
 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.2E 
Determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations.

Determine
AN INVERSE FUNCTION, WHEN IT EXISTS, FOR A GIVEN FUNCTION OVER ITS DOMAIN OR A SUBSET OF ITS DOMAIN
Represent
THE INVERSE OF A FUNCTION USING MULTIPLE REPRESENTATIONS
Including, but not limited to:
 Inverse of a function – function that undoes the original function. When composed f(f ^{–1}(x)) = x and f ^{–1}(f(x)) = x.
 Characteristics of inverse functions
 Domain of the function becomes an appropriate range of the inverse function.
 Range of the function becomes an appropriate domain of the inverse function.
 Composed as f(f ^{–1}(x)) = x and f ^{–1}(f(x)) = x
 Multiple representations
 Inverse function notation
 When a function f(x) has an inverse that is also a function, the inverse can be written with f ^{–1}(x).
 For the function f(x) = x + 4, the inverse function is f ^{–1}(x) = x – 4.
 For the function g(x) = x^{2}:
 If the restricted domain of g(x) is x ≥ 0, then the inverse function is g^{}^{1}(x) = .
 If the restricted domain of g(x) is x ≤ 0, then the inverse function is g^{}^{1}(x) = –.
 Algebraic
 The inverse of a function can be found algebraically by:
 Writing the original function in “y =” form
 Interchanging the x and y variables
 Solving for y
 A function’s inverse can be confirmed algebraically if both of the following are true: f(f ^{–1}(x)) = x and f ^{–1}(f(x)) = x.
 Tabular
 From the table of values for a given function, the tabular values of the inverse function can be found by switching the x and yvalues of each ordered pair.
 Graphical
 The graphs of a function and its inverse are reflections over the line y = x.
 Verbal description of the relationships between the domain and range of a function and its inverse
 Restrictions on the domain of the original function to maintain functionality
 Inverse functions over a subset of the domain of the original function
Note(s):
 Grade Level(s):
 Algebra II analyzed the relationship between functions and inverses, such as quadratic and square root, or logarithmic and exponential, including necessary restrictions on the domain.
 Precalculus extends the analysis of inverses to include other types of functions, such as trigonometric and others.
 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.
 D1 – Interpret multiple representations of equations and relationships.
 D2 – Translate among multiple representations of equations and relationships.
 III Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VI Statistical Reasoning
 B2 – Select and apply appropriate visual representations of data.
 C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
 VII. Functions
 B1 – Understand and analyze features of a function.
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.2F 
Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Graph
EXPONENTIAL AND LOGARITHMIC 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 EXPONENTIAL AND LOGARITHMIC, FUNCTIONS 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
 Exponential functions: f(x) = 2^{x}, f(x) = e^{x}, f(x) = 10^{x}
 Logarithmic functions: f(x) = log_{2}(x), f(x) = ln(x), f(x) = logx
 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 EXPONENTIAL AND LOGARITHMIC FUNCTIONS SUCH AS DOMAIN, RANGE, 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
 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
 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 , –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 AND LOGARITHMIC 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) → –∞.
 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 EXPONENTIAL AND LOGARITHMIC 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.)
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.5G 
Use the properties of logarithms to evaluate or transform logarithmic expressions.

Use
THE PROPERTIES OF LOGARITHMS
Including, but not limited to:
 Connection of logarithms to exponents: log_{b}x = y b^{y} = x
 Common logarithms (base 10): log x = y 10^{y} = x
 Natural logarithms (base e): ln x = y e^{y} = x
 Logarithms of a product: log_{b}(xy) = log_{b}x + log_{b}y
 Logarithms of a quotient: log_{b} = log_{b}x – log_{b}y
 Power rule of logarithms: log_{b}(x^{r}) = r • log_{b}x
 Change of base property: log_{b}x =
To Transform, To Evaluate
LOGARITHMIC EXPRESSIONS
Including, but not limited to:
 Evaluating logarithmic expressions
 Changing to exponential notation
 With technology
 Transforming logarithmic expressions
 Numerical expressions
 Algebraic expressions
Note(s):
 Grade Level(s):
 Algebra I simplified expressions using the laws (properties) of exponents, including integral and rational exponents.
 Algebra II rewrote exponential equations to logarithmic equations and vice versa.
 Algebra II formulated and solved exponential and logarithmic equations.
 Precalculus applies the properties of logarithms to transform expressions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 I. Numeric Reasoning
 B1 – Perform computations with real and complex numbers.
 II. Algebraic Reasoning
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 VII. Functions
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.5H 
Generate and solve logarithmic equations in mathematical and realworld problems.

Generate, Solve
LOGARITHMIC EQUATIONS IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Solution strategies
 Solving logarithmic equations algebraically
 Simplifying expressions on both sides of an equation by writing them as single logarithms
 Rewriting logarithmic equations in exponential form
 Extraneous solutions
 Solving logarithmic equations with technology
 Various situations
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra II rewrote exponential equations to logarithmic equations and vice versa.
 Algebra II formulated and solved exponential and logarithmic equations.
 Precalculus applies the properties of logarithms to simplify expressions and solve equations.
 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.5I 
Generate and solve exponential equations in mathematical and realworld problems.

Generate, Solve
EXPONENTIAL EQUATIONS IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Various solution strategies
 Solving exponential equations algebraically
 Simplifying expressions on both sides of an equation
 Rewriting exponential equations in logarithmic form
 Solving exponential equations with technology
 Various situations
 Mathematical and realworld problem situations
 Exponential growth
 Exponential decay
 Other exponential behavior
Note(s):
 Grade Level(s):
 Algebra I analyzed and investigated quadratic and exponential functions and their applications.
 Algebra II analyzed and investigated logarithmic, exponential, absolute value, rational, square root, cube root, and cubic functions.
 Algebra II formulated and solved exponential and logarithmic equations.
 Algebra I and Algebra II analyzed and described the effects of transformations on the parent functions with changes in a, b, c, and d parameters.
 Precalculus extends these skills to generate and solve exponential equations in mathematical and realworld situations.
 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
