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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 10: Exponential and Logarithmic Functions and Equations SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address the inverse relationships of exponential functions and logarithmic functions, transformations of logarithmic functions, and key attributes of logarithmic functions. Student expectations also address connections between exponential and logarithmic equations and formulating, solving, and justifying solutions to exponential and logarithmic equations for problem situations. 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 Unit 06, Unit 09, and Unit 11, students analyzed exponential functions and applied laws of exponents in mathematical and real-world problem situations. In Algebra II Unit 04, students applied laws of exponents. In Unit 09, students analyzed transformations and key attributes of exponential functions. Students also solved and applied exponential equations in mathematical and real-world problem situations.

During this Unit
Students describe and analyze the inverse relationship between the exponential and logarithmic functions, including the restriction(s) on domain/range, and graph and write the inverse functions using notation such as f–1(x). Students graph the function f(x) = logb(x) where b is 2, 10, and e, and analyze the key attributes such as domain, range, intercepts, and asymptotic behavior. Students determine the effects on the key attributes on the graph of f(x) = logb(x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x – c) for specific positive and negative real values of a, c, and d, and investigate parameter changes and key attributes in terms of real-world problem situations. Students make connections between exponential and logarithmic equations by rewriting exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations. Students use this understanding to solve single logarithmic equations having real solutions and justify the reasonableness of the solutions. Students formulate exponential and logarithmic equations that model real-world situations, solve the equations, and determine the reasonableness of the solution in terms of the problem situation.

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

After this Unit
In Unit 11, students will distinguish between linear, quadratic, and exponential functions to model problem situations and apply logarithms to solve exponential equations as necessary. In Algebra II Unit 12, students will review concepts of exponential and logarithmic functions. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving exponential and logarithmic functions and equations

In Algebra II, analyzing the inverse relationship between exponential and logarithmic functions and graphing, transforming, and analyzing key attributes of logarithmic functions are identified in STAAR Readiness Standards 2A.2A, 2A.2C, and 2A.5A. Solving exponential equations and single logarithmic equations are identified in STAAR Readiness Standard 2A.5D. These Readiness Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses and STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. Graphing and writing inverse functions using notation are identified in STAAR Supporting Standard 2A.2B and subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Writing exponential equations as logarithmic equations and writing logarithmic equations as exponential equations are identified in STAAR Supporting Standard 2A.5C. Formulating exponential and logarithmic equations to model problem situations and determining the reasonableness of solutions are identified in STAAR Supporting Standards 2A.5B and 2A.5E. These Supporting Standards are subsumed under STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, C3, D1, D2; V. Statistical Reasoning A1, C2; VI. Functions A2, B1, B2, C1, C2; 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 (2000), Principles and Standards for School Mathematics, 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). Research found in National Council of Teachers of Mathematics also states, “Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone” (as cited by NCTM, 2009, p. 41). This unit places particular emphasis on multiple representations. State and national mathematics standards support such an approach. The Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions, such as algebraic, tabular, graphical, and verbal descriptions. This skill is mirrored in the Principles and Standards for School Mathematics (1989). Specifically, this work calls for instructional programs that enable all students to understand relations and functions and select, convert flexibly among, and use various representations for them. More recently, according to National Council of Teachers of Mathematics (2007), the importance of multiple representations has been highlighted in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. According to this resource, students should be able to translate among verbal, tabular, graphical, and algebraic representations of functions and describe how aspects of a function appear in different representations as early as Grade 8. Also, in research summaries such as Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement, such concept development is even cited among strategies that increase student achievement. Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences has been statistically shown to improve student performance on standardized measures of progress (Marzano, Pickering & Pollock, 2001).

Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development.
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. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2007). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. 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 judgements 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 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 relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Exponential functions are characterized by a rate of change that is proportional to the value of the function and can be used to describe, model, and make predictions about problem situations.
• Logarithmic functions are characterized as inverses of exponential functions and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can be modeled by …
• exponential functions?
• logarithmic functions?
• What graphs, key attributes, and characteristics are unique to …
• exponential functions?
• logarithmic functions?
• What patterns of covariation are associated with …
• exponential functions?
• logarithmic functions?
• How can the key attributes of exponential and logarithmic functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of exponential and logarithmic functions?
• How can the key attributes of exponential and logarithmic functions be used to make predictions and critical judgments?
• 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?
• What relationships exist between a function and its inverse?
• How are the key attributes of a function related to the key attributes of its inverse?
• How can the inverse of a function be determined and represented?
• How can function composition be used to analyze relationships between functions?
• 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
• x- and y-intercept(s)
• Zeros
• Asymptotes
• Functions
• Exponential
• Logarithmic
• Inverse
• Relations and Generalizations
• Patterns, Operations, and Properties
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• 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 judgements 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 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 relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Exponential functions are characterized by a rate of change that is proportional to the value of the function and can be used to describe, model, and make predictions about problem situations.
• Logarithmic functions are characterized as inverses of exponential functions and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can be modeled by …
• exponential functions?
• logarithmic functions?
• What graphs, key attributes, and characteristics are unique to …
• exponential functions?
• logarithmic functions?
• What patterns of covariation are associated with …
• exponential functions?
• logarithmic functions?
• How can the key attributes of exponential and logarithmic functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of exponential and logarithmic functions?
• What are the real-world meanings of the key attributes of exponential and logarithmic function models?
• How can the key attributes of exponential and logarithmic functions 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?
• 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 does knowing more than one solution strategy build mathematical flexibility?
• 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 exponential and logarithmic equations?
• How does the given information and/or representation influence the selection of an efficient method for writing exponential and logarithmic equations?
• What methods can be used to solve exponential and logarithmic equations?
• How does the structure of the equation influence the selection of an efficient method for solving exponential and logarithmic equations?
• How can the solutions to exponential and logarithmic equations be determined and represented?
• How are properties and operational understandings used to transform exponential and logarithmic equations?
• What connections exist between corresponding exponential and logarithmic equations?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Asymptotes
• Functions and Equations
• Exponential
• Logarithmic
• Patterns, Operations, and Properties
• Relations and Generalizations
• 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, when rewriting an exponential equation to its logarithmic equation form, may confuse the position of the argument and exponent. Students may rewrite the equation y = bx → logb(x) = y rather than y = bx → logb(y) = x.

Unit Vocabulary

• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• 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
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Interval notation – notation in which the solution is represented by a continuous interval
• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and f –1(f(x)) = x.
• Range – set of output values for the dependent variable over which the function is defined
• Set notation – notation in which the solution is represented by a set of values
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• 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:

 Decreasing Dependent Evaluate Exponent Exponential regression Function notation Horizontal compression Horizontal shift Horizontal stretch Increasing Independent Logarithmic regression Parameter change Parent function Reflection Vertical compression Vertical shift Vertical stretch y = x line
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 II 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.
• 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
2A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – 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.
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:
• 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.
2A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
2A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

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

Including, but not limited to:

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

Note(s):

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
2A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
2A.2 Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:
2A.2A

Graph the functions f(x)=, f(x)=1/x, f(x)=x3, f(x)=, f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.

Graph

THE FUNCTIONS f(x) = bx AND f(x) = logb(x) WHERE b IS 2, 10, AND e

Including, but not limited to:

• Representations of functions, including graphs, tables, and algebraic generalizations
• Exponential, f(x) = bx, where b is 2, 10, and e
• Logarithmic, f(x) = logb(x), where b is 2, 10, and e
• Connections between representations of families of functions
• Comparison of similarities and differences of families of functions

Analyze

THE KEY ATTRIBUTES OF THE FUNCTIONS SUCH AS DOMAIN, RANGE, INTERCEPTS, AND ASYMPTOTIC BEHAVIOR

Including, but not limited to:

• Domain and range of the function
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Representation for domain and range
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, y ∈ Ζ
• Set notation – notation in which the solution is represented by a set of values
• Braces are used to enclose the set.
• Solution is read as “The set of x such that x is an element of …”
• Ex: {x|x ∈ ℜx < 5}
• Ex: {x|x ∈ ℜ}
• Ex: {y|y ∈ ℜ, –3 < y ≤ 6}
• Ex: {y|y ∈ Ζy ≥ 0}
• Interval notation – notation in which the solution is represented by a continuous interval
• Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
• Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
• Ex: (–, 5)
• Ex: (–, )
• Ex: (–3, 6]
• Ex: [0, ∞)
• Domain and range of the function versus domain and range of the contextual situation
• Key attributes of functions
• Intercepts/Zeros
• 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
• y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Use key attributes to recognize and sketch graphs
• Application of key attributes to real-world problem situations

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = bx and their key attributes.
• Precalculus will study polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
2A.2B Graph and write the inverse of a function using notation such as f -1(x).
Supporting Standard

Graph, Write

THE INVERSE OF A FUNCTION USING NOTATION SUCH AS f –1 (x)

Including, but not limited to:

• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and  f –1(f(x)) = x.
• Inverse functions
• Exponential and logarithmic
• Inverses of functions on graphs
• Symmetric to yx
• Inverses of functions in tables
• Interchange independent (x) and dependent (y) coordinates in ordered pairs
• Inverses of functions in equation notation
• Interchange independent (x) and dependent (y) variables in the equation, then solve for y
• Inverses of functions in function notation
• f –1(x) represents the inverse of the function f(x).

Note(s):

• Algebra II introduces inverse of a function.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
2A.2C

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

Describe, Analyze

THE RELATIONSHIP BETWEEN A FUNCTION AND ITS INVERSE (LOGARITHMIC AND EXPONENTIAL), INCLUDING THE RESTRICTION(S) ON DOMAIN, WHICH WILL RESTRICT ITS RANGE

Including, but not limited to:

• Relationships between functions and their inverses
• All inverses of functions are relations.
• Inverses of one-to-one functions are functions.
• Inverses of functions that are not one-to-one can be made functions by restricting the domain of the original function, f(x).
• Characteristics of inverse relations
• Interchange of independent (x) and dependent (y) coordinates in ordered pairs
• Reflection over y = x
• Domain and range of the function versus domain and range of the inverse of the given function
• Functionality of the inverse of the given function
• Exponential function and logarithmic function, f(x) = bx and g(x) = logb(x) where b is 2, 10, and e

Note(s):

• Algebra I determined if relations represented a function.
• Algebra II introduces inverse of a function and restricting domain to maintain functionality.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
2A.5 Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:
2A.5A Determine the effects on the key attributes on the graphs of f(x) = bx and f(x) = logb(x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d.

Determine

THE EFFECTS ON THE KEY ATTRIBUTES ON THE GRAPHS OF f(x) = bx AND f(x) = logb(x) WHERE b IS 2, 10, AND e WHEN f(x) IS REPLACED BY af(x), f(x) + d, and f(xc) FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, c, AND d

Including, but not limited to:

• General form of the power function
• Exponential functions, f(x) = bx, where b is 2, 10, and e
• f(x) = 2x; f(x) = 10x; f(x) = ex
• Logarithmic functions, y = logb(x), where b is 2, 10, and e
• f(x) = log2(x); f(x) = log10(x) or f(x) = log(x); f(x) = loge(x) or f(x) = ln(x)
• Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Key attributes
• Intercepts
• Asymptotes
• Effects on the graphs of f(x) = bx and y = logb(x) when parameters a, b, c, and d are changed in f(x) = ab(xc) + d and f(x) = a • logb(xc) + d
• Effects on the graphs of f(x) = 2x and f(x) = log2(x) when f(x) is replaced by af(x) with and without technology
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the x-axis
• Effects on the graphs of f(x) = 10x and f(x) = log(x), when f(x) is replaced by f(xc) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left for values of c < 0 by |c| units
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
• Horizontal shift right for values of c > 0 by |c| units
• For f(x – 2), c = 2, and the function moves to the right two units.
• Effects on the graphs of f(x) = ex and f(x) = ln(x), when f(x) is replaced by f(x) + d with and without technology
• d = 0, no vertical shift
• Vertical shift down for values of d < 0 by |d| units
• Vertical shift up for values of d > 0 by |d| units
• Connections between the critical attributes of transformed functions and f(x) = bx and y = logb(x)
• Determination of parameter changes given a graphical or algebraic representation
• Determination of a graphical representation given the algebraic representation or parameter changes
• Determination of an algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects on the domain and range by the parameter changes
• Effects of multiple parameter changes
• Mathematical problem situation
• Effects of parameter changes in real-world problem situations

Note(s):

• Algebra I determined effects on the graphs of the parent functions, f(x) = x and 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.
• Algebra II continues to investigate the exponential parent function and introduces logarithmic parent function and transformations of both functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
2A.5B

Formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation.

Supporting Standard

Formulate

EXPONENTIAL AND LOGARITHMIC EQUATIONS THAT MODEL REAL-WORLD SITUATIONS

Including, but not limited to:

• Data collection activities with and without technology
• Data modeled by exponential functions
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Data modeled by logarithmic functions
• Real-world problem situations
• Real-world problem situations modeled by exponential functions
• Exponential growth
• Exponential decay
• Real-world problem situations modeled by logarithmic functions
• Representations of exponential and logarithmic equations
• Tables/graphs
• Verbal descriptions
• Technology methods
• Transformations of f(x) = bx and y = logb(x)
• Exponential regression
• Logarithmic regression

Note(s):

• Algebra II introduces formulating exponential and logarithmic equations.
• 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.5C Rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations.
Supporting Standard

Rewrite

EXPONENTIAL EQUATIONS AS THEIR CORRESPONDING LOGARITHMIC EQUATIONS AND LOGARITHMIC EQUATIONS AS THEIR CORRESPONDING EXPONENTIAL EQUATIONS

Including, but not limited to:

• Laws (properties) of exponents
• Product of powers (multiplication when bases are the same): ama = am+n
• Quotient of powers (division when bases are the same): = am-n
• Power to a power: (am)n = amn
• Negative exponents: a-n =
• Zero exponent: a0 = 1
• Laws (properties) of logarithms
• Product: logb(pq) = logb(p) + logb(q); ln(pq) = ln(p) + ln(q)
• Quotient: logb = logb(p) – logb(q); ln = ln(p) – ln(q)
• Power: log(p)q = q bull; logb(p); ln(p)q = q • ln(p)
• Reciprocal: logb = –logb(p); ln = –ln(p)
• Log of base: logb(b) = 1; ln(e) = 1
• Log of 1: logb(1) = 0; ln(1) = 0
• Exponential equations to corresponding logarithmic equations
• bx = A → logb(A) = x
• Logarithmic equations to corresponding exponential equations
• logb(A) = xbx = A

Note(s):

• Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = abx.
• Algebra II introduces logarithms.
• Algebra II connects between exponential equations and logarithmic equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
2A.5D Solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions.

Solve

EXPONENTIAL EQUATIONS OF THE FORM y = abx WHERE a IS A NONZERO REAL NUMBER AND b IS GREATER THAN ZERO AND NOT EQUAL TO ONE

Including, but not limited to:

• Exponential equation, y = abx
• a – initial value at x = 0
• b – common ratio
• Solving exponential equations
• Application of laws (properties) of exponents
• Application of logarithms as necessary
• Real-world problem situations modeled by exponential functions
• Exponential growth
• f(x) = abx, where b > 1
• f(x) = aekx, where k > 0
• Exponential decay
• f(x) = abx, where 0 < b < 1
• f(x) = aekx, where k < 0

Solve

SINGLE LOGARITHMIC EQUATIONS HAVING REAL SOLUTIONS

Including, but not limited to:

• Single logarithmic equation, y = logb(x)
• x – argument
• b – base
• y – exponent
• Solving logarithmic equations
• Transformation to exponential form as necessary
• Real-world problem situations modeled by logarithmic functions

Note(s):

• Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = abx.
• Algebra II solves exponential equations algebraically.
• Algebra II introduces logarithms and solving logarithmic equations.
• Precalculus will use properties of logarithms to solve equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
2A.5E Determine the reasonableness of a solution to a logarithmic equation.
Supporting Standard

Determine

THE REASONABLENESS OF A SOLUTION TO A LOGARITHMIC EQUATION

Including, but not limited to:

• Justification of solutions to logarithmic equations with and without technology
• Verbal description
• Tables
• Graphs
• Substitution of solutions into original functions
• Justification of reasonableness of solutions in terms of mathematical and real-world problem situations or data collections

Note(s):

• Algebra II introduces logarithms and solving logarithmic equations.
• Various mathematical process standards will be applied to this student expectation 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.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.4. Justify the solution.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• 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.