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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 03: Polynomial and Power Functions, Equations, and Inequalities SUGGESTED DURATION : 13 days

#### Unit Overview

This unit bundles student expectations that address graphs, attributes, and transformations of polynomial and power functions and application of polynomial and power functions in mathematical and real-world problem situations. Polynomial 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 Units 01 and 06, students simplified polynomials, applied laws of exponents, and factored quadratic expressions. In Algebra II Units 04, students also simplified polynomials, applied the laws of exponents, and factored quadratic, cubic, and quartic expressions. In both Algebra I and Algebra II, students performed operations with polynomials, including addition, subtraction, multiplication, and division. In Algebra I Units 01 – 04, 07, and 08 students completed an in-depth study of linear and quadratic functions and equations. In Algebra II Units 06, 07, 08, and 09, students completed an in-depth study of square root, rational, cubic, and cube root functions and equations. Students applied transformations to various functions, including vertical shifts, horizontal shifts, and vertical compressions and stretches. In Precalculus Unit 01, students analyzed the key features of various function types, including polynomial and power functions. In Unit 02, students investigated and applied composition of functions.

During this unit, students graph polynomial and power functions and analyze their key features, including general shape, number of bends, and end behavior. Students develop strategies that can be used to determine these key features based on analysis of the function type and coefficients. Students determine and analyze the key features of polynomial and power functions, including domain, range, symmetry, relative maximum, relative minimum, zeros, and intervals of increasing and decreasing behavior, in mathematical and real-world problem problems. Students analyze and describe end behavior of polynomial and power functions using infinity notation in mathematical and real-world problems. Students graph polynomial and power functions 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 situations modeled by polynomial and power functions to solve real-world problems. Students solve polynomial equations with real coefficients using graphs, tables, and algebraic methods to determine real and complex roots, including factoring, the quadratic formula, and synthetic substitution and division. Students solve polynomial inequalities with real coefficients using graphs, tables, and algebraic methods, including solving the related polynomial equation and testing the intervals between the solutions. Students write the solution sets for polynomial inequalities in interval notation. When presented with problems involving polynomial equations and inequalities, students determine the appropriate representations, key features, and various methods needed to solve the problems.

After this unit, in Precalculus Unit 04, students will continue to work with polynomial expressions and operations when studying rational functions. Additionally, students will encounter these topics again in college mathematics courses where polynomial functions play a central role.

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): II. Algebraic Reasoning C1, D1, D2; VII. Functions B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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 polynomial functions. The Texas College and Career Readiness Standards (2009) extend the reach of polynomial function analysis to include complex numbers, calling for students to be able to define and give examples of complex numbers and to perform computations with real and complex numbers.

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/collegereadiness/crs.pdf

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

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?

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?

Transformations 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?

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

• Why are equations and inequalities used to model problem situations?
• How are equations and inequalities used to model problem situations?
• What methods can be used to solve equations and inequalities?
• Why is it essential to solve equations and inequalities using various methods?
• How can solutions to equations and inequalities be represented?
• How do the representations of solutions to equations and solutions to inequalities compare?
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

• Evaluate
• Expressions
• Multiple Representations
• Patterns/Rules

Functions

• Attributes of Functions
• Non-Linear Functions

Geometric Reasoning

• Transformations

Associated Mathematical Processes

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

Transformations of polynomial and power functions can be used to determine graphs and equations of representative functions in problem situations.

• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by af(x), for specific values of a?
• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by f(bx), for specific values of b?
• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by f(x - c) for specific values of c?
• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by f(x) + d, for specific values of d?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Evaluate
• Expressions
• Inequalities
• Multiple Representations
• Patterns/Rules
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Geometric Reasoning

• Transformations

Associated Mathematical Processes

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

Polynomial functions have key attributes, including domain, range, symmetry, relative extrema, zeros, intervals over which the function is increasing or decreasing, and end behavior.

• What are the key attributes of polynomial functions?
• How can the key attributes of a polynomial function be determined from multiple representations of the function?
• How can knowledge of the key attributes of a polynomial function be used to sketch the graph of the function?
• How can the domain and range of a polynomial function be determined and described?
• How can the relative extrema of a polynomial function be determined and described?
• How can the zeros of a polynomial function be determined and described?
• How can the intervals where the polynomial function is increasing and decreasing be determined and described?
• How can the end behavior of a polynomial function be determined and described using infinity notation?
• How do even and odd functions compare graphically and symbolically?

Power functions have key attributes, including domain, range, symmetry, relative extrema, zeros, intervals over which the function is increasing or decreasing, and end behavior.

• What are the key attributes of power functions?
• How can the key attributes of a power function be determined from multiple representations of the function?
• How can knowledge of the key attributes of a power function be used to sketch the graph of the function?
• How can the domain and range of a power function be determined and described?
• How can the zero of a power function be determined and described?
• How can the intervals where the power function is increasing and decreasing be determined and described?
• How can the end behavior of a power function be determined and described using infinity notation?
• How do even and odd functions compare graphically and symbolically?

Transformations of polynomial and power functions can be used to determine graphs and equations of representative functions in problem situations.

• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by af(x), for specific values of a?
• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by f(bx), for specific values of b?
• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by f(x - c) for specific values of c?
• What are the effects of changes on the graph of polynomial and power functions when f(x) is replaced by f(x) + d, for specific values of d?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Expressions
• Inequalities
• Multiple Representations
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

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

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

• How are polynomial equations used to model problem situations?
• What methods can be used to solve polynomial equations?
• What are the advantages and disadvantages of various methods used to solve polynomial equations?
• What methods can be used to justify the reasonableness of solutions?
• What is the relationship between the degree of a polynomial equation and the number of solutions to the equation?
• What types of solutions can a polynomial equation have?

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

• How are polynomial inequalities used to model problem situations?
• What methods can be used to solve polynomial inequalities?
• What are the advantages and disadvantages of various methods used to solve polynomial inequalities?
• What methods can be used to justify the reasonableness of solutions?
• How are the solutions of polynomial equations related to the solutions of corresponding polynomial inequalities?
• How can solutions to polynomial inequalities be represented?

Polynomial and power 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 polynomial and power function models?
• What key attributes identify a polynomial and power function model?
• How does the domain and range of the function compare to the domain and range of the problem situation?
• What are the connections between the key attributes of a polynomial and power function model and the real-world problem situation?
• How can polynomial and power function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students might think that a vertical compression is equivalent to a horizontal stretch and that a vertical stretch is equivalent to a horizontal compression. While this relationship appears to hold for certain polynomial functions (i.e., f(x) = x2 and f(x) = x3), it is not true in general. For example, consider the function f(x) = x34x under the transformations g(x) = 3f(x), g(x) = f(3x), g(x) =  f(x), and g(x) = .

Underdeveloped Concepts:

• Some students might confuse elements of the polynomial division algorithm with the synthetic division algorithm. For example, when dividing a polynomial by (x – 3), some students might write 3 to the left of the array, rather than 3. Additionally, some students might subtract terms vertically when performing synthetic division, rather than adding these terms.
• Some students may need to be reminded to include placeholder terms for polynomial or synthetic division if either a dividend or divisor is missing terms of a certain degree.
• Some students may have difficulty simplifying complex expressions that arise when the quadratic formula yields negative radicands.
• Some students may have difficulty when multiplying trinomials.
• Some students may not know to include only positive numbers when computing a regression for a power function.

#### Unit Vocabulary

• Covariation – pattern of related change between two variables in a function

Related Vocabulary:

 Bends Complex roots Compression Conjugate pairs Continuous Cubic Decreasing Degree Domain End behavior Equation Even function Factoring Imaginary roots Increasing Inequality Inequality notation Interval notation Leading coefficient Maximum Minimum Multiplicity Odd function Polynomial Polynomial division Power Quadratic Quadratic formula Quartic Range Real roots Reflection Roots Set notation Stretch Symmetry Synthetic division Synthetic substitution Transformations Translation Turning points x-intercepts Zeros
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 Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• 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)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
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:
• IX. Communication and Representation
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.

Graph

POLYNOMIAL AND POWER 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:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 POLYNOMIAL AND POWER FUNCTIONS AND THEIR TRANSFORMATIONS, INCLUDING af(x), f(x) + d, f(x – c), f(bx) FOR SPECIFIC VALUES OF abc, AND d, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• General form of parent function
• Polynomial functions: f(x) = anxnan-1xn-1 +...+ a2x2a1x + a0, where n is a positive integer (e.g., 2x5 – 7x3 + 11x + 6, etc.)
• Power functions: f(x) = axn, where n is a real number (e.g., f(x) = x2, f(x) = x3, f(x) = x4, f(x) = x0.5, f(x) = x2.3, f(x) = x-0.5, etc.)
• Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters abc, 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(x – c)
• 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:
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 POLYNOMIAL AND POWER FUNCTIONS SUCH AS DOMAIN, RANGE, SYMMETRY, RELATIVE MAXIMUM, RELATIVE MINIMUM, ZEROS, 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
• Patterns in the nth differences
• Polynomial functions
• Power 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 ≥ 0}
• {y| 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
• Intervals where the function is increasing or decreasing
• Represented with inequality notation, –1 <  ≤ 3
• Represented with set notation, {x|x  , –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:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 POLYNOMIAL AND POWER 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
• Polynomial: f(x) = anxn + an-1xn-1 + an-2xn-2 +...+a0x0, where n is a positive integer
• The leading coefficient (an) determines the right end behavior.
• Ex: If an > 0, as x → ∞ (on the right), f(x) → ∞.
• Ex: If an < 0, as x → ∞ (on the right), f(x) → –∞.
• The degree of the polynomial (n) determines whether the left and right end behaviors are the same or different.
• Ex: When an > 0, if n is even, then as x → ∞ (on the right), f(x) → ∞, and as → –∞ (on the left), f(x) → ∞.
• Ex: When an > 0, if n is odd, then as → ∞ (on the right), f(x) → ∞, and as → –∞ (on the left), f(x) → –∞.
• Power: f(x) = axn, where n is a real number
• Ex: If a > 0 and n > 0, as x → ∞ (on the right), f(x) → ∞.
• Ex: If a > 0 and n < 0, as x → ∞ (on the right), f(x) → 0.
• 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:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 POLYNOMIAL AND POWER 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
• Polynomial functions (e.g., area, volume, motion, etc.)
• Power functions

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:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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.5J Solve polynomial equations with real coefficients by applying a variety of techniques in mathematical and real-world problems.

Solve

POLYNOMIAL EQUATIONS WITH REAL COEFFICIENTS BY APPLYING A VARIETY OF TECHNIQUES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Various methods to solve polynomial equations
• Graphs
• Tables
• Algebraic methods
• Solving equations by taking square roots
• Solving quadratic equations using absolute value
• x2 = 25, |x| = 5; therefore, x = ±5
• Completing the square
• Factoring
• Synthetic substitution (synthetic division)
• Technology
• Types of solutions
• Real roots
• Non-real roots (imaginary or complex)
• Double roots (repeated roots)
• Mathematical problem situations
• Real-world problem situations
• Maximization or minimization
• Finding maximum area or volume, given constraints
• Maximizing revenue or profit
• Minimizing cost

Note(s):

• Algebra I and Algebra II solved quadratic equations using a variety of methods.
• Algebra II analyzed the graphs of cubic functions and determined the linear and quadratic factors of polynomials of degree three and four.
• Precalculus applies these skills, along with technology, to solve polynomial equations of higher degrees.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.5K Solve polynomial inequalities with real coefficients by applying a variety of techniques and write the solution set of the polynomial inequality in interval notation in mathematical and real-world problems.

Solve

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

Including, but not limited to:

• Various methods to solve polynomial inequalities
• Graphs
• Tables
• Algebraic methods
• Solving inequalities by taking square roots
• Solving quadratic inequalities using absolute value
• x2 ≤ 25, |x| ≤ 5; therefore, –5 ≤ x ≤ 5
• Completing the square
• Factoring
• Synthetic substitution (and/or synthetic division)
• Technology
• Relating solutions of polynomial inequalities to the solutions of the related polynomial equations
• Testing the intervals between the solutions
• 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 A POLYNOMIAL INEQUALITY IN INTERVAL NOTATION BY APPLYING A VARIETY OF TECHNIQUES 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 I and Algebra II solved quadratic equations using a variety of methods.
• Algebra II solved quadratic inequalities.
• Algebra II analyzed the graphs of cubic functions and determined the linear and quadratic factors of polynomials of degree three and four.
• Precalculus applies these skills, along with technology, to solve polynomial inequalities of higher degrees.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections