
 Bold black text in italics: Knowledge and Skills Statement (TEKS)
 Bold black text: Student Expectation (TEKS)
 Bold red text in italics: Student Expectation identified by TEA as a Readiness Standard for STAAR
 Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR
 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)

2A.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


2A.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:

2A.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

2A.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

2A.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

2A.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

2A.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:

2A.1G 
Display, explain, or 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

2A.2 
Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:


2A.2A 
Graph the functions f(x)=, f(x)=1/x, f(x)=x^{3}, f(x)=, f(x)=b^{x}, f(x)=x, and f(x)=log_{b} (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.
Readiness Standard

Graph
THE FUNCTIONS f(x)=
Including, but not limited to:
 Representations of functions, including graphs, tables, and algebraic generalizations
 Rational (reciprocal of x), f(x) =
 Connections between representations of families of functions
 Comparison of similarities and differences of families of functions
Analyze
THE KEY ATTRIBUTES OF THE FUNCTIONS SUCH AS DOMAIN, RANGE, INTERCEPTS, SYMMETRIES, AND ASYMPTOTIC BEHAVIOR
Including, but not limited to:
 Domain and range of the function
 Domain – set of input values for the independent variable over which the function is defined
 Continuous function – function whose values are continuous or unbroken over the specified domain
 Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
 Range – set of output values for the dependent variable over which the function is defined
 Representation for domain and range
 Verbal description
 Ex: x is all real numbers less than five.
 Ex: x is all real numbers.
 Ex: y is all real numbers greater than –3 and less than or equal to 6.
 Ex: y is all integers greater than or equal to zero.
 Inequality notation – notation in which the solution is represented by an inequality statement
 Ex: x < 5, x
 Ex: x
 Ex: –3 < y ≤ 6, x
 Ex: y ≥ 0, y
 Set notation – notation in which the solution is represented by a set of values
 Braces are used to enclose the set.
 Solution is read as “The set of x such that x is an element of …”
 Ex: x < 5, x
 Ex: x
 Ex: –3 < y ≤ 6, x
 Ex: y ≥ 0, y
 Interval notation – notation in which the solution is represented by a continuous interval
 Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
 Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
 Ex: (–, 5)
 Ex: (–, )
 Ex: (–3, 6]
 Domain and range of the function versus domain and range of the contextual situation
 Key attributes of functions
 Intercepts/Zeros
 xintercept(s) – x coordinate of a point at which the relation crosses the xaxis, meaning the y coordinate equals zero, (x, 0)
 Zeros – the value(s) of x such that the y value of the relation equals zero
 yintercept(s) – y coordinate of a point at which the relation crosses the yaxis, meaning the x coordinate equals zero, (0, y)
 Symmetries
 Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
 Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
 Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
 Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
 Use key attributes to recognize and sketch graphs
 Application of key attributes to realworld problem situations
Note(s):
 Grade Level(s):
 The notation represents the set of real numbers, and the notation represents the set of integers.
 Algebra I studied parent functions f(x) = x, f(x) = x^{2}, and f(x) = b^{x} and their key attributes.
 Precalculus will study polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III. Geometric Reasoning
 B1 – Identify and apply transformations to figures.
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A1 – Recognize whether a relation is a function.
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.6 
Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:


2A.6G 
Analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x  c), and f(x) + d for specific positive and negative real values of a, b, c, and d.
Supporting Standard

Analyze
THE EFFECT ON THE GRAPHS OF f(x) = WHEN f(x) IS REPLACED BY af(x), f(bx), f(x – c), AND f(x) + d FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, b, c, AND d
Including, but not limited to:
 General form of the rational function
 Rational function
 f(x) =
 Representations with and without technology
 Graphs
 Tables
 Verbal descriptions
 Algebraic generalizations
 Effects on the graph of f(x) = , when parameters a, b, c, and d are changed in or
 Effects on the graph of f(x) = , when f(x) is replaced by af(x) with and without technology
 a ≠ 0
 a > 1, the graph stretches vertically
 0 < a < 1, the graph compresses vertically
 Opposite of a reflects vertically over the xaxis
 Effects on the graph of f(x) = , when f(x) is replaced by f(bx) with and without technology
 b ≠ 0
 b > 1, the graph compresses horizontally
 0 < b < 1, the graph stretches horizontally
 Opposite of b reflects horizontally over the yaxis
 Effects on the graph of f(x) = , when f(x) is replaced by f(x – c) with and without technology
 c = 0, no horizontal shift
 Horizontal shift left for values of c < 0 by c units
 For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
 Horizontal shift right for values of c > 0 by c units
 For f(x – 2), c = 2, and the function moves to the right two units
 Effects on the graph of f(x) = , when f(x) is replaced by f(x) + d with and without technology
 d = 0, no vertical shift
 Vertical shift down for values of d < 0 by d units
 Vertical shift up for values of d > 0 by d units
 Connections between the critical attributes of transformed function and f(x) =
 Determination of parameter changes given a graphical or algebraic representation
 Determination of a graphical representation given the algebraic representation or parameter changes
 Determination of an algebraic representation given the graphical representation or parameter changes
 Descriptions of the effects on the domain and range by the parameter changes
 Descriptions of the effects on the asymptotes by the parameter changes
 Effects of multiple parameter changes
 Mathematical problem situations
 Realworld problem situation
Note(s):
 Grade Level(s):
 Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x^{2} when f(x) is replaced by af(x),f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d.
 Algebra II introduces the rational function and its transformations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III. Geometric Reasoning
 B1 – Identify and apply transformations to figures.
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.6H 
Formulate rational equations that model realworld situations.
Supporting Standard

Formulate
RATIONAL EQUATIONS THAT MODEL REALWORLD SITUATIONS
Including, but not limited to:
 Rational equations composed of linear or quadratic funcitons
 Data collection activities with and without technology
 Data modeled by rational functions
 Realworld problem situations
 Realworld problem situations modeled by rational functions
 Data tables
 Technology methods
 Transformations of f(x) =
Note(s):
 Grade Level(s):
 Algebra II introduces the rational equation and its applications.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.
 B2 – Algebraically construct and analyze new functions.
 C2 – Develop a function to model a situation.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.6I 
Solve rational equations that have real solutions.
Readiness Standard

Solve
RATIONAL EQUATIONS THAT HAVE REAL SOLUTIONS
Including, but not limited to:
 Rational equations composed of linear or quadratic funcitons
 Limited to real solutions
 Methods for solving rational equations with and without technology
 Graphs
 Algebraic methods
 Solving processes
 Identification of domain restrictions; denominator ≠ 0
 Methods to solve
 Application of cross products for proportional problems
 Multiplication by least common denominator
 Determination of least common denominator
 Multiplication of least common denominator to eliminate fractions
 Transformation of equation to solve for unknown
 Justifications of solutions with and without technology
 Graphs
 Substitution of solutions into original functions
 Removal of extraneous solutions
 Realworld problem situations modeled by rational functions
 Justification of reasonableness of solutions in terms of realworld problem situations or data collections
Note(s):
 Grade Level(s):
 Algebra II introduces the rational equation and its applications.
 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
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
 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.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.6J 
Determine the reasonableness of a solution to a rational equation.
Supporting Standard

Determine
THE REASONABLENESS OF A SOLUTION TO A RATIONAL EQUATION
Including, but not limited to:
 Rational equations composed of linear or quadratic functions
 Justification of solutions to rational equations with and without technology
 Verbal description
 Tables
 Graphs
 Substitution of solutions into original functions
 Justification of reasonableness of solutions in terms of realworld problem situations or data collections
Note(s):
 Grade Level(s):
 Algebra II introduces rational equations and solving rational equations.
 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
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
 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.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.6K 
Determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation.
Supporting Standard

Determine
THE ASYMPTOTIC RESTRICTIONS ON THE DOMAIN OF A RATIONAL FUNCTION
Including, but not limited to:
 Discontinuity in rational functions
 Discontinuity – characteristic of a function where it is not continuous at some point along its graph; place where there is a break in the continuous curve of the function
 Asymptote – a line that is approached and may or may not be crossed
 Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
 Discontinuity where the denominator cannot equal zero
 Determination of vertical asymptotes by setting the denominator ≠ 0
 Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve. A horizontal asymptote describes the long run behavior of the rational function.
 If the degree of the numerator is less than the degree of denominator, the horizontal asymptote is f(x) = 0.
 If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is f(x) = where p is the coefficient of the highest degreed term of the numerator and q is the coefficient of the highest degreed term of the denominator.
 Oblique (slant) asymptote – nonvertical and nonhorizontal line approached by the curve as the function approaches positive or negative infinity. Oblique (slant) asymptotes may be crossed by the curve.
 If the degree of the numerator is more than the degree of the denominator, then the oblique asymptote is of the form y = mx + b determined by the quotient of the numerator and denominator through long division.
 Point (removable) discontinuity – hole or discontinuity in the graph of a rational function generated when a factor in the denominator, that should create a vertical asymptote, reduces out with an equivalent factor in the numerator
 Determination of canceled factor
 Determination of xvalue in canceled factor that would create a zero in the denominator
 Calculation of the corresponding yvalue of the point discontinuity using the reduced rational function
 Graphical analysis using discontinuity
 Domain and range
 Limitations from discontinuities
 Vertical asymptote(s) restrictions on domain
 Horizontal asymptote restrictions on range
 Point(s) of discontinuity restrictions on domain and range
 End behavior
 Single and compound inequality statements to identify domain and range
 Analyzing graph of function in regions formed on graph
 Point tested in regions
 Symmetry
 Intercepts
 Appropriate curve sketched in each region
Represent
DOMAIN AND RANGE USING INTERVAL NOTATION, INEQUALITIES, AND SET NOTATION
Including, but not limited to:
 Inequality notation
 Ex: x < 5 or x > 8?
 Ex: –3 < y < 6
 Ex: x < –3 or 0 < x < 2 or x > 4
 Set notation
 Ex: {xx , x < 5 or x > 8}
 Ex: {yy , –3 < y < 6}
 Ex: {xx , x < –3 or 0 < x < 2 or x > 4}
 Interval notation
 Ex: (–∞,5) (8,∞)
 Ex: (–3, 6)
 Ex: (–∞,–3) (0,2) (4,∞)
Note(s):
 Grade Level(s):
 Algebra II introduces the rational function and its attributes.
 Precalculus will continue to investigate rational functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 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

2A.6L 
Formulate and solve equations involving inverse variation.
Readiness Standard

Formulate
EQUATIONS INVOLVING INVERSE VARIATION
Including, but not limited to:
 Characteristics of variation
 Constant of variation
 Particular equation to represent variation
 Types of variation
 Direct variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
 y varies directly as x
 General equation: y = kx
 Connection of direct variation to linear functions
 Inverse variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y =
 y varies inversely as x
 General equation: y =
 Connection of inverse variation to rational functions
 Realworld problem situations involving variation
 Reasonableness of solutions mathematically and in context of realworld problem situations
Solve
EQUATIONS INVOLVING INVERSE VARIATION
Including, but not limited to:
 Methods for solving variation equations with and without technology
 Graphs
 Algebraic methods
 Solving processes
 Determination of a particular equation to represent the problem
 Direct variation, y = kx
 Inverse variation, y =
 Transformation of equation to solve for unknown
 Justification of solutions with and without technology
 Substitution of solutions into original functions
 Realworld problem situations modeled by rational functions
 Justification of reasonableness of solutions in terms of realworld problem situations or data collections
Note(s):
 Grade Level(s):
 Prior grade levels studied direct variation and proportionality.
 Algebra II introduces inverse variation and its applications in problem situations.
 Precalculus will continue to investigate rational functions.
 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
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 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.
 VII. Functions
 B2 – Algebraically construct and analyze new functions.
 C2 – Develop a function to model a situation.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.7 
Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:


2A.7F 
Determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two.
Readiness Standard

Determine
THE SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF RATIONAL EXPRESSIONS WITH INTEGRAL EXPONENTS OF DEGREE ONE AND OF DEGREE TWO
Including, but not limited to:
 Equivalent rational expressions
 Multiplication by a fractional form of 1
 Simplification of rational expressions
 Factorization of numerator and denominator
 Cancellation or dividing out of common factors
 Operations with rational expressions
 Sum and difference
 Common denominator (CD) for both terms by calculating equivalent expressions as needed
 Combination of numerators by addition/subtraction over single common denominator (CD)
 Simplification of answer
 Product and quotient
 Inversion of divisors and conversion to multiplication
 Factorization of numerators and denominators
 Cancellation or dividing out of common factors in numerator and denominator
 Multiplication of remaining numerators and denominators
Note(s):
 Grade Level(s):
 Previous grade levels simplified and performed operations on fractions.
 Algebra II simplifies and performs operations on rational expressions involving variables.
 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).
 D1 – Interpret multiple representations of equations and relationships.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
