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:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

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) = , f(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
Including, but not limited to:
 Representations of functions, including graphs, tables, and algebraic generalizations
 Square root, f(x) =
 Rational (reciprocal of x), f(x) =
 Cubic, f(x) = x^{3}
 Cube root, f(x) =
 Exponential, f(x) = b^{x}, where b is 2, 10, and e
 Absolute value, f(x) = x
 Logarithmic, f(x) = log_{b}(x), where b is 2, 10, and e
 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, ASYMPTOTIC BEHAVIOR, AND MAXIMUM AND MINIMUM GIVEN AN INTERVAL, WHEN APPLICABLE
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 ∈ ℜ
 Ex: x ∈ ℜ
 Ex: –3 < y ≤ 6, y ∈ ℜ
 Ex: y ≥ 0, y ∈ Ζ
 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: {xx ∈ ℜ, x < 5}
 Ex: {xx ∈ ℜ}
 Ex: {yy ∈ ℜ, –3 < y ≤ 6}
 Ex: {yy ∈ Ζ, y ≥ 0}
 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]
 Ex: [0, ∞)
 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
 Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
 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.
 Maximum and minimum (extrema)
 Relative maximum – largest ycoordinate, or value, a function takes over a given interval of the curve
 Relative minimum – smallest ycoordinate, or value, a function takes over a given interval of 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 ℜ represents the set of real numbers, and the notation Ζ 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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

2A.2C 
Describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range.
Readiness Standard

Describe, Analyze
THE RELATIONSHIP BETWEEN A FUNCTION AND ITS INVERSE (QUADRATIC AND SQUARE ROOT, LOGARITHMIC AND EXPONENTIAL), INCLUDING THE RESTRICTION(S) ON DOMAIN, WHICH WILL RESTRICT ITS RANGE
Including, but not limited to:
 Relationships between functions and their inverses
 All inverses of functions are relations.
 Inverses of onetoone functions are functions.
 Inverses of functions that are not onetoone can be made functions by restricting the domain of the original function, f(x).
 Characteristics of inverse relations
 Interchange of independent (x) and dependent (y) coordinates in ordered pairs
 Reflection over y = x
 Domain and range of the function versus domain and range of the inverse of the given function
 Functionality of the inverse of the given function
 Quadratic function and square root function, f(x) = x^{2} and f(x) =
 Restrictions on domain when using positive square root
 Restrictions on domain when using negative square root
 Cubic function and cube root function, f(x) = x^{3} and g(x) =
 Exponential function and logarithmic function, f(x) = b^{x} and g(x) = log_{b}(x) where b is 2, 10, and e
Note(s):
 Grade Level(s):
 Algebra I determined if relations represented a function.
 Algebra II introduces inverse of a function and restricting domain to maintain functionality.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.B. Algebraic Reasoning – Manipulating expressions
 II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

2A.3 
Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to:


2A.3B 
Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.
Readiness Standard

Note(s):
 Grade Level(s):
 Algebra I solved systems of two linear equations in two variables using graphs, tables, and algebraic methods.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

2A.3C 
Solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation.
Supporting Standard

Solve
SYSTEMS OF TWO EQUATIONS IN TWO VARIABLES CONSISTING OF A LINEAR EQUATION AND A QUADRATIC EQUATION, ALGEBRAICALLY
Including, but not limited to:
 Two equations in two variables
 One linear equation
 One quadratic equation
 Methods for solving systems of equations consisting of one linear equation and one quadratic equation
 Tables
 Graphs
 Identification of possible solutions in terms of points of intersection
 Algebraic methods
 Substitution of linear equation into quadratic
 Solve by factoring
 Solve by quadratic formula
 Solve by completing the square
Note(s):
 Grade Level(s):
 Algebra I solved systems of two linear equations in two variables using graphs, tables, and algebraic methods.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

2A.3F 
Solve systems of two or more linear inequalities in two variables.
Supporting Standard

Solve
SYSTEMS OF TWO OR MORE LINEAR INEQUALITIES IN TWO VARIABLES
Including, but not limited to:
 Systems of linear inequalities in two variables
 Two variables or unknowns
 Two or more inequalities
 Method for solving system of inequalities
 Graphical analysis of system
 Graphing of each function
 Shading of inequality region for each function
 Representation of the solution as points in the solution region
Note(s):
 Grade Level(s):
 Algebra I solved systems of two linear inequalities in two variables using graphs, tables, and algebraic methods.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

2A.4 
Quadratic and square root functions, equations, and inequalities. The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:


2A.4B 
Write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
Readiness Standard

Write
THE EQUATION OF A PARABOLA USING GIVEN ATTRIBUTES, INCLUDING VERTEX, FOCUS, DIRECTRIX, AXIS OF SYMMETRY, AND DIRECTION OF OPENING
Including, but not limited to:
 Parabola – the locus of points, P, such that the distance from P to a point F (the focus) is equal to the distance from P to a line q(the directrix)
 Quadratic equation representations
 Standard form
 Vertical axis of symmetry: y = ax^{2} + bx + c
 Horizontal axis of symmetry: x = ay^{2} + by + c
 Vertex form
 Vertical axis of symmetry: y = a(x – h)^{2} + k
 Horizontal axis of symmetry: x = a(y – k)^{2} + h
 Parabola (conic form)
 Vertical axis of symmetry: (x – h)^{2} = 4p(y – k)
 Horizontal axis of symmetry: (y – k)^{2} = 4p(x – h)
 Connection between a and p in the vertex form and parabola (conic form)
 a =
 Attributes of a parabola
 Vertex: (h, k)
 Axis of symmetry
 Vertical axis of symmetry for a parabola that opens up or down: x = h
 Horizontal axis of symmetry for a parabola that opens to the right or to the left: y = k
 Positive value of a or p, the parabola opens up or to the right
 Negative value of a or p, the parabola opens down or to the left
 p – distance from vertex to directrix or distance from vertex to focus
 Directrix – horizontal or vertical line not passing through the focus whose distance from the vertex is p and is perpendicular to the axis of symmetry
 Focus – point not on the directrix whose distance from the vertex is p and lies on the axis of symmetry
Note(s):
 Grade Level(s):
 Algebra I wrote quadratic equations in vertex form (f(x) = a(x – h)^{2} + k), and rewrote from vertex form to standard form (f(x) = ax^{2} + bx + c).
 Precalculus will address parabolas as conic sections.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

2A.4C 
Determine the effect on the graph of f(x) = when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x  c) for specific positive and negative values of a, b, c, and d.
Readiness Standard

Determine
THE EFFECT ON THE GRAPH OF f(x) = WHEN f(x) IS REPLACED BY af(x), f(x) + d, f(bx), AND f(x – c) FOR SPECIFIC POSITIVE AND NEGATIVE VALUES OF a, b, c, AND d
Including, but not limited to:
 General form of the square root 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 f(x) =
 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 or right by c units
 Left shift when c < 0
 For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
 Right shift when c > 0
 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 up or down by d units
 Down shift when d < 0
 Up shift when d > 0
 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
 Effects of multiple parameter changes
 Mathematical problem situation
 Realworld problem situations
Note(s):
 Grade Level(s):
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.

2A.4F 
Solve quadratic and square root equations.
Readiness Standard

Solve
QUADRATIC AND SQUARE ROOT EQUATIONS
Including, but not limited to:
 Methods for solving quadratic equations with and without technology
 Tables
 Zeros – the value(s) of x such that the y value of the relation equals zero
 Domain values with equal range values
 Graphs
 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
 Algebraic methods
 Factoring
 Solving equations by taking square roots
 Solving quadratic equations using absolute value
 x^{2} = 25, x = 5; therefore, x = ±5
 Completing the square
 Quadratic formula, x =
 The discriminant, b^{2} – 4ac, can be used to analyze types of solutions for quadratic equations.
 b^{2} – 4ac = 0, one rational double root
 b^{2} – 4ac > 0 and perfect square, two rational roots
 b^{2} – 4ac > 0 and not perfect square, two irrational roots (conjugates)
 b^{2} – 4ac < 0, two imaginary roots (conjugates)
 Connections between solutions and roots of quadratic equations to the zeros and xintercepts of the related function
 Complex number system
 Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
 Imaginary number – a number in the form of bi where b is a real number and i =
 i^{2} = –1
 i =
 Complex conjugates – complex numbers having the same real part but an opposite imaginary part
 Operations with complex numbers, with and without technology
 Complex solutions for quadratic equations
 One real solution
 Two real solutions
 Two rational roots
 Two irrational root conjugates
 Two complex solutions
 Two complex root conjugates
 Methods for solving square root equations with and without technology
 Tables
 Zeros – the values of x such that the y value of the relation equals zero
 Domain values with equal range values
 Graphs
 xintercept(s) – xcoordinate 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
 Algebraic methods
 Identification of extraneous solutions
 Reasonableness of solutions
Note(s):
 Grade Level(s):
 Algebra I solved quadratic equations having real solutions using tables, graphs, factoring, completing the square, and the quadratic formula.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

2A.5 
Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:


2A.5A 
Determine the effects on the key attributes on the graphs of f(x) = b^{x} and f(x) = log_{b}(x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x  c) for specific positive and negative real values of a, c, and d.
Readiness Standard

Determine
THE EFFECTS ON THE KEY ATTRIBUTES ON THE GRAPHS OF f(x) = b^{x} AND f(x) = log_{b}(x) WHERE b IS 2, 10, AND e WHEN f(x) IS REPLACED BY af(x), f(x) + d, and f(x – c) FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, c, AND d
Including, but not limited to:
 General form of the power function
 Exponential functions, f(x) = b^{x}, where b is 2, 10, and e
 f(x) = 2^{x}; f(x) = 10^{x}; f(x) = e^{x}
 Logarithmic functions, y = log_{b}(x), where b is 2, 10, and e
 f(x) = log_{2}(x); f(x) = log_{10}(x) or f(x) = log(x); f(x) = log_{e}(x) or f(x) = ln(x)
 Representations with and without technology
 Graphs
 Tables
 Verbal descriptions
 Algebraic generalizations
 Key attributes
 Effects on the graphs of f(x) = b^{x} and y = log_{b}(x) when parameters a, b, c, and d are changed in f(x) = a • b^{(x – c)} + d and f(x) = a • log_{b}(x – c) + d
 Effects on the graphs of f(x) = 2^{x} and f(x) = log_{2}(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 graphs of f(x) = 10^{x} and f(x) = log(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 graphs of f(x) = e^{x} and f(x) = ln(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 functions and f(x) = b^{x} and y = log_{b}(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
 Effects of multiple parameter changes
 Mathematical problem situation
 Effects of parameter changes in realworld problem situations
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 continues to investigate the exponential parent function and introduces logarithmic parent function and transformations of both functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.

2A.5D 
Solve exponential equations of the form y = ab^{x} where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions.
Readiness Standard

Solve
EXPONENTIAL EQUATIONS OF THE FORM y = ab^{x} WHERE a IS A NONZERO REAL NUMBER AND b IS GREATER THAN ZERO AND NOT EQUAL TO ONE
Including, but not limited to:
 Exponential equation, y = ab^{x}
 a – initial value at x = 0
 b – common ratio
 Solving exponential equations
 Application of laws (properties) of exponents
 Application of logarithms as necessary
 Realworld problem situations modeled by exponential functions
 Exponential growth
 f(x) = ab^{x}, where b > 1
 f(x) = ae^{k}^{x}, where k > 0
 Exponential decay
 f(x) = ab^{x}, where 0 < b < 1
 f(x) = ae^{k}^{x}, where k < 0
Solve
SINGLE LOGARITHMIC EQUATIONS HAVING REAL SOLUTIONS
Including, but not limited to:
 Single logarithmic equation, y = log_{b}(x)
 x – argument
 b – base
 y – exponent
 Solving logarithmic equations
 Transformation to exponential form as necessary
 Realworld problem situations modeled by logarithmic functions
Note(s):
 Grade Level(s):
 Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = a • b^{x}.
 Algebra II solves exponential equations algebraically.
 Algebra II introduces logarithms and solving logarithmic equations.
 Precalculus will use properties of logarithms to solve equations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

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.6B 
Solve cube root equations that have real roots.
Supporting Standard

Solve
CUBE ROOT EQUATIONS THAT HAVE REAL ROOTS
Including, but not limited to:
 Application of laws (properties) of exponents
 Application of cube roots to solve cubic equations
 Applications of cubics to solve cube root equations
 Reasonableness of solutions
 Substitution of solutions into original problem
 Graphical analysis
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra II introduces cubic and cube root functions and solving cube root equations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

2A.6E 
Solve absolute value linear equations.
Readiness Standard

Solve
ABSOLUTE VALUE LINEAR EQUATIONS
Including, but not limited to:
 Methods for solving absolute value linear equations with and without technology
 Graphs
 Algebraic methods
 Solving process
 Transform the equation so that the absolute value expression is on one side of the equation and all other variable terms and constants are on the other side of the equation.
 Separate the equation into two parts divided by “or”:
 Expression inside the absolute value equal to the other side of the equation
 Expression inside the absolute value equal to the opposite of the other side of the equation
 x = 5 → x = 5 or x = –5
 Extraneous solution – solution derived by solving the equation algebraically that is not a true solution of the equation and will not be valid when substituted back into the original equation
 Solving absolute value equations involves separating the absolute value into both the possible positive value inside the absolute and the possible negative value inside the absolute. In the case of x = 2, The x value can be either positive or negative 2. However, this is not a reversible situation, x = 2 but x ≠ –2.
 Justification of solutions with and without technology
 Graphs
 Substitution of solutions into original functions
 Extraneous solutions
 Realworld problem situations modeled by absolute value functions
 Justification of reasonableness of solutions in terms of the realworld problem situations or data collections
Note(s):
 Grade Level(s):
 Grade 6 defined absolute value and identified the absolute value of a number.
 Algebra II introduces the absolute value equation and its applications.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

2A.6F 
Solve absolute value linear inequalities.
Supporting Standard

Solve
ABSOLUTE VALUE LINEAR INEQUALITIES
Including, but not limited to:
 Methods for solving absolute value linear inequalities with and without technology
 Graphs
 Algebraic methods
 Solving process
 Isolation of absolute expression on one side of the inequality
 Separation of the inequality into two parts
 Greater than (>) or greater than or equal to (≥)
 First part: expression inside the absolute value set greater than or greater than or equal to other side of the inequality
 Second part: expression inside the absolute value set less than or less than or equal to the opposite of the other side of the inequality
 Parts separated by “ or ”
 Representation of solutions
 Symbolic notation
 Interval notation
 Graphical notation
 Less than (<) or less than or equal to (≤)
 First part: expression inside the absolute value set less than or less than or equal to other side of the inequality
 Second part: expression inside the absolute value set greater than or greater than or equal to the opposite of the other side of the inequality
 Parts separated by “ and ”
 Representation of solutions
 Symbolic notation
 Interval notation
 Graphical notation
 Justification of solutions of absolute value inequalities with and without technology
 Graphs
 Substitution of solutions into original functions
 Removal of extraneous solutions
Note(s):
 Grade Level(s):
 Grade 6 defined absolute value and identified the absolute value of a number.
 Algebra II introduces absolute value inequalities.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

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:
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.

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 functions
 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:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

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 one 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 ∈, ℜ, x < 5 or x > 8}
 Ex: {yy ∈, ℜ, –3 < y < 6}
 Ex: {xx ∈, ℜ, 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:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.

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:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.

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.7D 
Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods.
Supporting Standard

Determine
THE LINEAR FACTORS OF A POLYNOMIAL FUNCTION OF DEGREE THREE AND OF DEGREE FOUR USING ALGEBRAIC METHODS
Including, but not limited to:
Note(s):
 Grade Level(s):
 Algebra I introduced factorization of polynomials of degree one and degree two.
 Algebra II introduces synthetic division of degree three and four polynomials by degree one polynomials.
 Algebra II introduces depression of polynomials to determine roots and factors of the polynomial.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.B. Algebraic Reasoning – Manipulating expressions
 II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.

2A.7E 
Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping.
Readiness Standard

Determine
LINEAR AND QUADRATIC FACTORS OF A POLYNOMIAL EXPRESSION OF DEGREE THREE AND OF DEGREE FOUR, INCLUDING FACTORING THE SUM AND DIFFERENCE OF TWO CUBES AND FACTORING BY GROUPING
Including, but not limited to:
 Determination of linear and quadratic factors by factorization
 Greatest common factor
 Difference of squares: a^{2} – b^{2} = (a + b)(a – b)
 Trinomials
 Sum of cubes: a^{3} + b^{3} = (a + b)(a^{2} – ab + b^{2})
 Difference of cubes: a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})
 Grouping methods
 Verify factorization by remultiplying the factors.
 Factor using nonalgebraic techinques to determine rational roots
Note(s):
 Grade Level(s):
 Algebra I introduced factorization of polynomials of degree one and degree two.
 Algebra II introduces factorization of polynomials of degree three and degree four.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.B. Algebraic Reasoning – Manipulating expressions
 II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.

2A.8 
Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to:


2A.8C 
Predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.
Readiness Standard

Predict, Make
DECISIONS AND CRITICAL JUDGMENTS FROM A GIVEN SET OF DATA USING LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Including, but not limited to:
 Mathematical and realworld problem situations modeled by linear, quadratic, and exponential functions and equations
 Predictions, decisions, and critical judgments from function models
 Justification of reasonableness of solutions in terms of mathematical and realworld problem situations
 Mathematical justification
 Substitution in original problem
 Justification for predictions using the coefficient of determination, r^{2}
Note(s):
 Grade Level(s):
 Algebra I introduced the linear, quadratic, and exponential functions.
 Algebra I introduced the correlation coefficient as a measure of the strength of linear association.
 Algebra I applied linear, quadratic, and exponential functions to model and make predictions in realworld problem situations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.2. Make connections between geometry, statistics, and probability.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
