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

#### Unit Overview

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.

After this unit, in Algebra II 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, C1, D1, D2; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

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

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

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

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

Transformation(s) of a parent function create a new function within that family of functions.

• Why are transformations of parent functions necessary?
• How do transformations affect a function?
• How can transformations be interpreted from various representations?
• Why does a transformation of a function create a new function?
• How do the attributes of an original function compare to the attributes of a transformed function?

Inverses of functions create new functions.

• What relationships and characteristics exist between a function and its inverse?

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

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Multiple Representations

Functions

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

Geometric Reasoning

• Transformations

Associated Mathematical Processes

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

Logarithmic functions have unique graphs and attributes.

• What representations can be used to represent a logarithmic function?
• How do the different bases of 2, 10, and e, affect the representations of the logarithmic function?
• What are the key attributes of a logarithmic function and how can they be determined?

Transformations of the logarithmic parent function, f(x) = bx, can be used to determine graphs and equations of representative logarithmic functions in problem situations.

• What are the effects of changes on the graph of f(x) = bx when f(x) is replaced by af(x), for specific positive and negative values of a?
• What are the effects of changes on the graph of f(x) = bx when f(x) is replaced by f(x) + d,for specific positive and negative values of d?
• What are the effects of changes on the graph of f(x) = bx when f(x) is replaced by f(c) for specific positive and negative values of c?

The inverse of a function can be determined from multiple representations.

• How can the inverse of a function be determined from the graph of the function?
• How can the inverse of a function be determined from a table of coordinate points of the function?
• How can the inverse of a function be determined from the equation of the function?
• How are a function and its inverse distinguished symbolically?
• How are function compositions related to inverse functions?
• How do the attributes of inverse functions compare to the attributes of original functions?

The domain and range of the inverse of a function may need to be restricted in order for the inverse to also be a function.

• When must the domain of an inverse function be restricted?
• How does the relationship between a function and its inverse, including the restriction(s) on the domain, affect the restriction(s) on its range?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Evaluate
• Expressions
• Multiple Representations
• Solution Representations
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

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

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

• How are real-world problem situations identified as ones that can be modeled by exponential and logarithmic equations?
• How are exponential and logarithmic equations used to model problem situations?
• What methods can be used to solve exponential and logarithmic equations?
• What are the advantages and disadvantages of various methods used to solve exponential and logarithmic equations?
• What methods can be used to justify the reasonableness of solutions to exponential and logarithmic equations?
• What causes extraneous solutions in exponential and logarithmic equations?
• How can extraneous solutions be identified in graphs, tables, and algebraic calculations?

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

• What representations can be used to display exponential function models?
• What key attributes identify an exponential function model?
• How does the domain and range of the function compare to the domain and range of the problem situation?
• What are the connections between the key attributes of an exponential function model and the real-world problem situation?
• How can exponential function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?

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

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

#### 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 Creator if your district has granted access to that tool.

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

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity

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

Apply

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
2A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

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

Select

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

Including, but not limited to:

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

Note(s):

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

Communicate

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

Including, but not limited to:

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

Note(s):

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

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

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
2A.1G Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.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.

Graph

THE FUNCTIONS f(x) = , f(x), f(x) = x3, f(x) = , f(x) = bx, f(x) = |x|, 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
• 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
• Ex: x
• Ex: –3 < y ≤ 6, x
• Ex: y ≥ 0, y
• 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  x < 5}
• Ex: {x|x  }
• Ex: {y|y  , –3 < y ≤ 6}
• Ex: {y|y  y ≥ 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]
• 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 represents the set of real numbers, and the notation 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:
• III. Geometric Reasoning
• B1 – Identify and apply transformations to figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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(x – c) 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 = logbx, where b is 2, 10, and e
• f(x) = log2x; f(x) = log10x or f(x) = log(x); f(x) = logex 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 = logbx when parameters ab, c, and d are changed in f(x) = a • b(x – c) + d and f(x) = a  logb(x – c) + d
• Effects on the graphs of f(x) = 2x and f(x)log2x, 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) = logx, when f(x) is replaced by f(x – c) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left for values of c < 0 by |c| units
• For f(+ 2) → f(– (–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(– 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 = logbx
• 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:
• III. Geometric Reasoning
• B1 – Identify and apply transformations to figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 = logbx
• 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): a• a= am+n
• Quotient of powers (division when bases are the same):  = am-n
• Power to a power: (am)= amn
• Negative exponents: a-n =
• Zero exponent: a= 1
• Laws (properties) of logarithms
• Product: logb(p • q)logbp +  logbq; ln(p • q)lnp + lnq
• Quotient: logb = logbp – logbq; ln = ln– lnq
• Power: logbpqqlogbp; lnpqlnp
• Reciprocal: logb = –logbp; ln = –lnp
• Log of base: logbb = 1; lne = 1
• Log of 1: logb1 = 0; ln1 = 0
• Exponential equations to corresponding logarithmic equations
• b= A → logb(A) = x
• Logarithmic equations to corresponding exponential equations
• logb(A) = x → b= A

Note(s):

• Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization,
f(x) = a • bx.
• 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:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 = logbx
• 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) = a • bx.
• 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:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections