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

#### Unit Overview

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 equation, 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 Algebra II 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.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Algebraic expressions (numbers, variables, and operational symbols) are the basic tools of algebra.

• Why are algebraic expressions the basic tools of algebra?
• How are algebraic expressions used to express mathematical ideas and model mathematical and real-world situations?
• What operations do algebraic expressions undergo?
• How can two expressions be related?
• Why are algebraic expressions evaluated?

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?

Proportional reasoning can be used to describe and solve problems in everyday life.

• Why can proportional reasoning be used to make predictions and comparisons in problem situations?
• How is proportional change distinguished from non-proportional change?
• How are ratios used in a proportional relationship?

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?

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
• Independent/Dependent
• Non-Linear Functions

Geometric Reasoning

• Transformations

Associated Mathematical Processes

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

Rational functions have unique graphs and attributes.

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

Asymptotes and point discontinuity(ies) are key attributes that are critical in defining a rational function.

• How do rational functions exhibit asymptotic behavior?
• What types of asymptotic behavior are modeled in rational functions?
• Why can point discontinuity be considered an asymptotic behavior?
• How do asymptotic behavior and point discontinuity affect the domain and range of a rational function?
• How can asymptotic behavior and point discontinuity to graph and/or write a representative rational function?
• What is the meaning of asymptotic behavior as reflected in a real-world problem situation?

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

• What are the effects of changes on the graph of      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 graph of      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 graph of      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 graph of      f(x) =  , when f(x) is replaced by f(x) + d, for specific positive and negative values of d?

Rational 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 the rational function model?
• What key attributes identify the rational parent function model?
• What are the connections between the key attributes of rational function models and the real-world problem situation?
• How can a rational function representation be used to interpret and make predictions and critical judgments in terms of the problem situation?
 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

• Division
• Exponents
• Multiplication
• Subtraction

Algebraic Reasoning

• Equations
• Equivalence
• Evaluate
• Expressions
• Multiple Representations
• Simplify
• Solution Representations
• Solve

Functions

• Attributes of Functions
• Inverse Variation
• Non-Linear Functions

Associated Mathematical Processes

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

Rational expressions can be added, subtracted, multiplied, or divided.

• How are terms identified and combined when adding and subtracting rational expressions?
• How is multiplication of rational expressions used to determine an equivalent expression?
• How is division of rational expressions used to determine an equivalent expression?
• Why must the denominator of a rational expression be evaluated?

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 rational equations?
• How are rational equations used to model problem situations?
• What methods can be used to solve rational equations?
• What are the advantages and disadvantages of various methods used to solve rational equations?
• What methods can be used to justify the reasonableness of solutions to rational equations?
• What causes extraneous solutions in rational equations?
• How can extraneous solutions be identified in graphs, tables, and algebraic calculations?

Proportional relationships, including inverse variation, can be used to model and solve problem situations.

• How do the characteristics of direct variation and inverse variation compare?
• Why are inverse variation equations considered a subset of rational equations?
• How are inverse variation equations formulated to model problem situations?
• What methods can be used to solve rational equations?

#### 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 (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)

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

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 < 5, x
• Ex: x
• Ex: –3 < y ≤ 6, x
• Ex: y ≥ 0, y
• 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
• 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 ab, 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(x – c) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left for values of c < 0 by |c| units
• For f(+ 2) → f(– (–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(– 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 funcitons
• 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 funcitons
• 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: < –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)
• 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