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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 08: Investigation and Application of Quadratic Functions SUGGESTED DURATION : 15 days

Unit Overview

Introduction
This unit bundles student expectations that address the analysis of the graphs of quadratic functions and their key characteristics and determines the effects of parameter changes of quadratic functions. Quadratic functions are used to model, investigate and make predictions. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Unit 01, students solved linear equations. In Unit 03, students were introduced to parameter changes on linear functions. In Units 06 and 07, students performed operations on second degree polynomials, factored second degree polynomials, and solved quadratic equations.

During this Unit
Students graph quadratic functions on the coordinate plane identifying key attributes, including y-intercept, x-intercept(s), zeros, maximum value, minimum value, vertex, and the equation of the axis of symmetry, when applicable. Students determine the domain and range, representing the domain and range using inequality notation and verbal descriptions. Students describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions and write quadratic functions when given real solutions and graphs of their related equations. Students write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form f(x) = a(xh)2 + k, and rewrite the equation from vertex form to standard form f(x) = ax2 + bx + c. Students formulate quadratic functions for real-world problem situations over an appropriate domain and range given various attributes, identify key attributes in terms of the problem situation, and justify the meaning of key attributes in terms of the problem situation. Students determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d and identify effects of parameter changes of quadratic functions in terms of the problem situation.

After this Unit
In Unit 11, students will review quadratic equations and functions and their applications in the real-world. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving quadratic functions and equations.

In Algebra I, analyzing quadratic functions, their key attributes, and transformations of the quadratic parent function are identified as STAAR Readiness Standards A.6A, A.7A, and A.7C and subsumed under STAAR Reporting Category 4: Quadratic Functions and Equations. Writing quadratic functions using various methods and describing the relationship between linear factors and zeros are identified as STAAR Supporting Standards A.6B, A.6C, A.7B, and A.8B and subsumed under STAAR Reporting Category 4: Quadratic Functions and Equations. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning C1; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to Algebra Standards for Grades 9 – 12 (2002) from the National Council of Teachers of Mathematics (NCTM), “Fluency with algebraic symbolism helps students represent and solve problems in many areas of the curriculum” (p. 300). According to Algebra Standards for Grades 9 – 12 (2000) from the NCTM, high school algebra also should provide students with insights into mathematical abstraction and structure. In Grades 9 – 12, students should develop an understanding of the algebraic properties that govern the manipulation of symbols in expressions, equations, and inequalities. They should become fluent in performing such manipulations by appropriate means to solve equations and inequalities, to generate equivalent forms of expressions or functions, or to prove general results. According to the National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics (2000), students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12:

“High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions.” (NCTM, 2002, p. 3)

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Algebra standards for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgments about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Quadratic functions are characterized by a rate of change that changes at a constant rate and can be used to describe, model, and make predictions about problem situations.
• What kinds of mathematical and real-world situations can be modeled by quadratic functions?
• What graphs, key attributes, and characteristics are unique to quadratic functions?
• What pattern of covariation is associated with quadratic functions?
• How can the key attributes of quadratic functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a quadratic function?
• What are the real-world meanings of the key attributes of a quadratic function model?
• How can the key attributes of a quadratic function be used to make predictions and critical judgments?
• What relationships exist between the linear factors of quadratic expressions and the zeros of their associated quadratic functions?
• What relationships exist between the algebraic forms of a quadratic function and the graph and key attributes of the function?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
• How can equations be used to represent relationships between quantities?
• What methods can be used to write quadratic equations?
• How does the given information and/or representation influence the selection of an efficient method for writing quadratic equations?
• How are properties and operational understandings used to transform quadratic equations?
• Functions, Equations, and Inequalities
• Attributes of functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Vertex
• Axis of symmetry
• Functions and Equations
• Patterns, Operations, and Properties
• Relations and Generalizations
• Statistical Relationships
• Regression methods
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgments about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Quadratic functions are characterized by a rate of change that changes at a constant rate and can be used to describe, model, and make predictions about problem situations.
• What kinds of mathematical and real-world situations can be modeled by quadratic functions?
• What graphs, key attributes, and characteristics are unique to quadratic functions?
• What pattern of covariation is associated with quadratic functions?
• How can the key attributes of quadratic functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a quadratic function?
• What are the real-world meanings of the key attributes of a quadratic function model?
• How can the key attributes of a quadratic function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How are functions …
• shifted?
• scaled?
• reflected?
• How do transformations affect the …
• representations
• key attributes
… of a function?
• Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
• How can equations be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write quadratic equations?
• How does the given information and/or representation influence the selection of an efficient method for writing quadratic equations?
• How are properties and operational understandings used to transform quadratic equations?
• Functions, Equations, and Inequalities
• Attributes of functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Vertex
• Axis of symmetry
• Functions and Equations
• Relations and Generalizations
• Statistical Relationships
• Regression methods
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that when evaluating f(x) = –x2, the negative is included in the square rather than understanding it is a coefficient of –1 that reflects the graph of the quadratic parent function over the x-axis.
• Some students may not connect that the root or solution of a quadratic equation set equal to zero is the same as the x-intercept(s) or zero(s) when the related quadratic function is graphed.

Unit Vocabulary

• Axis of symmetry – line passing through the vertex of a parabola that divides the parabola into two congruent halves
• 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.
• Domain – set of input values for the independent variable over which the function is defined
• Inequality notation – notation in which the solution is represented by an inequality statement
• Maximum – graph opens downward, negative a value
• Minimum – graph opens upward, positive a value
• Parent functions – set of basic functions from which related functions are derived by transformations
• Quadratic expression – a second-degree polynomial expression that can be described by ax2 + bx + c, where a ≠ 0
• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Range – set of output values for the dependent variable over which the function is defined
• Symmetric points – the image and pre-image points reflected across the axis of symmetry of the parabola
• Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
• x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

 Area Dilation Finite difference Horizontal compression Horizontal shift Horizontal stretch Linear factors Parabola Parameter change Quadratic equation Reflection Roots Solutions Square roots Standard form Vertex form Vertical compression Vertical shift Vertical stretch
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway –Resources Aligned to Algebra I Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
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:
• X. Connections
A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving 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 non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
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.

Determine, Represent

THE DOMAIN AND RANGE OF QUADRATIC FUNCTIONS USING INEQUALITIES

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• 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 real-world problem situations
• Reasonable domain and range for the real-world problem situation
• Comparison of domain and range of function model to appropriate domain and range for real-world 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 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) = ax2+ bx + c).
Supporting Standard

Write

EQUATIONS OF QUADRATIC FUNCTIONS GIVEN THE VERTEX AND ANOTHER POINT ON THE GRAPH IN VERTEX FORM (f(x) = a(xh)2 + k)

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 , and the graph of the function is always parabolic or U-shaped
• 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(xh)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) = ax2 + bx + c)

Including, but not limited to:

• Vertex form: f(x) = a(x – h)2 + k
• Standard form: f(x) = ax2 + bx + c
• Mathematical problem situations
• Real-world problem situations

Note(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 second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Representations of quadratic functions
• Graphs
• Algebraic generalizations
• Comparisons of quadratic equations (0 = ax2 + bx + c) and quadratic functions (y = ax2 + bx + c)
• Comparisons of zeros/x-intercepts of quadratic functions and solutions/roots of quadratic equations
• Comparisons of solutions/roots and factors of the quadratic equation
• Solutions/roots: r1 and r2
• Factors: (xr1)(xr2)
• Quadratic equation: 0 = (xr1)(xr2)
• Zeros/x-intercepts and factors of the quadratic function
• Zeros/x-intercepts: z1 and z2 or (z1, 0) and (z2, 0)
• Factors: (xz1)(xz2)
• Quadratic function: f(x) = (xz1)(xz2)
• Multiple functions with the same solutions are possible depending on scalar multiples or the “a” value.

Note(s):

• Algebra I introduces writing quadratic functions from solutions/roots and zeros/x-intercepts.
• 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 x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry.

Graph

QUADRATIC FUNCTIONS ON THE COORDINATE PLANE

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Graphs of quadratic functions with and without technology
• Algebraic generalizations
• Real-world problem situations involving quadratic functions

Use

THE GRAPH OF A QUADRATIC FUNCTION TO IDENTIFY KEY ATTRIBUTES, IF POSSIBLE, INCLUDING x-INTERCEPT, y-INTERCEPT, 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) = ax2 + bx + c
• Vertex form: f(x) = a(xh)2 + k
• Characteristics of quadratic functions
• Intercepts/Zeros
• x-intercept – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• Zeros – the value(s) of x such that the y value of the relation equals zero
• y-intercept – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0,y)
• Denoted as c in f(x) = ax2 + bx + c
• Denoted as ah2 + k in f(x) = a(xh)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 pre-image points reflected across the axis of symmetry of the parabola
• Real-world problem situations involving quadratic functions
• Analysis and conclusions for real-world problem situations using key attributes

Note(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 second-degree polynomial expression that can be described by ax2 + bx + c, where a ≠ 0
• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic, or U-shaped
• Comparisons of quadratic equations (0 = ax2 + bx + c) and quadratic functions (y = ax2 + bx + c)
• Comparisons of zeros/x-intercepts of quadratic functions and solutions/roots of quadratic equations
• Comparisons of solutions/roots and linear factors of the quadratic equation
• Solutions/roots: r1 and r2
• Linear factors: (xr1)(xr2)
• Quadratic equation: 0 = (xr1)(xr2)
• Comparisons of zeros/x-intercepts and factors of the quadratic function
• Zeros/x-intercepts: z1 and z2 or (z1, 0) and (z2, 0)
• Linear factors: (xz1)(xz2)
• Quadratic function: f(x) = (xz1)(xz2)
• When a quadratic equation is set equal to zero, solutions and roots of the quadratic equation are equal to the zeros and x-intercepts of the quadratic function.

Note(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) = x2 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.

Determine

THE EFFECTS ON THE GRAPH OF THE PARENT FUNCTION f(x) = x2 WHEN f(x) IS REPLACED BY af(x), f(x) + d, f(xc), 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)
• y = x2
• f(x) = x2
• 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) = x2
• Effects on the graph of the quadratic parent function f(x) = x2 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 (x-axis)
• Effects on the graph of the quadratic parent function f(x) = x2 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 y-axis
• Effects on the graph of the quadratic parent function f(x) = x2 when f(x) is replaced by f(xc)
• 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) = x2 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 x-and y-intercepts
• For af(x) and f(bx), there are no changes to the x-and y-intercepts..
• For f(xc),
• If c < 0, then the x-intercept shifts or translates left by |c| units, (c, 0).
• If c > 0, then the x-intercept shifts or translates right by |c| units (c, 0).
• For f(x) + d,
• If d < 0, then the y-intercept shifts or translates up by |d| units (0, d).
• If d > 0, then the y-intercept 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):

• 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 real-world 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 real-world problems.
Supporting Standard

Write

QUADRATIC FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Data representations
• Real-world problem situations
• Data collections using technology
• Technology to determine a function model using quadratic regression

To Estimate, To Make

SOLUTIONS AND PREDICTIONS FOR REAL-WORLD PROBLEMS

Including, but not limited to:

• Data representations
• Real-world 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 real-world problem situations
• Vertex (maximum, minimum)
• Intercepts (y-intercepts, x-intercepts, zeros)

Note(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 