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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 04: Rational Functions, Equations, and Inequalities SUGGESTED DURATION : 12 days

#### Unit Overview

Introduction
This unit bundles student expectations that address graphs, attributes, and transformations of rational functions and application of rational functions in mathematical and real-world problem situations. Discontinuities and asymptotic behavior are analyzed and described. Rational equations and inequalities are also addressed. These topics are studied using multiple representations, including graphical, tabular, verbal, and algebraic methods. 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 Unit 06, students determined the quotient of polynomial expressions of degree one and two. In Algebra II Unit 04, students determined the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two. In Algebra II Unit 08, students determined the sum, difference, product, and quotient of rational expressions with numerators and denominators of degree one and two. Additionally, in Algebra II Unit 08, students studied rational functions extensively, including their graphs, key attributes, transformations, equations, and real-world applications, as well as inverse variation.

During this Unit
Students graph rational functions, including f(x) = and f(x) = , and their transformations, including af(x), f(x) + d, f(x – c), and f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems. Students analyze the behavior of rational functions around their horizontal asymptotes and analyze and describe the end behavior of rational functions using infinity notation based on their equations. Students determine various types of discontinuities in rational functions in the interval (–∞, ∞), including infinite discontinuities (vertical asymptotes) and removable discontinuities, using verbal, symbolic, tabular, and graphical representations. Students describe the left-sided and right-sided behavior of the graph of the function around the discontinuities, including limitations of the graphing calculator as it relates to the behavior of the function around the discontinuities. Students analyze graphs of rational functions that contain oblique asymptotes, describe the behavior of the function around these asymptotes (using verbal, symbolic, tabular, and graphical methods), and determine the equations of these oblique asymptotes using polynomial division. Students determine and analyze the key features of rational functions (including domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, intervals over which the function is increasing or decreasing, end behavior, and discontinuities) in mathematical and real-world problems. Students also make connections between interrelated representations to algebraically construct a model for the function. Students solve rational equations using algebraic methods (factoring, quadratic formula, etc.), graphs, and tables in both mathematical and real-world problems. Students solve rational inequalities with real coefficients by solving the related rational equation, determining discontinuities in the related function, and testing the intervals between the solutions and points of discontinuity (numerically, graphically, and/or with tables) in both mathematical and real-world problems. Students write the solution set of rational inequalities in interval notation.

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

After this Unit
In Unit 05 and Unit 08, students will identify discontinuities and asymptotic behavior in exponential, logarithmic, and trigonometric functions. In subsequent courses in mathematics, these concepts will continue to be applied in problem situations involving rational functions, equations and inequalities.

Function analysis serves as the foundation for college readiness. Analyzing, representing, and modeling with functions are emphasized in the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, C2, C3, D1, D2; VI. Functions A2, B1, B2, C1, C2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to a 2007 report published by the Mathematical Association of America, a “fundamental experience” for students entering college mathematics courses is to address problems in real-world situations by creating and interpreting mathematical models. Functions provide a way to quantitatively study the relationships and change in numerous real-world phenomena; this, coupled with the applicability of functions to many mathematical topics, make functions one of the most important topics in high school mathematics (Cooney, Beckmann, & Lloyd, 2010). In Texas, the importance of these skills is emphasized in the Texas College and Career Readiness Standards (2009), which call for students to be able to understand and analyze features of a function to model real-world situations. Algebraic models allow us to efficiently visualize and analyze the vast amount of interconnected information that is contained in a functional relationship; these tools are particularly helpful as the mathematical models become increasingly complex (National Research Council, 2005). Additionally, research argues that students need both a strong conceptual understanding of functions, as well as procedural fluency; as such, good instruction must include “a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions” (NRC, 2005, p. 353). Lastly, students need to be involved in metacognitive engagement in mathematics as they problem solve and reflect on their solutions and strategies; this is particularly important as students transition into more abstract mathematics, where fewer “clues” may exist warning students of a mathematical misstep (NRC, 2005). In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) notes the necessity for high school students to understand and compare the properties and classes of functions, including rational functions. In the AP Calculus Course Description, the College Board (2012) states that mathematics designed for college-bound students should involve analysis and understanding of elementary functions, including rational functions. Students must be familiar with the properties, algebra, graphs, and language of these functions. More specifically, students should have a basic understanding of asymptotes in terms of graphical behavior so that this knowledge can later be extended to describing asymptotic behavior in terms of limits involving infinity.

Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Katz, Victor J. (Ed.). (2007). Algebra: Gateway to a technological future. Mathematical Association of America.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.
National Research Council. (2005). How Students Learn: Mathematics in the Classroom. M.S. Donovan & J.D. Bransford (Eds.). Washington, DC: The National Academies Press.
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.
• 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?
• 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 (including algebraically, graphically, verbally, and numerically) 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
• Functions
• Rational
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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 (including algebraically, graphically, verbally, and numerically) 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
• Minimum or maximum value
• Asymptotes
• Symmetries
• Increasing or decreasing
• End behavior
• Discontinuities
• Functions
• Rational
• Relations and Generalizations
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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 represented in various ways (including algebraically, graphically, verbally, and numerically) 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?
• Inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the inequality.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can inequalities be used to represent relationships between quantities?
• How do solutions to inequalities differ from solutions to equations?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write rational inequalities?
• What methods can be used to solve rational inequalities?
• How does the structure of the inequality influence the selection of an efficient method for solving rational inequalities?
• How can the solutions to rational inequalities be determined and represented?
• How are properties and operational understandings used to transform rational inequalities?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Asymptotes
• Symmetries
• Increasing or decreasing
• End behavior
• Discontinuities
• Functions
• Rational
• Patterns, Operations, and Properties
• Relations and Generalizations
• Algebraic Reasoning
• Inequalities
• 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 instead of 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 have misconceptions about the nature of the graph of a rational function when using a graphing calculator, where the calculator obscures the details or hidden behavior of a function.
• Some students may think that rational functions must have a vertical asymptote. Although many rational functions do have vertical asymptotes, some may only have one or more removable discontinuities.
• Some students may forget to consider the undefined values in a rational equation when solving a rational inequality. They may only consider the zeros of the corresponding rational equation when determining test intervals.

Underdeveloped Concepts:

• Some students may think that when zeros of an expression occur in the denominator of a function, they always produce vertical asymptotes. However, if an x-value makes both the numerator and denominator equal to zero, this value indicates a removable discontinuity and not a vertical asymptote.
• Some students may forget to consider the presence of extraneous solutions when solving rational equations.

#### Unit Vocabulary

• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Covariation – pattern of related change between two variables in a function
• Infinite discontinuities – values of x where vertical asymptotes occur. Specifically, if a function has an infinite discontinuity at x = c, as xc, f(x) → ±∞
• Jump discontinuities – values or intervals of x where a function “jumps” (or skips, or disconnects). If a function has a jump discontinuity at x = c, then the function approaches a specific y-value on the left of x = c (or, when x < c), but approaches a different y-value on the right side of x = c (or, when x > c).
• Removable discontinuities – values or intervals of x where a function has a “hole” in the graph. If a function has a removable discontinuity at x = c, then the function approaches the same specific y-value on both the left and right of x = c, even though f(c) is not the same (or undefined).

Related Vocabulary:

 compression decreasing degree denominator discontinuity domain end behavior equation even function extraneous solution factor factoring fraction horizontal asymptote increasing inequality notation infinity notation interval notation leading coefficient least common denominator left-sided behavior maximum minimum numerator oblique asymptote odd function polynomial polynomial division quotient range rational equation rational function rational inequality reflection right-sided behavior root set notation solution solution set stretch symmetry transformation translation undefined vertical asymptote x-intercept zero
Unit Assessment Items System Resources Other Resources

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

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

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

Specificity Legend:

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

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

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

Note(s):

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

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

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

Note(s):

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

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

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

Note(s):

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

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

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

Note(s):

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

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

Note(s):

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

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
P.2F

Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Note(s):

• Algebra II graphed various types of functions, including square root, cube root, absolute value, and rational functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

Graph

RATIONAL FUNCTIONS

Including, but not limited to:

• Graphs of the parent functions
• Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations
• Various methods for graphing
• Curve sketching
• Plotting points from a table of values
• Transformations of parent functions (parameter changes abc, and d)
• Using graphing technology

Note(s):

• Algebra II graphed various types of functions, including square root, cube root, absolute value, and rational functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
P.2G

Graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems.

Graph

FUNCTIONS, INCLUDING RATIONAL AND THEIR TRANSFORMATIONS, INCLUDING af(x), f(x) + d, f(xc), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• General form of parent function
• Rational functions: f(x) = • Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters a, b, c, and d on graphs
• Effects of a on f(x) in af(x)
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the horizontal axis (x-axis)
• Effects of d on f(x) in f(x) + d
• d = 0, no vertical shift
• Translation, vertical shift up or down by |d| units
• Effects of c on f(x) in f(xc)
• c = 0, no horizontal shift
• Translation, horizontal shift left or right by |c| units
• Effects of b on f(x) in f(bx)
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the vertical axis or y-axis
• Combined transformations of parent functions
• Transforming a portion of a graph
• Illustrating the results of transformations of the stated functions in mathematical problems using a variety of representations
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra II graphed transformations of various types of functions, including square root, cube, cube root, absolute value, rational, exponential, and logarithmic functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VI.B.2. Algebraically construct and analyze new functions.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
P.2I

Determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing.

Determine, Analyze

THE KEY FEATURES OF RATIONAL FUNCTIONS SUCH AS DOMAIN, RANGE, SYMMETRY, RELATIVE MAXIMUM, RELATIVE MINIMUM, ZEROS, ASYMPTOTES, AND INTERVALS OVER WHICH THE FUNCTION IS INCREASING OR DECREASING

Including, but not limited to:

• Covariation – pattern of related change between two variables in a function
• Multiplicative patterns
• Rational functions
• Domain and range
• Represented as a set of values
• {0, 1, 2, 3, 4}
• Represented verbally
• All real numbers greater than or equal to zero
• All real numbers less than one
• Represented with inequality notation
• x ≥ 0
• y < 1
• Represented with set notation
• {x|x ∈ ℜ, x ≥ 0}
• {y|y ∈ ℜ, y < 1}
• Represented with interval notation
• [0, ∞)
• (–∞, 1)
• Symmetry
• Reflectional
• Rotational
• Symmetric with respect to the origin (180° rotational symmetry)
• Relative extrema
• Relative maximum
• Relative minimum
• Zeros
• Roots/solutions
• x-intercepts
• Asymptotes
• Vertical asymptotes (x = h)
• Horizontal asymptotes (y = k)
• Slant asymptotes (y = mx + b)
• Intervals where the function is increasing or decreasing
• Represented with inequality notation, –1 <  ≤ 3
• Represented with set notation, {x|x ∈ ℜ, –1 < x ≤ 3}
• Represented with interval notation, (–1, 3]
• Connections among multiple representations of key features
• Graphs
• Tables
• Algebraic
• Verbal

Note(s):

• Algebra II analyzed functions according to key attributes, such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum values over an interval.
• Precalculus extends the analysis of key attributes of functions to include zeros and intervals where the function is increasing or decreasing.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
P.2J

Analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems.

Analyze, Describe

END BEHAVIOR OF FUNCTIONS, INCLUDING RATIONAL FUNCTIONS, USING INFINITY NOTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Describing end behavior with infinity notation
• Right end behavior
• As x → ∞ (or as x approaches infinity) the function becomes infinitely large; f(x) → ∞.
• As x → ∞ (or as x approaches infinity) the function becomes infinitely small; f(x) → –∞.
• As x → ∞ (or as x approaches infinity) the function approaches a constant value, cf(x) → c.
• Left end behavior
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely large; f(x) → ∞.
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely small; f(x) → –∞.
• As x → –∞ (or as x approaches negative infinity) the function approaches a constant value, cf(x) → c.
• Determining end behavior from multiple representations
• Tables: evaluating the function for extreme negative (left end) and positive (right end) values of x
• Graphs: analyzing behavior on the left and right sides of the graph
• Determining end behavior from analysis of the function type and the constants used
• Rational: f(x) = , where p(x) and q(x) are polynomials in terms of xq(x) ≠ 0
• Ex: If the degree of p(x) is greater than the degree of q(x), as x → –∞, f(x) → ±∞ on the left, and as x → ∞, f(x) → ± on the right.
• Ex: If the degree of p(x) is less than the degree of q(x), as x → –∞, f(x) → 0 on the left, and as x → ∞, f(x) → 0 on the right.
• Ex: If the degree of p(x) and q(x) are the same, as x → –∞, f(x) → k on the left, and as x → ∞, f(x) → k on the right, where k is a constant determined by the leading coefficients of p(x) and q(x).
• Interpreting end behavior in real-world situations

Note(s):

• Algebra II analyzed the domains and ranges of quadratic, square root, exponential, logarithmic, and rational functions.
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends analysis of domain, range, and asymptotic restrictions to determine the end behavior of functions and describes this behavior using infinity notation.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.2K Analyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes.

Analyze

CHARACTERISTICS OF RATIONAL FUNCTIONS

Including, but not limited to:

• Analyzing algebraically rational functions of the form f(x) = , where p(x) and q(x) are polynomials in x, q(x) ≠ 0
• Degrees (highest power of x) of p(x) and q(x)
• Leading coefficients of p(x) and q(x)
• Factors of p(x) and q(x)
• Zeros of p(x) and q(x) (or where p(x) = 0 and q(x) = 0)
• Quotient of p(x) and q(x) (or p(x) ÷ q(x), obtained through long division of polynomials)
• Determining discontinuities algebraically, for rational functions of the form f(x) = , where p(x) and q(x) are polynomials in x, q(x) ≠ 0
• Behavior of function around discontinuities
• Determining whether a discontinuity is a removable discontinuity or a non-removable discontinuity
• Non-removable discontinuities
• Vertical asymptotes
• Vertical asymptotes occur at values of x where q(x) = 0, but p(x) ≠ 0.
• Removable discontinuities
• Removable discontinuities occur at values of x where both p(x) = 0 and q(x) = 0.
• Determining end behavior algebraically, for rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, q(x) ≠ 0
• Horizontal asymptotes
• If the degree of p(x) is less than the degree of q(x), f(x) has a horizontal asymptote at y = 0.
• If the degrees of p(x) and q(x) are the same, f(x) has a horizontal asymptote at y = k where k is a constant determined by the leading coefficients of p(x) and q(x).
• Oblique (or slant) asymptotes
• If the degree of p(x) is one more than the degree of q(x), then f(x) has an oblique asymptote of the form y = mx + b determined by the quotient of p(x) and q(x) through long division.

Analyze

THE BEHAVIOR OF THE RATIONAL FUNCTION AROUND THE ASYMPTOTES, INCLUDING HORIZONTAL, VERTICAL, AND OBLIQUE ASYMPTOTES

Including, but not limited to:

• Behavior around horizontal asymptotes
• Verbal and symbolic
• If a rational function f(x) has a horizontal asymptote at y = k, then as the x-values increase or decrease without bound (or as x → ±∞), the y-values of the function approach k (or f(x) → k).
• Tabular
• Graphical
• Behavior around vertical asymptotes
• Verbal and symbolic
• If a rational function f(x) has a vertical asymptote at x = h, then as x-values approach h (or as xh), the y-values of the function either increase or decrease without bound (or f(x) → ±∞).
• Tabular
• Graphical
• Behavior around oblique (or slant) asymptotes
• Verbal and symbolic
• If a rational function f(x) has an oblique (or slant) linear asymptote at y = mx + b, then as x-values increase or decrease without bound (or as x → ±∞), the y-values of the function approach the line y = mx + b (or f(x) → mx + b).
• Tabular
• Graphical

Note(s):

• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Algebra II determined the quotient of polynomials using algebraic methods.
• Precalculus analyzes the behavior of rational functions and describes this behavior around asymptotes.
• Precalculus uses the quotient of polynomials to determine oblique (or other) asymptotes of rational functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.2L Determine various types of discontinuities in the interval (-∞, ∞) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities.

Determine

VARIOUS TYPES OF DISCONTINUITIES IN THE INTERVAL (–∞, ∞) AS THEY RELATE TO FUNCTIONS

Including, but not limited to:

• Determining whether a discontinuity is a removable discontinuity or a non-removable discontinuity
• Behavior of function around discontinuities
• Non-removable discontinuities
• Jump discontinuities – values or intervals of x where a function “jumps” (or skips, or disconnects). If a function has a jump discontinuity at x = c, then the function approaches a specific y-value on the left of x = c (or when x < c), but approaches a different y-value on the right side of x = c (or when x > c).
• Graphical
• Tabular
• Algebraic
• Piecewise defined functions
• Evaluate both parts of the function to the left and right at breaks in the domain, then check to see if the values agree.
• General functions
• Jump discontinuities can occur at values of x where the function is not defined due to limits on the domain.
• Infinite discontinuities – values of x where vertical asymptotes occur, function has an infinite discontinuity at x = c, as xc, f(x) → ±∞
• Graphical
• Tabular
• Algebraic
• Rational functions
• For rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, vertical asymptotes (or infinite discontinuities) occur at values of x where q(x) = 0 but p(x) ≠ 0.
• Trigonometric functions
• For trigonometric functions, vertical asymptotes can occur at values of x where the function is undefined.
• Removable discontinuities – values or intervals of x where a function has a “hole” in the graph. If a function has a removable discontinuity at x = c, then the function approaches the same specific y-value on both the left and right of x = c, even though f(c) is not the same (or undefined).
• Graphical
• Tabular
• Algebraic
• Rational functions
• For rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, removable discontinuities occur at values of x where both p(x) = 0 and q(x) = 0.

Explore

THE LIMITATIONS OF THE GRAPHING CALCULATOR AS IT RELATES TO THE BEHAVIOR OF THE FUNCTION AROUND DISCONTINUITIES

Including, but not limited to:

• Tables
• Hidden behavior of a function
• Because tables show only discrete values of x and y, the tables often do not fully describe the behavior of a function.
• Values of x that get skipped
• Because tables default to integer values of x and y, the tables often skip important features of a function that occur at the rational (decimal) values in between.
• Values of x where a function is undefined
• While tables can locate values of x where a function is undefined, tables do not identify the type of discontinuity that has occurred.
• Graphing functions with graphing calculators
• Evaluating functions at specific x-values
• Setting a window
• Screen width = (maximum x-value) – (minimum x-value)
• Resolution = number of pixels in the screen width
• Δx = (screen width) ÷ (resolution)
• Behavior of calculator graphs around discontinuities
• Jump discontinuities
• Infinite discontinuities
• Removable discontinuities

Note(s):

• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends the idea of domain restrictions to include various types of discontinuities: removable, infinite, and jump.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
P.2M Describe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities.

Describe

THE LEFT-SIDED BEHAVIOR AND THE RIGHT-SIDED BEHAVIOR OF THE GRAPH OF A FUNCTION AROUND DISCONTINUITIES

Including, but not limited to:

• Verbal and symbolic
• Left-sided behavior near a discontinuity at x = c
• Words: As x approaches c from the left
• Symbols: xc
• Right-sided behavior near a discontinuity at x = c
• Words: As x approaches c from the right
• Symbols: xc+
• Function behavior near a discontinuity at x = c
• As x approaches c (from the left or right), the function values can approach a constant, k.
• Words: As x approaches c, the function approaches k.
• Symbols: As xc, f(x) → k (or yk)
• As x approaches c (from the left or right), the function values can continue to increase without limit.
• Words: As x approaches c, the function approaches infinity.
• Symbols: As xc, f(x) → ∞ (or y → ∞)
• As x approaches c (from the left or right), the function values can continue to decrease without limit.
• Words: As x approaches c, the function approaches negative infinity.
• Symbols: As xc, f(x) → –∞ (or y → –∞)
• Graphical
• Left-sided behavior near a discontinuity at x = c
• Move along the graph on the interval x < c from left to right
• Right-sided behavior near a discontinuity at x = c
• Move along the graph on the interval x > c from right to left
• Tabular
• Left-sided behavior near a discontinuity at x = c
• Consider values in the table where x < c, such as c – 0.1, c – 0.01, c – 0.001, etc.
• Right-sided behavior near a discontinuity at x = c
• Consider values in the table where x > c, such as c + 0.1, c + 0.01, c + 0.001, etc.
• Use left- and right-sided behavior of a function to determine whether a discontinuity is a removable discontinuity or a non-removable discontinuity

Note(s):

• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends the concept of domain restrictions around asymptotes to include other types of discontinuities and analyzes the left-sided and right-sided behavior of functions near these discontinuities.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.2N

Analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems.

Analyze, To Solve

SITUATIONS MODELED BY FUNCTIONS, INCLUDING RATIONAL FUNCTIONS

Including, but not limited to:

• Models that represent problem situations
• Understanding the meaning of the variables (both independent and dependent)
• Evaluating the function when independent quantities (x-values) are given
• Solving equations when dependent quantities (y-values) are given
• Appropriateness of given models for a situation
• Analyzing the attributes of a problem situation
• Determining which type of function models the situation
• Determining a function to model the situation
• Using transformations
• Using attributes of functions
• Using technology
• Describing the reasonable domain and range values
• Comparing the behavior of the function and the real-world relationship
• Rational functions (e.g., averages, temperature/pressure/volume relationships (Boyle’s Law), etc.)
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value

Note(s):

• Algebra II analyzed situations involving exponential, logarithmic, and rational functions.
• Precalculus extends function analysis to include polynomial and power functions and expects students to solve real-world problems and interpret solutions to those problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VI.C.2. Develop a function to model a situation.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
P.5 Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:
P.5L Solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems.

Solve

RATIONAL INEQUALITIES WITH REAL COEFFICIENTS BY APPLYING A VARIETY OF TECHNIQUES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Relating solutions of rational inequalities to the solutions of the related rational equations
• Solving related rational equations using algebraic methods
• Factoring
• Solving related rational equations with technology
• Graphs
• Tables
• Relating solutions of rational inequalities to the discontinuities of the related rational functions
• Identifying types of discontinuities
• Vertical asymptotes
• Removable discontinuities
• Locating the discontinuities of rational functions using algebraic methods
• Finding values where the denominator of a rational expression is equal to zero
• Factoring
• Checking to see if these values are also zeros of the numerator of the rational expression
• Locating discontinuities of rational functions with technology
• Graphs
• Tables
• Testing the intervals between the solutions and points of discontinuity
• Evaluating the expression to determine whether values satisfy the inequality
• Analyzing graphs to determine whether values satisfy the inequality
• Using tables to determine whether values satisfy the inequality

Write

THE SOLUTION SET OF THE RATIONAL INEQUALITY IN INTERVAL NOTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Using brackets for closed intervals
• Using parentheses for open intervals
• Using parentheses and the infinity symbol for boundless intervals
• Using the symbol for set union to describe solution sets with more than one interval

Note(s):

• Algebra II determined the sum, difference, product, and quotient of rational expressions.
• Algebra II formulated and solved rational equations with real solutions.
• Precalculus extends these skills to solve rational inequalities.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences. 