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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 02: Introduction to Functions SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address an introductory exploration of various representations of relations, including functionality, domain and range, function notation, and evaluation of the function for specific domain values. 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 Grade 6, students identified independent and dependent quantities from tables and graphs and wrote equations to represent the relationships. In Grade 8, students were introduced to functions as proportional and non-proportional relationships using sets of ordered pairs, tables, mappings, and graphs.

During this Unit
Students identify relations and determine if relations represented verbally, tabularly, graphically, and symbolically define a function. Students identify domain and range (continuous and discrete) of functions and represent the domain and range using inequality notation and verbal descriptions. Students express functions in function notation. Students evaluate functions in function notation given one or more elements in their domains. Students explore real-world problem situations, identify the domain and range of problem situations, express representative functions for problem situations using function notation, and evaluate functions for specified domains in problem situations. Problems in this unit incorporate linear, quadratic, and exponential functions, since those are the functions studied in Algebra I.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Algebra I

After this Unit
In Algebra I, students will continue to represent functions, identify attributes of functions, and apply function notation in a more in-depth study of linear, quadratic, and exponential functions. The concepts in this unit will also be applied in subsequent mathematics courses.

In Algebra I, determining domain and range of linear functions is identified as STAAR Readiness Standard A.2A and part of STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Determining domain and range of quadratic functions is identified as STAAR Readiness Standard A.6A and part of STAAR Reporting Category 4: Quadratic Functions and Equations. Determining domain and range of exponential functions is identified as STAAR Supporting Standard A.9A and part of STAAR Reporting Category 5: Exponential Functions and Equations. Deciding whether relations define functions is identified as STAAR Supporting Standards A.12A, and evaluating functions written in function notation is identified as STAAR Supporting Standards A.12B. Both STAAR Supporting Standards A.12A and A.12B are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, D1, D2; V. Statistical Reasoning A1, C2; VI. Functions A1, B1; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions, Grades 9 – 12 (2010), foundations for functions begin in elementary when studying patterns and using informal notation to represent variables. It continues in middle school by developing rules to represent tabular data with an understanding of relationships in bivariate data. In high school students further analyze these relationships and develop an understanding of families of functions and their characteristics. The National Council of Teachers of Mathematics (NCTM), the Texas Education Agency (TEA) in the math professional development modules and Algebra 1 EOC Success, and other mathematics research Algebra should take a functional approach. According to Navigating through Algebra in Grades 9 – 12 (2002), “much of what has traditionally been Algebra I in secondary schools is expected content for the middle grades. It is imperative then that a broadening and deepening of mathematics content take place in high school. New topics…such as classes of functions and using technology on symbolic expressions are emerging in the high school curriculum” (NCTM, p. v). Additionally, Algebra in a Technological World (1995) states, the high school algebra curriculum should undergo “a shift in perspective from algebra as skills for transforming, simplifying, and solving symbolic expressions to algebra as a way to express and analyze relationships” (NCTM, p. v). By beginning formal algebra with real-life situations that are naturally algebraic, students understand that formal algebra is not only a manipulation of symbols, but also a logical way to approach mathematical situations in an effort to make sense of them. Experiencing real-life functional situations and their characteristics helps build algebraic habits of mind (Driscoll, 1999). Through careful instruction, teachers connect real-life with algebraic representation and build conceptual understanding before delving into algebraic manipulation. If students completely develop solving equations using symbolic manipulation before they develop a solid conceptual foundation for their work, they will be unable to do more than symbolic manipulation (National Research Council, 1998).

Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6 – 10. Portsmouth, VA: Heinemann.
National Council of Teachers of Mathematics. (1995). Curriculum and evaluation standards for school mathematics: Algebra in a technological world. 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.
National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Functions, Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (1998). High school mathematics at work: Essays and examples for the education of all students. Washington, DC: National Academy Press.
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 judgements about the relationship?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• What kinds of mathematical and real-world situations can be modeled by …
• linear functions?
• exponential functions?
• How can the domain and range of linear, quadratic, and exponential functions be …
• determined?
• analyzed?
• described?
• How can the graph of the function and the domain and range be used to determine if the function is linear, quadratic or exponential?
• How can domain and range be used to describe the behavior of linear, quadratic, and exponential functions?
• What relationships exist between the mathematical and real-world meanings of the domain and range of linear, quadratic, and exponential function models?
• How can domain and range be used to make predictions and critical judgments about the problem situation?
• 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, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Functions and Equations
• Linear
• Exponential
• Relations and Generalizations
• Number and Algebraic Methods
• Relations and Functions
• 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 all relations are functions rather than that relations are functions only if each element of the input is paired with exactly one element of the output.
• Some students may think functions can only be expressed in function notation, such as f(x) = mx + b or h(x) = ax2 + bx + c, and not recognize equations without function notation, such as y = mx + b or w = ax2 + bx + c, as valid representations of the same functional relationships.
• Some students may think that variables represented as letters identify an object rather than the number or quantity of objects.

#### Unit Vocabulary

• 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
• Function – relation in which each element of the domain (x) is paired with exactly one element of the range (y)
• Function notation – notation that describes a specific function such as f(x) = x
• Inequality notation – notation in which the solution is represented by an inequality statement
• Range – set of output values for the dependent variable over which the function is defined
• Relation – a set of ordered pairs (x, y) where the x is associated with a specific y

Related Vocabulary:

 Decreasing function Dependent variable Exponential function Increasing function Independent variable Linear function Quadratic function Representations
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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
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:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
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:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
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:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
A.2 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:
A.2A Determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities.

Determine

THE DOMAIN AND RANGE OF A LINEAR FUNCTION IN MATHEMATICAL PROBLEMS AND REASONABLE DOMAIN AND RANGE VALUES FOR REAL-WORLD SITUATIONS, BOTH CONTINUOUS AND DISCRETE

Represent

THE DOMAIN AND RANGE OF A LINEAR FUNCTION USING INEQUALITIES

Including, but not limited to:

• Domain and range of linear 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
• 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, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, yΖ
• Domain and range of linear functions in real-world problem situations
• Reasonable domain and range for real-world problem situations
• Comparison of domain and range of function model to appropriate domain and range for a real-world problem situation

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Grade 6 identified independent and dependent quantities.
• Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
• 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 introduce representing domain and range using interval and set notation.
• Precalculus will introduce piecewise functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
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:

• 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:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
A.9 Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, 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.9A Determine the domain and range of exponential functions of the form f(x) = abx and represent the domain and range using inequalities.
Supporting Standard

Determine, Represent

THE DOMAIN AND RANGE OF EXPONENTIAL FUNCTIONS OF THE FORM f(x) = abUSING INEQUALITIES

Including, but not limited to:

• Domain and range of exponential 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 values in real-world problem situations
• Reasonable domain and range for the real-world problem situation
• Comparison of domain and range of a 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, yΖ

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 exponential 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 exponential functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
A.12 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:
A.12A Decide whether relations represented verbally, tabularly, graphically, and symbolically define a function.
Supporting Standard

Decide

WHETHER RELATIONS REPRESENTED VERBALLY, TABULARLY, GRAPHICALLY, AND SYMBOLICALLY DEFINE A FUNCTION

Including, but not limited to:

• Relation – a set of ordered pairs (x, y) where the x is associated with a specific y
• Function – relation in which each element of the domain (x) is paired with exactly one element of the range (y)
• Identification of functions
• Verbally
• Set of points
• Tabularly
• Graphically (Vertical line test – a vertical line must intersect the graph at one and only one point.)
• Symbolically
• Applications to mathematical problems
• Applications to real-world problems
• Applications to data collection and analysis

Note(s):

• Grade 8 developed the basic foundation of functions.
• Algebra I extends linear functions and introduces quadratic and exponential functions.
• Algebra II will extend to inverses of functions and restricting domains as needed to maintain functionality.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.1. Recognize if a relation is a function.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
A.12B Evaluate functions, expressed in function notation, given one or more elements in their domains.
Supporting Standard

Evaluate

FUNCTIONS, EXPRESSED IN FUNCTION NOTATION, GIVEN ONE OR MORE ELEMENTS IN THEIR DOMAINS

Including, but not limited to:

• Function – relation in which each element of the domain (x) is paired with exactly one element of the range (y)
• Function notation – notation that describes a specific function such as f(x) = x
• Letters are used to name specific functions, such as f, g, h, etc.
• x is the input or domain.
• f(x) = 2x + 3 is the rule for the specific function.
• f(x) is the range at a domain of x.
• f(x) is the output or range from the rule, when the input is x.
• Benefits of function notation
• Naming of function
• f(x) names function f.
• g(x) names function g.
• Designating value to be evaluated in the function
• h(–5) means to find the rule h, substitute –5 in for the variable, and simplify.
• Evaluation of functions in function notation
• Identification of specified function rule
• Substitution of domain value into the identified rule
• Simplification of the numeric expression
• Application of function notation in real-world problem situations

Note(s): 