
 Bold black text in italics: Knowledge and Skills Statement (TEKS)
 Bold black text: Student Expectation (TEKS)
 Strikethrough: 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: Unitspecific 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:

P.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:

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 realworld 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 a, b, c, and d)
 Using graphing technology
Note(s):
 Grade Level(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, piecewisedefined, 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 realworld 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 a, b, c, AND d, IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 General form of parent function
 Polynomial functions: f(x) = a_{n}x^{n} + an_{1}x^{n}^{1} +...+ a_{2}x^{2} + a_{1}x + a_{0}, where n is a positive integer (e.g., 2x^{5} – 7x^{3} + 11x + 6, etc.)
 Power functions: f(x) = ax^{n}, where n is a real number (e.g., f(x) = x^{2}, f(x) = x^{3}, f(x) = x^{4}, f(x) = x^{0.5}, f(x) = x^{2.3}, f(x) = x^{0.5}, etc.)
 Representations with and without technology
 Graphs
 Verbal descriptions
 Algebraic generalizations (including equation and function notation)
 Changes in parameters 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 (xaxis)
 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 yaxis
 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
 Realworld problem situations
Note(s):
 Grade Level(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, piecewisedefined, 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 n^{th} differences
 Polynomial functions
 Power functions
 Domain and range
 Represented as a set of values
 Represented verbally
 All real numbers greater than or equal to zero
 All real numbers less than one
 Represented with inequality notation
 Represented with set notation
 {xx , x ≥ 0}
 {yy , y < 1}
 Represented with interval notation
 Symmetry
 Reflectional
 Rotational
 Symmetric with respect to the origin (180° rotational symmetry)
 Relative extrema
 Relative maximum
 Relative minimum
 Zeros
 Roots/solutions
 xintercepts
 Intervals where the function is increasing or decreasing
 Represented with inequality notation, –1 < x ≤ 3
 Represented with set notation, {xx , –1 < x ≤ 3}
 Represented with interval notation, (–1, 3]
 Connections among multiple representations of key features
 Graphs
 Tables
 Algebraic
 Verbal
Note(s):
 Grade Level(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 realworld problems.

Analyze, Describe
END BEHAVIOR OF FUNCTIONS, INCLUDING POLYNOMIAL AND POWER FUNCTIONS, USING INFINITY NOTATION IN MATHEMATICAL AND REALWORLD 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, c; f(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, c; f(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) = a_{n}x^{n} + a_{n}_{1}x^{n1} + a_{n}_{2}x^{n2} +...+a_{0}x^{0}, where n is a positive integer
 The leading coefficient (a_{n}) determines the right end behavior.
 Ex: If a_{n} > 0, as x → ∞ (on the right), f(x) → ∞.
 Ex: If a_{n} < 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 a_{n} > 0, if n is even, then as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → ∞.
 Ex: When a_{n} > 0, if n is odd, then as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → –∞.
 Power: f(x) = ax^{n}, 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 realworld situations
Note(s):
 Grade Level(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 realworld 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 (xvalues) are given
 Solving equations when dependent quantities (yvalues) 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 realworld relationship
 Polynomial functions (e.g., area, volume, motion, etc.)
 Power functions
Note(s):
 Grade Level(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 realworld 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 realworld problems.

Solve
POLYNOMIAL EQUATIONS WITH REAL COEFFICIENTS BY APPLYING A VARIETY OF TECHNIQUES IN MATHEMATICAL AND REALWORLD 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
 x^{2} = 25, x = 5; therefore, x = ±5
 Completing the square
 Factoring
 Quadratic formula
 Synthetic substitution (synthetic division)
 Technology
 Types of solutions
 Real roots
 Nonreal roots (imaginary or complex)
 Double roots (repeated roots)
 Mathematical problem situations
 Realworld problem situations
 Maximization or minimization
 Finding maximum area or volume, given constraints
 Maximizing revenue or profit
 Minimizing cost
Note(s):
 Grade Level(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 realworld problems.

Solve
POLYNOMIAL INEQUALITIES WITH REAL COEFFICIENTS BY APPLYING A VARIETY OF TECHNIQUES IN MATHEMATICAL AND REALWORLD 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
 x^{2} ≤ 25, x ≤ 5; therefore, –5 ≤ x ≤ 5
 Completing the square
 Factoring
 Quadratic formula
 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 REALWORLD 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):
 Grade Level(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
