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


A.1A 
Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

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:

A.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.
Process Standard

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

A.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.
Process Standard

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

A.1D 
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

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

A.1E 
Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

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

A.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

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:

A.1G 
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

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

A.6 
Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to:


A.6A 
Determine the domain and range of quadratic functions and represent the domain and range using inequalities.
Readiness Standard

Determine, Represent
THE DOMAIN AND RANGE OF QUADRATIC FUNCTIONS USING INEQUALITIES
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2 }+ bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Domain and range of quadratic functions in mathematical problem situations
 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
 Domain and range of quadratic functions in realworld problem situations
 Reasonable domain and range for the realworld problem situation
 Comparison of domain and range of function model to appropriate domain and range for realworld problem situation
 Inequality representations
 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
 Ex: x ∈ ℜ
 Ex: –3 < y ≤ 6
 Ex: y ≥ 0, y ∈ Ζ
Note(s):
 Grade Level(s):
 Grade 6 identified independent and dependent quantities.
 Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
 Algebra I introduces quadratic functions.
 Algebra I introduces the concept of domain and range of a function.
 Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
 Algebra II will extend the concept of domain and range.
 Algebra II will introduce representing domain and range using interval and set notation.
 Algebra II will continue to investigate quadratic functions.
 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

A.6B 
Write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x  h)^{2}+ k), and rewrite the equation from vertex form to standard form (f(x) = ax^{2}+ bx + c).
Supporting Standard

Write
EQUATIONS OF QUADRATIC FUNCTIONS GIVEN THE VERTEX AND ANOTHER POINT ON THE GRAPH IN VERTEX FORM (f(x) = a(x – h)^{2} + k)
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 , and the graph of the function is always parabolic or Ushaped
 Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
 Determination of an algebraic representation for a quadratic function in vertex form, y = a(x – h)^{2} + k
 Given vertex (h, k)
 Given point (x, y) or (x, f(x))
Rewrite
THE EQUATION FROM VERTEX FORM TO STANDARD FORM (f(x) = ax^{2 }+ bx + c)
Including, but not limited to:
 Vertex form: f(x) = a(x – h)^{2} + k
 Standard form: f(x) = ax^{2 }+ bx + c
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra I introduces writing quadratic functions in vertex and standard form.
 Algebra II will transform quadratic functions from standard to vertex form.
 Algebra II will write equations of parabolas given the vertex and other attributes.
 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

A.6C 
Write quadratic functions when given real solutions and graphs of their related equations.
Supporting Standard

Write
QUADRATIC FUNCTIONS WHEN GIVEN REAL SOLUTIONS AND GRAPHS OF THEIR RELATED EQUATIONS
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Representations of quadratic functions
 Graphs
 Algebraic generalizations
 Comparisons of quadratic equations (0 = ax^{2} + bx + c) and quadratic functions (y = ax^{2} + bx + c)
 Comparisons of zeros/xintercepts of quadratic functions and solutions/roots of quadratic equations
 Comparisons of solutions/roots and factors of the quadratic equation
 Solutions/roots: r_{1} and r_{2}
 Factors: (x – r_{1})(x – r_{2})
 Quadratic equation: 0 = (x – r_{1})(x – r_{2})
 Zeros/xintercepts and factors of the quadratic function
 Zeros/xintercepts: z_{1} and z_{2} or (z_{1}, 0) and (z_{2}, 0)
 Factors: (x – z_{1})(x – z_{2})
 Quadratic function: f(x) = (x – z_{1})(x – z_{2})
 Multiple functions with the same solutions are possible depending on scalar multiples or the “a” value.
Note(s):
 Grade Level(s):
 Algebra I introduces writing quadratic functions from solutions/roots and zeros/xintercepts.
 Algebra II will write quadratic functions given three points on the parabola.
 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

A.7 
Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations. The student is expected to:


A.7A 
Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including xintercept, yintercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry.
Readiness Standard

Graph
QUADRATIC FUNCTIONS ON THE COORDINATE PLANE
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Graphs of quadratic functions with and without technology
 Algebraic generalizations
 Realworld problem situations involving quadratic functions
Use
THE GRAPH OF A QUADRATIC FUNCTION TO IDENTIFY KEY ATTRIBUTES, IF POSSIBLE, INCLUDING xINTERCEPT, yINTERCEPT, ZEROS, MAXIMUM VALUE, MINIMUM VALUES, VERTEX, AND THE EQUATION OF THE AXIS OF SYMMETRY
Including, but not limited to:
 Representation of quadratic functions
 Standard form: f(x) = ax^{2} + bx + c
 Vertex form: f(x) = a(x – h)^{2} + k
 Characteristics of quadratic functions
 Intercepts/Zeros
 xintercept – 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 – y coordinate of a point at which the relation crosses the yaxis, meaning the x coordinate equals zero, (0,y)
 Denoted as c in f(x) = ax^{2} + bx + c
 Denoted as ah^{2} + k in f(x) = a(x – h)^{2} + k
 Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
 Graphically, the maximum or minimum point of the parabola
 Algebraically x = and solving for y
 From standard form: x = and solving for y
 From vertex form: (h, k)
 Maximum – graph opens downward, negative a value
 Minimum – graph opens upward, positive a value
 Axis of symmetry – line passing through the vertex of a parabola that divides the parabola into two congruent halves
 Equation of the axis of symmetry
 From standard form: x =
 Vertex form: x = h
 Symmetric points – the image and preimage points reflected across the axis of symmetry of the parabola
 Realworld problem situations involving quadratic functions
 Analysis and conclusions for realworld problem situations using key attributes
Note(s):
 Grade Level(s):
 Algebra I introduces quadratic functions.
 Algebra I introduces the key attributes of a quadratic function.
 Algebra II will continue to investigate and apply quadratic functions and equations and write the equation of a parabola from given attributes, including direction of opening.
 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.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.7B 
Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions.
Supporting Standard

Describe
THE RELATIONSHIP BETWEEN THE LINEAR FACTORS OF QUADRATIC EXPRESSIONS AND THE ZEROS OF THEIR ASSOCIATED QUADRATIC FUNCTIONS
Including, but not limited to:
 Quadratic expression – a seconddegree polynomial expression that can be described by ax^{2} + bx + c, where a ≠ 0
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic, or Ushaped
 Comparisons of quadratic equations (0 = ax^{2} + bx + c) and quadratic functions (y = ax^{2} + bx + c)
 Comparisons of zeros/xintercepts of quadratic functions and solutions/roots of quadratic equations
 Comparisons of solutions/roots and linear factors of the quadratic equation
 Solutions/roots: r_{1} and r_{2}
 Linear factors: (x – r_{1})(x – r_{2})
 Quadratic equation: 0 = (x – r_{1})(x – r_{2})
 Comparisons of zeros/xintercepts and factors of the quadratic function
 Zeros/xintercepts: z_{1} and z_{2} or (z_{1}, 0) and (z_{2}, 0)
 Linear factors: (x – z_{1})(x – z_{2})
 Quadratic function: f(x) = (x – z_{1})(x – z_{2})
 When a quadratic equation is set equal to zero, solutions and roots of the quadratic equation are equal to the zeros and xintercepts of the quadratic function.
Note(s):
 Grade Level(s):
 Algebra I introduces quadratic functions.
 Algebra I introduces the connections between the linear factors of a quadratic expression and the zeros of a quadratic function.
 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

A.7C 
Determine the effects on the graph of the parent function 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.
Readiness Standard

Determine
THE EFFECTS ON THE GRAPH OF THE PARENT FUNCTION 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
Including, but not limited to:
 Parent functions – set of basic functions from which related functions are derived by transformations
 General form of quadratic parent function (including equation and function notation)
 Multiple representations
 Graphs
 Tables
 Verbal descriptions
 Algebraic generalizations (including equation and function notation)
 Changes in parameters a, b, c, and d on the graph of the parent function f(x) = x^{2}
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by af(x)
 a ≠ 0
 a > 1, stretches the graph vertically or makes the graph more narrow
 0 < a < 1, compresses the graph vertically or makes the graph wider
 Opposite of a reflects the graph vertically over the horizontal axis (xaxis)
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by f(bx)
 b ≠ 0
 b > 1, the graph compresses horizontally or makes the graph more narrow
 0 < b < 1, the graph stretches horizontally or makes the graph wider
 b < 0, reflects horizontally over the yaxis
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by f(x – c)
 c = 0, no horizontal shift or translation
 Horizontal shift or translation left or right by c units
 Left shift or translation when c < 0
 For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left or shifts left or translates left two units.
 Right shift or translation when c > 0
 For f(x – 2), c = 2, and the function moves to the right or shifts right or translates right two units
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by f(x) + d
 d = 0, no vertical shift or translation
 Vertical shift or translation up or down
 Shift or translation down when d < 0
 For f(x) – 2, d = –2, and the function moves down or shifts down or translates down two units.
 Shift or translation up when d > 0
 For f(x) + 2, d = 2, and the function moves up or shifts up or translates up two units.
 Generalizations of parameter changes to f(x) = x on the xand yintercepts
 For af(x) and f(bx), there are no changes to the xand yintercepts..
 For f(x – c),
 If c < 0, then the xintercept shifts or translates left by c units, (c, 0).
 If c > 0, then the xintercept shifts or translates right by c units (c, 0).
 For f(x) + d,
 If d < 0, then the yintercept shifts or translates up by d units (0, d).
 If d > 0, then the yintercept shifts or translates down by d units (0, d).
 Graphical representation given the algebraic representation or parameter changes
 Algebraic representation given the graphical representation or parameter changes
 Descriptions of the effects on the domain and range by the parameter changes
 Combined parameter changes
Note(s):
 Grade Level(s):
 Algebra I introduces effects of parameter changes a, b, c, and d on the quadratic parent function.
 Algebra II will extend effects of parameter changes to other parent 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

A.8 
Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on realworld data. The student is expected to:


A.8B 
Write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for realworld problems.
Supporting Standard

Write
QUADRATIC FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Data representations
 Realworld problem situations
 Data collections using technology
 Technology to determine a function model using quadratic regression
To Estimate, To Make
SOLUTIONS AND PREDICTIONS FOR REALWORLD PROBLEMS
Including, but not limited to:
 Data representations
 Realworld problem situations
 Data collections using technology
 Technology to determine a function model using quadratic regression
 Technology to determine key attributes of functions specific to the realworld problem situations
 Vertex (maximum, minimum)
 Intercepts (yintercepts, xintercepts, zeros)
Note(s):
 Grade Level(s):
 Algebra I introduces writing quadratic functions using technology to reasonably fit data.
 Algebra II will continue to investigate and apply quadratic equations.
 Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 I. Numeric reasoning
 C1 – Use estimation to check for errors and reasonableness of solutions.
 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
