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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 08: Rational Functions and Equations SUGGESTED DURATION : 15 days

Unit Overview

Introduction
This unit bundles student expectations that address transformations, characteristics, and applications of rational functions. The unit addresses simplifying and performing operations with rational expressions and rational equations, including equations with rational exponents and inverse variation. 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 and 04, students formulated, solved, and justified solutions to linear equations, including direct variation. In Algebra II Unit 01, students explored parent functions and key attributes of functions, including the rational parent function. In Units 04 – 06, students used the rules of exponents to solve equations involving rational exponents.

During this Unit
Students graph the function f(x) = , and analyze the key attributes such as domain/range, intercepts, symmetries, and asymptotic behavior, determining the asymptotic restrictions on the domain of a rational function and representing domain and range using interval notation, inequalities, and set notation. Students analyze the effect on the graph of 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 determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two. Students solve rational equations that have real solutions and determine the reasonableness of the solutions. In real-world situations, students formulate rational equations, including inverse variation equations, solve equations, and justify solution(s) in terms of the problem situation.

After this Unit
In Unit 12, students will review rational and inverse variation functions and equations and their applications in the real-world. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving rational and inverse variation functions and equations.

In Algebra II, graphing and analyzing key attributes of rational functions are identified in STAAR Readiness Standards 2A.2A and subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Simplifying and performing operations with rational expressions are identified in STAAR Readiness Standards 2A.7F and subsumed under STAAR Reporting Category 1: Numbers and Algebraic Methods. Solving and applying rational and inverse variation functions and equations are identified in STAAR Readiness Standards 2A.6I and 2A.6L and subsumed under STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. Identifying, describing, and analyzing transformations of rational functions and asymptotic behaviors of rational functions are identified in STAAR Supporting Standards 2A.6G and 2A.6K and subsumed under STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. Formulating rational equations to model problem situations and justifying the reasonableness of solutions to rational equations are identified in STAAR Supporting Standards 2A.6H and 2A.6J 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, B1, C1, D1, D2; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to the National Council of Teachers of Mathematics (NCTM, 2000), Principles and Standards for School Mathematics, 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 city in Focus In High School Mathematics: Reasoning and Sense Making 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” (NCTM, 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, 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 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.
• Rational functions can be written as ratios of two polynomial functions, have rates of change that are influenced by the polynomial functions within these ratios, and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can rational functions model?
• What graphs, key attributes, and characteristics are unique to rational functions?
• What patterns of covariation are associated with rational functions?
• How can the key attributes of rational functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of rational functions?
• What are the real-world meanings of the key attributes of rational function models?
• How can the key attributes of rational functions be used to make predictions and critical judgments?
• What relationships exist between the algebraic forms of a rational function and the graph and key attributes of the 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 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, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Asymptotes
• Symmetries
• Functions
• Rational
• Relations and Generalizations
• 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.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy? Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically? 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)
• The ability to represent quantities in various forms develops the understanding of equivalence and allows for working flexibly with algebraic expressions in order to communicate and reason about quantities.
• How can expressions be used to represent situations?
• What mathematical conventions are used when representing expressions? Why?
• How can it be determined if two expressions are equivalent?
• How are properties and operational understandings used to generate equivalent expressions?
• Why can it be useful to simplify expressions?
• Why can it be useful to factor expressions?
• How does the structure of the expression influence the selection of an efficient method for simplifying rational expressions?
• Equations 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.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can equations be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What types of rational equations produce extraneous solutions, and why are the solutions considered extraneous?
• What methods can be used to write …
• rational equations?
• equations involving inverse variation?
• How does the given information and/or representation influence the selection of an efficient method for writing rational equations?
• What methods can be used to solve …
• rational equations?
• equations involving inverse variation?
• How does the structure of the equation influence the selection of an efficient method for solving rational equations?
• How can the solutions to rational equations be determined and represented?
• How are properties and operational understandings used to transform rational equations?
• Functions, Equations, and Inequalities
• Functions and Equations
• Rational
• Inverse variation
• Patterns, Operations, and Properties
• Relations and Generalizations
• Number and Algebraic Methods
• Expressions
• Rational
• Patterns, Operations, and Properties
• 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 an asymptote is a line that can never be crossed rather than a line that is approached. Although a vertical asymptote cannot be crossed, a horizontal asymptote can be crossed and approached in another section of the graph.
• Some students may think that when zeros of an expression occur in the denominator of the function, it always produces a vertical asymptote rather than understanding that if an x-value makes both the numerator and the denominator equal to zero, it indicates a removable discontinuity, not a vertical asymptote.
• Some students may have misconceptions about the nature of the graph of a function when using a graphing calculator, rather than understanding that sometimes the calculator obscures the details or hidden behavior of a function.

Unit Vocabulary

• Asymptote – a line that is approached and may or may not be crossed
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Continuous function –function whose values are continuous or unbroken over the specified domain
• Direct variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
• Discontinuity – characteristic of a function where it is not continuous at some point along its graph; place where there is a break in the continuous curve of the function
• 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
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• 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 variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = • Point (removable) discontinuity – hole or discontinuity in the graph of a rational function generated when a factor in the denominator, that should create a vertical asymptote, reduces out with an equivalent factor in the numerator
• Range –set of output values for the dependent variable over which the function is defined
• 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
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• 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:

 Constant of variation Extraneous solutions Horizontal compression Horizontal shift Horizontal stretch Least common denominator Parameter change Parent functions Rational exponents Rational equation Rational function Reciprocal of x Variation 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

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# Unit Level Taught Directly TEKS Unit Level Specificity

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.

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.
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) Including, but not limited to:

• Representations of functions, including graphs, tables, and algebraic generalizations
• Rational (reciprocal of x), 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 ASYMPTOTIC BEHAVIOR, 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
• 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.
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Use key attributes to recognize and sketch graphs
• Application of key attributes to real-world problem situations

Note(s):

• The notation represents the set of real numbers, and the notation represents the set of integers.
• Algebra I studied parent functions f(x) = xf(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.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.6G Analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(- 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) = 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 rational function
• Rational function
• f(x) = • Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Effects on the graph of f(x) = , when parameters a, b, c, and d are changed in or • Effects on the graph of 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 graph of 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 graph of 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 graph of 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) = • 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
• Descriptions of the effects on the asymptotes by the parameter changes
• Effects of multiple parameter changes
• Mathematical problem situations
• 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 rational 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.6H Formulate rational equations that model real-world situations.
Supporting Standard

Formulate

RATIONAL EQUATIONS THAT MODEL REAL-WORLD SITUATIONS

Including, but not limited to:

• Rational equations composed of linear or quadratic functions
• Data collection activities with and without technology
• Data modeled by rational functions
• Real-world problem situations
• Real-world problem situations modeled by rational functions
• Data tables
• Technology methods
• Transformations of f(x) = Note(s):

• Algebra II introduces the rational equation and its applications.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.6I Solve rational equations that have real solutions.

Solve

RATIONAL EQUATIONS THAT HAVE REAL SOLUTIONS

Including, but not limited to:

• Rational equations composed of linear or quadratic functions
• Limited to real solutions
• Methods for solving rational equations with and without technology
• Graphs
• Algebraic methods
• Solving processes
• Identification of domain restrictions; denominator ≠ 0
• Methods to solve
• Application of cross products for proportional problems
• Multiplication by least common denominator
• Determination of least common denominator
• Multiplication of least common denominator to eliminate fractions
• Transformation of equation to solve for unknown
• Justifications of solutions with and without technology
• Graphs
• Substitution of solutions into original functions
• Removal of extraneous solutions
• Real-world problem situations modeled by rational functions
• Justification of reasonableness of solutions in terms of real-world problem situations or data collections

Note(s):

• Algebra II introduces the rational equation and its applications.
• 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.”
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• 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.6J Determine the reasonableness of a solution to a rational equation.
Supporting Standard

Determine

THE REASONABLENESS OF A SOLUTION TO A RATIONAL EQUATION

Including, but not limited to:

• Rational equations composed of linear or quadratic functions
• Justification of solutions to rational equations with and without technology
• Verbal description
• Tables
• Graphs
• Substitution of solutions into original functions
• Justification of reasonableness of solutions in terms of real-world problem situations or data collections

Note(s):

• Algebra II introduces rational equations and solving rational 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.”
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• 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.6K Determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation.
Supporting Standard

Determine

THE ASYMPTOTIC RESTRICTIONS ON THE DOMAIN OF A RATIONAL FUNCTION

Including, but not limited to:

• Discontinuity in rational functions
• Discontinuity – characteristic of a function where it is not continuous at some point along its graph; place where there is a break in the continuous curve of the function
• Asymptote – a line that is approached and may or may not be crossed
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• Discontinuity where the denominator cannot equal zero
• Determination of vertical asymptotes by setting the denominator ≠ 0
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve. A horizontal asymptote describes the long run behavior of the rational function.
• If the degree of the numerator is less than the degree of denominator, the horizontal asymptote is f(x) = 0.
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is f(x) = where p is the coefficient of the highest degreed term of the numerator and q is the coefficient of the highest degreed term of the denominator.
• Oblique (slant) asymptote – non-vertical and non-horizontal line approached by the curve as the function approaches positive or negative infinity. Oblique (slant) asymptotes may be crossed by the curve.
• If the degree of the numerator is one more than the degree of the denominator, then the oblique asymptote is of the form y = mx + b determined by the quotient of the numerator and denominator through long division.
• Point (removable) discontinuity – hole or discontinuity in the graph of a rational function generated when a factor in the denominator, that should create a vertical asymptote, reduces out with an equivalent factor in the numerator
• Determination of canceled factor
• Determination of x-value in canceled factor that would create a zero in the denominator
• Calculation of the corresponding y-value of the point discontinuity using the reduced rational function
• Graphical analysis using discontinuity
• Domain and range
• Limitations from discontinuities
• Vertical asymptote(s) restrictions on domain
• Horizontal asymptote restrictions on range
• Point(s) of discontinuity restrictions on domain and range
• End behavior
• Single and compound inequality statements to identify domain and range
• Analyzing graph of function in regions formed on graph
• Point tested in regions
• Symmetry
• Intercepts
• Appropriate curve sketched in each region

Represent

DOMAIN AND RANGE USING INTERVAL NOTATION, INEQUALITIES, AND SET NOTATION

Including, but not limited to:

• Inequality notation
• Ex: x < 5 or x > 8?
• Ex: –3 < y < 6
• Ex: x < –3 or 0 < x< 2 or x > 4
• Set notation
• Ex: {x|x ∈, ℜ, x < 5 or x > 8}
• Ex: {y|y ∈, ℜ, –3 < y < 6}
• Ex: {x|x ∈, ℜ, x < –3 or 0 < x < 2 or x > 4}
• Interval notation
• Ex: (–∞,5) ∪ (8,∞)
• Ex: (–3, 6)
• Ex: (–∞,–3) ∪ (0,2) ∪ (4,∞)

Note(s):

• Algebra II introduces the rational function and its attributes.
• Precalculus will continue to investigate rational functions.
• 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
• 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.6L Formulate and solve equations involving inverse variation.

Formulate

EQUATIONS INVOLVING INVERSE VARIATION

Including, but not limited to:

• Characteristics of variation
• Constant of variation
• Particular equation to represent variation
• Types of variation
• Direct variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
• y varies directly as x
• General equation: y = kx
• Connection of direct variation to linear functions
• Inverse variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = • y varies inversely as x
• General equation: y = • Connection of inverse variation to rational functions
• Real-world problem situations involving variation
• Reasonableness of solutions mathematically and in context of real-world problem situations

Solve

EQUATIONS INVOLVING INVERSE VARIATION

Including, but not limited to:

• Methods for solving variation equations with and without technology
• Graphs
• Algebraic methods
• Solving processes
• Determination of a particular equation to represent the problem
• Direct variation, y = kx
• Inverse variation, y = • Transformation of equation to solve for unknown
• Justification of solutions with and without technology
• Substitution of solutions into original functions
• Real-world problem situations modeled by rational functions
• Justification of reasonableness of solutions in terms of real-world problem situations or data collections

Note(s):

• Prior grade levels studied direct variation and proportionality.
• Algebra II introduces inverse variation and its applications in problem situations.
• Precalculus will continue to investigate rational functions.
• 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.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• 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.7F Determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two.

Determine

THE SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF RATIONAL EXPRESSIONS WITH INTEGRAL EXPONENTS OF DEGREE ONE AND OF DEGREE TWO

Including, but not limited to:

• Equivalent rational expressions
• Multiplication by a fractional form of 1
• Simplification of rational expressions
• Factorization of numerator and denominator
• Cancellation or dividing out of common factors
• Operations with rational expressions
• Sum and difference
• Common denominator (CD) for both terms by calculating equivalent expressions as needed
• Combination of numerators by addition/subtraction over single common denominator (CD)
• Simplification of answer
• Product and quotient
• Inversion of divisors and conversion to multiplication
• Factorization of numerators and denominators
• Cancellation or dividing out of common factors in numerator and denominator
• Multiplication of remaining numerators and denominators

Note(s):

• Previous grade levels simplified and performed operations on fractions.
• Algebra II simplifies and performs operations on rational expressions involving variables.
• 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
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• D1 – Interpret multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections 