
• Ex: (–, )
• Ex: (–3, 6]
• Ex: [0, ∞)
• Domain and range of the function versus domain and range of the contextual situation
• Key attributes of functions
• Intercepts/Zeros
• x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• Zeros – the value(s) of x such that the y value of the relation equals zero
• y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Symmetries
• Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
• Maximum and minimum (extrema)
• Relative maximum – largest y-coordinate, or value, a function takes over a given interval of the curve
• Relative minimum – smallest y-coordinate, or value, a function takes over a given interval of the curve
• Use key attributes to recognize and sketch graphs
• Application of key attributes to real-world problem situations

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = bx and their key attributes.
• Precalculus will study polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
2A.6 Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:
2A.6C Analyze the effect on the graphs of f(x)=|x| when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d.
Supporting Standard

Analyze

THE EFFECT ON THE GRAPHS OF f(x) = |x| WHEN f(x) IS REPLACED BY af(x), f(bx), f(xc), AND f(x) + d FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, b, c, AND d

Including, but not limited to:

• General form of the absolute value function
• Absolute value
• f(x) = |x|
• Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Effects on the graph of f(x) = |x| when parameters a, b, c, and d are changed in f(x) = a|b(xc)| + d
• Effects on the graph of f(x) = |x| when f(x) is replaced by af(x) with and without technology
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the x-axis
• Effects on the graph of f(x) = |x|, when f(x) is replaced by f(bx) with and without technology
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the y-axis
• Effects on the graph of f(x) = |x|, when f(x) is replaced by f(xc) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left for values of c < 0 by |c| units
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
• Horizontal shift right for values of c > 0 by |c| units
• For f(x – 2), c = 2, and the function moves to the right two units
• Effects on the graph of f(x) = |x|, when f(x) is replaced by f(x) + d with and without technology
• d = 0, no vertical shift
• Vertical shift down for values of d < 0 by |d| units
• Vertical shift up for values of d > 0 by |d| units
• Connections between the critical attributes of transformed function and f(x) = |x|
• Determination of parameter changes given a graphical or algebraic representation
• Determination of a graphical representation given the algebraic representation or parameter changes
• Determination of an algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects on the domain and range by the parameter changes
• Effects of multiple parameter changes
• Mathematical problem situation
• Real-world problem situation

Note(s):

• Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x – c), f(bx) for specific values of abc, and d.
• Algebra II introduces the absolute value function and its transformations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B1 – Identify and apply transformations to figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.6D Formulate absolute value linear equations.
Supporting Standard

Formulate

ABSOLUTE VALUE LINEAR EQUATIONS

Including, but not limited to:

• Data collection activities with and without technology
• Data modeled by absolute value functions
• Real-world problem situations
• Real-world problem situations modeled by absolute value functions
• Data tables
• Technology methods
• Transformations of f(x) = |x|

Note(s):

• Grade 6 defined absolute value and identified the absolute value of a number.
• Algebra II introduces the absolute value equation and its applications.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VI.C.2. Develop a function to model a situation.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
2A.6E Solve absolute value linear equations.

Solve

ABSOLUTE VALUE LINEAR EQUATIONS

Including, but not limited to:

• Methods for solving absolute value linear equations with and without technology
• Graphs
• Algebraic methods
• Solving process
• Transform the equation so that the absolute value expression is on one side of the equation and all other variable terms and constants are on the other side of the equation.
• Separate the equation into two parts divided by “or”:
• Expression inside the absolute value equal to the other side of the equation
• Expression inside the absolute value equal to the opposite of the other side of the equation
• |x| = 5 → x = 5 or x = –5
• Extraneous solution – solution derived by solving the equation algebraically that is not a true solution of the equation and will not be valid when substituted back into the original equation
• Solving absolute value equations involves separating the absolute value into both the possible positive value inside the absolute and the possible negative value inside the absolute. In the case of |x| = 2, The x value can be either positive or negative 2. However, this is not a reversible situation, |x| = 2 but |x| ≠ –2.
• Justification of solutions with and without technology
• Graphs
• Substitution of solutions into original functions
• Extraneous solutions
• Real-world problem situations modeled by absolute value functions
• Justification of reasonableness of solutions in terms of the real-world problem situations or data collections

Note(s):

• Grade 6 defined absolute value and identified the absolute value of a number.
• Algebra II introduces the absolute value equation and its applications.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
2A.6F Solve absolute value linear inequalities.
Supporting Standard

Solve

ABSOLUTE VALUE LINEAR INEQUALITIES

Including, but not limited to:

• Methods for solving absolute value linear inequalities with and without technology
• Graphs
• Algebraic methods
• Solving process
• Isolation of absolute expression on one side of the inequality
• Separation of the inequality into two parts
• Greater than (>) or greater than or equal to (≥)
• First part: expression inside the absolute value set greater than or greater than or equal to other side of the inequality
• Second part: expression inside the absolute value set less than or less than or equal to the opposite of the other side of the inequality
• Parts separated by “ or ”
• Representation of solutions
• Symbolic notation
• Interval notation
• Graphical notation
• Less than (<) or less than or equal to (≤)
• First part: expression inside the absolute value set less than or less than or equal to other side of the inequality
• Second part: expression inside the absolute value set greater than or greater than or equal to the opposite of the other side of the inequality
• Parts separated by “ and ”
• Representation of solutions
• Symbolic notation
• Interval notation
• Graphical notation
• Justification of solutions of absolute value inequalities with and without technology
• Graphs
• Substitution of solutions into original functions
• Removal of extraneous solutions

Note(s):

• Grade 6 defined absolute value and identified the absolute value of a number.
• Algebra II introduces absolute value inequalities.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.