
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
strikethrough.

Legend:  Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
 Unitspecific 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.

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.2D 
Describe symmetry of graphs of even and odd functions.

Describe
SYMMETRY OF GRAPHS OF EVEN AND ODD FUNCTIONS
Including, but not limited to:
 Symmetry of graphs
 Even functions – functions of the form f(x) with the property that f(–x) = f(x)
 Reflectional symmetry
 Graphs of even functions are symmetric with respect to the yaxis.
 Ordered pairs
 If (a, b) is an ordered pair on the graph of an even function, then (–a, b) is also on the graph.
 Algebraic properties
 If f(x) is an even function, then f(–x) = f(x).
 Odd functions – functions of the form f(x) with the property that f(–x) = –f(x)
 Rotational symmetry
 Graphs of odd functions are symmetric with respect to the origin (or have 180° rotational symmetry with respect to the origin).
 Ordered pairs
 If (a, b) is an ordered pair on the graph of an odd function, then (–a, –b) is also on the graph.
 Algebraic properties
 If f(x) is an odd function, then f(–x) = –f(x).
Note(s):
 Grade Level(s):
 Algebra II identified symmetry in parabolas.
 Precalculus uses symmetry to describe general functions as being even or odd.
 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.2F 
Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Graph
EXPONENTIAL, LOGARITHMIC, RATIONAL, POLYNOMIAL, POWER, AND PIECEWISE DEFINED FUNCTIONS, INCLUDING STEP FUNCTIONS
Including, but not limited to:
 Graphs of the parent functions
 Graphs of piecewise defined functions
 Functions specifically defined over given domains
 Functions that exhibit piecewise behavior
 Step 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 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.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 EXPONENTIAL, LOGARITHMIC, RATIONAL, POLYNOMIAL, POWER, 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
Including, but not limited to:
 Covariation – pattern of related change between two variables in a function
 Multiplicative patterns
 Exponential functions
 Logarithmic functions
 Rational functions
 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 ∈ ℜ, x ≥ 0}
 {yy ∈ ℜ, 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
 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 < x ≤ 3
 Represented with set notation, {xx ∈ ℜ, –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 EXPONENTIAL, LOGARITHMIC, RATIONAL, 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
 Exponential: f(x) = ab^{x}
 Ex: When a > 0 and b > 1, as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → 0.
 Ex: When a > 0 and 0 < b < 1, as x → ∞ (on the right), f(x) → 0, and as x → –∞ (on the left), f(x) → ∞.
 Logarithmic: f(x) = alog_{b}(x)
 Ex: When a > 0 and b > 1, as x → ∞ (on the right), f(x) → ∞.
 Ex: When a > 0 and b > 1, as x → 0 (on the left), f(x) → –∞.
 Rational: f(x) = , where p(x) and q(x) are polynomials in terms of x, q(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).
 Polynomial: f(x) = a_{n}x^{n} + a_{n}_{–1}x^{n}^{–1 }+ ... + a_{2}x^{2} + a_{1}x + a_{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.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 nonremovable discontinuity
 Behavior of function around discontinuities
 Nonremovable 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 yvalue on the left of x = c (or when x < c), but approaches a different yvalue 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 x → c, 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 yvalue 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 xvalues
 Setting a window
 Screen width = (maximum xvalue) – (minimum xvalue)
 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):
 Grade Level(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:
 VII. Functions
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.2M 
Describe the leftsided behavior and the rightsided behavior of the graph of a function around discontinuities.

Describe
THE LEFTSIDED BEHAVIOR AND THE RIGHTSIDED BEHAVIOR OF THE GRAPH OF A FUNCTION AROUND DISCONTINUITIES
Including, but not limited to:
 Verbal and symbolic
 Leftsided behavior near a discontinuity at x = c
 Words: As x approaches c from the left
 Symbols: x → c^{–}
 Rightsided behavior near a discontinuity at x = c
 Words: As x approaches c from the right
 Symbols: x → c^{+}
 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 x → c, f(x) → k (or y → k)
 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 x → c, 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 x → c, f(x) → –∞ (or y → –∞)
 Graphical
 Leftsided behavior near a discontinuity at x = c
 Move along the graph on the interval x < c from left to right
 Rightsided behavior near a discontinuity at x = c
 Move along the graph on the interval x > c from right to left
 Tabular
 Leftsided 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.
 Rightsided 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 rightsided behavior of a function to determine whether a discontinuity is a removable discontinuity or a nonremovable discontinuity
Note(s):
 Grade Level(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 leftsided and rightsided 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:
 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 EXPONENTIAL, LOGARITHMIC, RATIONAL, 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
 Exponential functions
 Exponential growth (e.g., accrued interest, population growth, etc.)
 Exponential decay (e.g., halflife, cooling rate, etc.)
 Logarithmic functions (e.g., pH, sound (decibel measures), earthquakes (Richter scale), etc.)
 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
 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
