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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 07: Cubic and Cube Root Functions and Equations SUGGESTED DURATION : 10 days

#### Unit Overview

This unit bundles student expectations that address transformations, characteristics, and applications of cubic and cube root functions, including inverse relationships between cube root and cubic functions. This unit also includes solving equations involving rational exponents and formulating, solving, and justifying the solutions to cube root equations. 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 II Unit 01, students investigated parent functions and their attributes. Students also analyzed inverse functions using various representations. In Algebra II Unit 04, students used the rules of exponents to solve equations involving rational exponents.

During this unit, students describe and analyze the inverse relationship between the cubic and cube root functions and graph and write the inverse functions using notation such as f -1 (x). Students graph the functions f(x) = x³ and  f(x) =  and analyze key attributes such as domain, range, intercepts, symmetries, and maximum and minimum given an interval. Students analyze the effect on the graphs of f(x) = x³ and f(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. Students investigate parameter changes and key attributes in terms of real-world problem situations. Students solve equations involving rational exponents that have real solutions, focusing on cubic and cube root equations. Students formulate and solve equations involving cubic and cube root equations for real-world situations and justify the solutions in terms of the problem situations.

After this unit, in Algebra II Unit 12, students will review cubic and cube root functions and equations and their real-world applications. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving cubic and cube root functions and equations.

In Algebra II, graphing, analyzing key attributes, and describing the inverse relationship of cubic and cube root functions are identified in STAAR Readiness Standards 2A.2A and 2A.2C and subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Solving equations involving rational exponents is identified in STAAR Readiness Standards 2A.7H and subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. Graphing and writing cubic and cube root functions as inverses of each other is identified in STAAR Supporting Standard 2A.2B and subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Analyzing transformations of cubic and cube root functions is identified in STAAR Supporting Standard 2A.6A and subsumed under STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. Solving cube root equations is identified in STAAR Supporting Standard 2A.6B and subsumed under STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. This unit supports the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, C1, D1, D2; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics (2000), students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12, “High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions” (NCTM, 2002, p. 3). Research from the National Council of Teachers of Mathematics (NCTM) also states, “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, 2000). 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 cited among strategies that increase student achievement. Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences have been statistically shown to improve student performance on standardized measures of progress (Marzano, Pickering & Pollock).

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Equations can model problem situations and be solved using various methods.

• Why are equations used to model problem situations?
• How are equations used to model problem situations?
• What methods can be used to solve equations?
• Why is it essential to solve equations using various methods?
• How can solutions to equations be represented?

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Transformation(s) of a parent function create a new function within that family of functions.

• Why are transformations of parent functions necessary?
• How do transformations affect a function?
• How can transformations be interpreted from various representations?
• Why does a transformation of a function create a new function?
• How do the attributes of an original function compare to the attributes of a transformed function?

Inverses of functions create new functions.

• What relationships and characteristics exist between a function and its inverse?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data, and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Algebraic Reasoning

• Multiple Representations

Functions

• Attributes of Functions
• Inverses of Functions
• Non-Linear Functions

Geometric Reasoning

• Transformations

Associated Mathematical Processes

• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Cubic and cube root functions have unique graphs and attributes.

• What representations can be used to represent cubic and cube root functions?
• What are the key attributes of cubic and cube root functions and how can they be determined from various representations?

The inverse of a function can be determined from multiple representations.

• How can the inverse of a function be determined from the graph of the function?
• How can the inverse of a function be determined from a table of coordinate points of the function?
• How can the inverse of a function be determined from the equation of the function?
• How are a function and its inverse distinguished symbolically?
• How do the attributes of inverse functions compare to the attributes of original functions?

Transformations of the cubic function, f(x) = x³, and cube root function,  f(x) = , can be used to determine graphs and equations of representative cubic and cube root functions in problem situations.

• What are the effects of changes on the graphs of  f(x) = x³ and  f(x) = , when f(x) is replaced by af(x), for specific positive and negative values of a?
• What are the effects of changes on the graphs of  f(x) = x³ and  f(x) =  , when f(x) is replaced by f(bx), for specific positive and negative values of b?
• What are the effects of changes on the graphs of  f(x) = x³ and  f(x) =  , when f(x) is replaced by   f(x c) for specific positive and negative values of c?
• What are the effects of changes on the graphs of  f(x) = x³ and f(x) =  , when f(x) is replaced by  f(x) + d, for specific positive and negative values of d?
 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.

Numeric Reasoning

• Exponents

Algebraic Reasoning

• Equations
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Equations can be used to model and solve mathematical and real-world problem situations.

• How are real-world problem situations identified as ones that can be modeled by cube root equations?
• How are cube root equations used to model problem situations?
• What methods can be used to solve cube root equations?
• What are the advantages and disadvantages of various methods used to solve cube root equations?
• What methods can be used to justify the reasonableness of solutions to cube root equations?

Cubic and cube root functions can be used to model real-world problem situations by analyzing collected data, key attributes, and various representations in order to interpret and make predictions and critical judgments.

• What representations can be used to display cubic and cube root function models?
• What key attributes identify the cubic and cube root parent function models?
• What are the connections between the key attributes of cubic and cube root function models and the real-world problem situation?
• How can cubic and cube root function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the cube root of a negative number is imaginary rather than understanding that the cube root of a negative number is just the negative number.

#### 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
• 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
• Inverse of a function – function that undoes the original function. When composed f(f--1(x)) = x and f--1(f(x)) = x.
• 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 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
• Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
• 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:

 Cube root equation Cube root function Cubic function Extraneous solution Horizontal shift Horizontal stretch Horizontal compression Parameter change Properties of exponents Rational exponents Root Solution Vertical compression Vertical shift Vertical stretch
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 Creator 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 (select CCRS from Standard Set dropdown menu)

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

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Bold red text in italics:  Student Expectation identified by TEA as a Readiness Standard for STAAR
• Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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:
• X. Connections
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:
• IX. Communication and Representation
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) = x3, f(x) =

Including, but not limited to:

• Representations of functions, including graphs, tables, and algebraic generalizations
• Cubic, f(x) = x3
• Cube root, f(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
• Ex: x
• Ex: –3 < y ≤ 6, x
• Ex: y ≥ 0, y
• 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  x < 5}
• Ex: {x|x  }
• Ex: {y|y  , –3 < y ≤ 6}
• Ex: {y|y  y ≥ 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]
• 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
• Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
• 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:
• III. Geometric Reasoning
• B1 – Identify and apply transformations to figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• 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.2B Graph and write the inverse of a function using notation such as f -1(x).
Supporting Standard

Graph, Write

THE INVERSE OF A FUNCTION USING NOTATION SUCH AS f –1 (x)

Including, but not limited to:

• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and  f –1(f(x)) = x.
• Inverse functions
• Cubic and cube root
• Inverses of functions on graphs
• Symmetric to yx
• Inverses of functions in tables
• Interchange independent (x) and dependent (y) coordinates in ordered pairs
• Inverses of functions in equation notation
• Interchange independent (x) and dependent (y) variables in the equation, then solve for y
• Inverses of functions in function notation
• f –1(x) represents the inverse of the function f(x).

Note(s):

• Algebra II introduces inverse of a function.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.2C

Describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range.

Describe, Analyze

THE RELATIONSHIP BETWEEN A FUNCTION AND ITS INVERSE

Including, but not limited to:

• Relationships between functions and their inverses
• All inverses of functions are relations.
• Inverses of one-to-one functions are functions.
• Inverses of functions that are not one-to-one can be made functions by restricting the domain of the original function, f(x).
• Characteristics of inverse relations
• Interchange of independent (x) and dependent (y) coordinates in ordered pairs
• Reflection over y = x
• Domain and range of the function versus domain and range of the inverse of the given function
• Cubic function and cube root function, f(x) = x3 and g(x) =

Note(s):

• Algebra I determined if relations represented a function.
• Algebra II introduces inverse of a function and restricting domain to maintain functionality.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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.6A Analyze the effect on the graphs of f(x) = x3 and f(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) = x3 AND f(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 cubic and cube root functions
• Cubic
• f(x) = x3
• Cube root
• f(x) =
• Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Effects on the graphs of f(x) = x3 and f(x) =  when parameters a, b, c, and d are changed in f(x) = a(b(xc))3 + d and f(x) =
• Effects on the graphs of f(x) = x3 and f(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 graphs of f(x) = x3 and f(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 graphs of f(x) = x3 and f(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 graphs of f(x) = x3, and f(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) = x3 and f(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(xc), f(bx) for specific values of a, b, c, and d.
• Algebra II introduces the cubic and cube root functions and their 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.6B Solve cube root equations that have real roots.
Supporting Standard

Solve

CUBE ROOT EQUATIONS THAT HAVE REAL ROOTS

Including, but not limited to:

• Application of laws (properties) of exponents
• Application of cube roots to solve cubic equations
• Applications of cubics to solve cube root equations
• Reasonableness of solutions
• Substitution of solutions into original problem
• Graphical analysis
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra II introduces cubic and cube root functions and solving cube root equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.7 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:
2A.7H Solve equations involving rational exponents.

Solve

EQUATIONS INVOLVING RATIONAL EXPONENTS

Including, but not limited to:

• Laws (properties) of exponents
• Product of powers (multiplication when bases are the same): am • an = am+n
• Quotient of powers (division when bases are the same): = am-n
• Power to a power: (am)n = amn
• Negative exponent: a-n =
• Rational exponent:
• Equations when bases are the same: am = anm = n
• Solving equations with rational exponents
• Isolation of base and power using properties of algebra
• Exponentiation of both sides by reciprocal of power of base
• If the denominator of the reciprocal power is even, then the variable must be represented using absolute value.
• Simplification to obtain solution
• Verification of solution
• Real-world problem situations modeled by equations involving rational exponents
• Justification of reasonableness of solutions in terms of real-world problem situations

Note(s):

• Prior grade levels simplified numeric expressions, including integral and rational exponents.
• Algebra II introduces equations involving rational exponents.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections