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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities SUGGESTED DURATION : 14 days

#### Unit Overview

This unit bundles student expectations that address writing equations of parabolas given various characteristics; writing quadratic functions given three points;  formulating, solving, and determining the reasonableness of solutions to a system of equations consisting of a linear equation and a quadratic equation; and  solving quadratic equations and inequalities. 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 Algebra I Units 07 and 08, students investigated quadratic functions and equations. Students also formulated quadratic models to represent problem situations and applied various methods to solve quadratic equations. In Algebra I Unit 05 and Algebra II Unit 03, students investigated systems of linear equations.

During this unit, students use a system of three equations in three variables to write quadratic functions given three specified points in a plane and justify the quadratic function using the graphing calculator. Students transform quadratic functions from standard form, f(x) = ax2 + bx + c, to vertex form, f(x) = a(x – h)2 + k, and identify attributes of f(x), including vertex, symmetries, maximum and minimum. Students write equations of parabolas from attributes including vertex, focus, directrix, axis of symmetry, and direction of opening. Students define the complex number system and its subsets as well as perform operations (addition, subtraction, multiplication) with complex numbers. Students solve quadratic equations using various methods, including graphing, factoring, completing the square, and the quadratic formula, and verify solutions by graphing and multiplying factors created by roots. Students solve quadratic inequalities graphically and algebraically. Students formulate quadratic equations from tables of data and real-world problem situations, solve the quadratic equations by a method of choice, and justify the solution in terms of the problem situation. Students formulate systems of equations consisting of two equations, the first linear and the second quadratic, solve the system algebraically, and determine the reasonableness of the solution in terms of the problem situation.

After this unit, in Algebra 2 Units 06, 07, 08 and 11, students will continue to apply the concepts of quadratic functions, equations, and inequalities. In subsequent mathematics courses, students will also continue to apply these concepts when quadratic functions, equations, and inequalities arise in problem situations.

According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions, Grades 9-12, understanding of the function concept is essential to describing and analyzing quantities which vary with respect to one another. According to research from the National Council of Teachers of Mathematics (2000), high school algebra should provide students with insights into mathematical abstraction and structure. High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with a higher degree of sophistication. Students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities.

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. (2011). Developing essential understanding of expressions, equations, and functions, grades 6-8. Reston, VA: National Council of Teachers of Mathematics, Inc.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

The complex number system is a way to encompass all number relationships.

• Why is the complex number system used to represent number relationships?
• How are different sets and subsets of numbers related in the complex number system?
• How are sets of numbers within the complex number system used in problem situations?

Equations and inequalities can model problem situations and be solved using various methods.

• Why are equations and inequalities used to model problem situations?
• How are equations and inequalities used to model problem situations?
• What methods can be used to solve equations and inequalities?
• Why is it essential to solve equations and inequalities using various methods?
• How can solutions to equations and inequalities be represented?
• How do the representations of solutions to equations and solutions to inequalities compare?

Relations are algebraic models that describe how two quantities relate to one another. Functions are a subset of relations.

• What are types of relations?
• How can relations be represented?
• Why do some relations not define a function?
• Why do some relations define a function?
• Why can function models describe how two variable quantities change in relation to one another?

Systems of equations can model problem situations and be solved using various methods.

• Why are systems of equations used to model problem situations?
• How are systems of equations used to model problem situations?
• What methods can be used to solve systems of equations?
• Why is it essential to solve systems of equations using various methods?
• How can solutions to systems of equations be represented?

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Algebraic Reasoning

• Equations
• Expressions
• Multiple Representations
• Solve
• Systems of Equations

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

A quadratic function can be determined using a three by three system of equations when given three points in a plane through which the function passes.

• How are the three points used to formulate a three by three system of equations?
• What methods can be used to solve the three by three system of equations?

A quadratic function in standard form, f(x) = ax2 + bx + c, can be transformed to vertex form, f(x) = a(xh)2 + k.

• How are quadratic equations transformed from standard form to vertex form?
• How are quadratic equations transformed from vertex form to standard form?
• What attributes can be determined from the standard form, f(x) = ax2 + bx + c?
• What attributes can be determined from the vertex form, f(x) = a(xh)2 + k?
 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.

Algebraic Reasoning

• Equations
• Multiple Representations
• Relations

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

The representative equation of a parabola can be determined by analyzing its attributes.

• How is a parabola defined as a locus of points? Explain.
• What are the attributes of a parabola?
• In what directions can a parabola open?
• How can the equation of a parabola be used to determine the way the parabola opens?
• Why does a parabola always represent a relation but not always represent a function?
• What is the connection between p in the formula (xh)2 = 4p(y k) and a in the formula y = a(xh)2 + k?
 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.

Numeric Reasoning

• Complex numbers
• Imaginary Numbers
• Multiplication
• Subtraction

Algebraic Reasoning

• Equations
• Equivalence
• Expressions
• Inequalities
• Simplify
• Solve

Associated Mathematical Processes

• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

The complex number system encompasses real numbers, imaginary numbers, and their subsets.

• How are the number systems interrelated?
• What makes up a complex number?
• What is an imaginary number?
• What operations can be performed with complex numbers?
• How are complex numbers applicable in quadratic functions and equations?

Equations and inequalities can be used to model and solve mathematical problem situations.

• What methods can be used to solve quadratic equations?
• What methods can be used to solve quadratic inequalities?
• What are the advantages and disadvantages of various methods used to solve quadratic equations and inequalities?
• What methods can be used to justify the reasonableness of solutions to quadratic equations and inequalities?
• How can roots and their factors be used to determine models for quadratic equations?
 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.

Algebraic Reasoning

• Equations
• Expressions
• Inequalities
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Quadratic functions can be used to model real-world problem situations by analyzing collected data, key attributes, and various representations in order to interpret and make predictions and critical judgments.

• What representations can be used to display quadratic function models?
• What key attributes identify a quadratic parent function model?
• What are the connections between the key attributes of a quadratic function model and the real-world problem situation?
• How can quadratic function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?

Equations and inequalities can be used to model and solve real-world problem situations.

• How are real-world problem situations identified as ones that can be modeled by quadratic equations and inequalities?
• How are quadratic equations used to model problem situations?
• How are quadratic inequalities used to model problem situations?
• What methods can be used to solve quadratic equations?
• What methods can be used to solve quadratic inequalities?
• What are the advantages and disadvantages of various methods used to solve quadratic equations and inequalities?
• What methods can be used to justify the reasonableness of solutions to quadratic equations and inequalities?
 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.

Algebraic Reasoning

• Equations
• Solve
• Systems of Equations

Functions

• Attributes of Functions
• Linear Functions
• Non-Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Systems of equations can be used to model and solve real-world problem situations.

• How are systems of equations used to model problem situations?
• What methods can be used to solve systems of equations?

Systems of equations in two variables consisting of a linear equation and a quadratic equation can be used to model real-world problem situations by analyzing the problem situation and various representations in order to interpret and make predictions and critical judgments.

• What representations can be used to display the system of equations?
• How can the representations of the system of equations be used to interpret and make predictions and critical judgments in terms of the problem situation?
• How can solutions be justified for reasonableness in terms of the problem situation?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the terms zeros, x-intercepts, roots, and solutions are all the same and can be used interchangeably rather than understanding that when an equation is set equal to zero, these will be equivalent, but not at other times. Roots and solutions pertain to equations, while x-intercepts and zeros pertain to functions.
• Some students may think that in order to be a complex number, the number must contain an imaginary part rather than that all numbers can be written in complex form, e.g., 25 can be written as 25 + 0i, and its conjugate is 25 – 0i.
• Some students may think that the x- and y-values in the solution to a system of two linear equations in two variables can never be equal in value.

#### Unit Vocabulary

• Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
• 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
• Imaginary number – a number in the form of bi where b is a real number and i
• |p| –  distance from vertex to directrix or distance from vertex to focus
• 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)
• x-intercept(s)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; the x-intercepts

Related Vocabulary:

 Axis of symmetry Completing the square Discriminant Factoring Fundamental Theorem of Algebra Gaussian method Horizontal shift Inverse matrix Locus of points Maximum Minimum Operations of complex numbers Quadratic equation Quadratic formula Quadratic function Quadratic inequality Quadratic regression rref Real numbers Roots Solutions Standard form, f(x)= ax2 + bx + c Substitution method Symmetric point Transformation Vertex Vertex form, f(x)= a(x - h)2 + k 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 Creator 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 (select CCRS from Standard Set dropdown menu)

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 II Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Bold red text in italics:  Student Expectation identified by TEA as a Readiness Standard for STAAR
• Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
2A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
2A.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.

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
2A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
2A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and 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
2A.1G Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.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.3A Formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic.

Formulate

SYSTEMS OF EQUATIONS, INCLUDING SYSTEMS CONSISTING OF THREE LINEAR EQUATIONS IN THREE VARIABLES AND SYSTEMS CONSISTING OF TWO EQUATIONS, THE FIRST LINEAR AND THE SECOND QUADRATIC

Including, but not limited to:

• Systems of linear equations
• Two equations in two variables
• Three equations in three variables
• Systems of one linear equation and one quadratic equation in two variables
• Real-world problem situations

Note(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. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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
• Methods for solving systems of equations consisting of one linear equation and one quadratic equation
• Tables
• Common points on 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 completing the square

Note(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. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.3D Determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables.
Supporting Standard

Determine

THE REASONABLENESS OF SOLUTIONS TO SYSTEMS OF A LINEAR EQUATION AND A QUADRATIC EQUATION IN TWO VARIABLES

Including, but not limited to:

• Types of equations in system
• Two equations in two variables
• One linear equation
• Justification of reasonableness of solutions to systems of equations
• Tables
• Graphs
• Substitution of solutions into original functions
• Restriction of solutions in terms of real-world problem situations
• Verbal description in terms of real-world problem situations

Note(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. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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.4A Write the quadratic function given three specified points in the plane.
Supporting Standard

Write

THE QUADRATIC FUNCTION GIVEN THREE SPECIFIED POINTS IN THE PLANE

Including, but not limited to:

• 3 x 3 system of three linear equations in three variables
• Determination of a linear system of three equations in three variables using the three points and the standard form of the quadratic function, ax2 + bx + c = y
• Methods for solving the linear system of three equations in three variables
• Substitution
• Gaussian elimination
• Graphing calculator technology
• Inverse matrix
• rref
• Quadratic regression using the graphing calculator
• Three points required
• Correlation of determination, or r2 value, closer to ±1, the better the fit of the regression equation

Note(s):

• Algebra I wrote quadractic functions given real solutions and graphs of their related equations.
• Algebra I formulated a quadratic function using technology.
• Algebra II solves systems of three linear equations in three variables using various methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4B Write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.

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 (the directrix)
• Standard form
• Vertical axis of symmetry: y = ax2 + bx + c
• Horizontal axis of symmetry: x = ay2 + by + c
• Vertex form
• Vertical axis of symmetry: y = a(x – h)2k
• Horizontal axis of symmetry: x = a(y – k)2h
• 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: (hk)
• 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):

• 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) = ax2 + bx + c).
• Precalculus will address parabolas as conic sections.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B2 – Identify the symmetries of a plane figure.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4D Transform a quadratic function f(x) = ax2 + bx + c to the form f(x) = a(x - h)2 + k to identify the different attributes of f(x).
Supporting Standard

Transform

A QUADRATIC FUNCTION f(x)= ax2 + bx + c TO THE FORM f(x)= a(x – h)2 + k

Including, but not limited to:

• Standard form: f(x) = ax2 + bx + c
• Vertex form: f(x) = a(x – h)2 + k
• Completing the square to transform from the standard form f(x) = ax2 + bx + c to vertex form f(x) = a(x – h)2 + k

To Identify

THE DIFFERENT ATTRIBUTES OF f(x)

Including, but not limited to:

• Attributes from the vertex form, f(x) = a(xh)2 + k
• Vertex of the function, (hk)
• Minimum point of function if a > 0
• Maximum point of function if a < 0
• Axis of symmetry, x = h
• Attributes from the standard form, f(x) = ax2bxc
• Vertex of the function,
• Minimum point of function if a > 0
• Maximum point of function if a < 0
• y-intercept, c
• Axis of symmetry, x

Note(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) = ax2 + bx + c).
• Algebra I solved quadratic equations having real solutions by completeing the square.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B2 – Identify the symmetries of a plane figure.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4E

Formulate quadratic and square root equations using technology given a table of data.

Supporting Standard

Formulate

QUADRATIC EQUATIONS USING TECHNOLOGY GIVEN A TABLE OF DATA

Including, but not limited to:

• Data collection activities with and without technology
• Data modeled by quadratic functions
• Real-world problem situations
• Real-world problem situations modeled by quadratic functions
• Data tables with at least three data points
• Technology methods
• Transformations of f(x) = x2
• Solving three by three matrix to determine ab, and c for f(x) = ax2 + bx + c

Note(s):

• Algebra I solved quadratic equations having real solutions using tables, graphs, factoring, completing the square, quadratic formula, and technology.
• Algebra I wrote, using technology, quadratic functions that provide a reasonable fit to date to estimate solutions and make predictions for real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4F

Solve quadratic and square root equations.

Solve

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
• x-intercept(s) – 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
• Algebraic methods
• Factoring
• Solving equations by taking square roots
• Solving quadratic equations using absolute value
• x2 = 25, therefore, x = ±5
• Completing the square
• The discriminant, b2 – 4ac, can be used to analyze types of solutions for quadratic equations.
• b2 – 4ac = 0, one rational double root
• b2 – 4ac > 0 and perfect square, two rational roots
• b2 – 4ac > 0 and not perfect square, two irrational roots (conjugates)
• b2 – 4ac < 0, two imaginary roots (conjugates)
• Connections between solutions and roots of quadratic equations to the zeros and x-intercepts 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 =
• 2 = –1
• i =
• Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• a + bi and a – bi
• Operations with complex numbers, with and without technology
• Complex solutions for quadratic equations
• One real solution
• One rational double root
•  Two real solutions
• Two rational roots
• Two irrational root conjugates
• Two complex solutions
• Two complex root conjugates
• Reasonableness of solutions

Note(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:
• I. Numeric Reasoning
• A2 – Define and give examples of complex numbers.
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
Supporting Standard

Solve

Including, but not limited to:

• Methods for solving quadratic inequalities with and without technology
• Graphs
• Tables
• Algebraic methods
• Factoring
• Solving inequalities by taking square roots
• Solving quadratic inequalities using absolute value
• x2 ≤ 25, |x| ≤ 5; therefore, –5 ≤ x ≤ 5
• Completing the square
• Testing and identifying acceptable regions on a number line
• Graphical analysis of solution sets for quadratic inequalities
• One-dimensional on a number line
• Two-dimensional on a coordinate plane
• Comparison of solution sets of equations and inequalities
• Comparison of one-dimensional solutions and two-dimensional solutions, e.g. intervals versus points
• Reasonableness of solutions

Note(s):

• Algebra I solved quadratic equations.
• Algebra II introduces quadratic inequalities.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.7 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:
2A.7A Add, subtract, and multiply complex numbers.
Supporting Standard

COMPLEX NUMBERS

Including, but not limited to:

• Complex number system
• The complex number system, C, is composed of real and imaginary numbers.
• Real numbers, , are composed of rational numbers, Q, and irrational numbers,  – Q.
• Rational numbers, Q, are composed of integers, , whole numbers, N  0, and natural numbers, N.
• Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
• Real part of a complex number, a
• Imaginary part of a complex number, b
• Imaginary number – a number in the fomr of bi where b is a real number and i =
• Imaginary number unit, i, is a number whose square equals –1; therefore, the  = i.
• If x is a non-negative, real number  = i.
• Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• When complex conjugates are added or multiplied the imaginary part equals 0.
• Operations with complex numbers
• Real parts combine with real parts and imaginary parts combine with imaginary parts.
• Multiplication of complex numbers
• Distribute and collect like terms.
• The imaginary unit, i, can only have a power of 1.
• Any i² units must be converted to –1.

Note(s):

• Algebra II introduces the system of complex numbers and operations with complex numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• D1 – Interpret multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections