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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 05: Exponential and Logarithmic Functions and Equations SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address inverses, graphs, attributes, and transformations of exponential and logarithmic functions and equations. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Algebra I Units 02, 09, and 10, students graphed and identified key attributes of exponential functions, wrote exponential functions to model both mathematical and real-world problem situations, and interpreted the meaning of a and b in f(x) = abx in problem situations. In Algebra II Units 01 and 09 – 11, students studied transformations, characteristics, and applications of exponential and logarithmic functions, as well as the connections between exponential and logarithmic functions. Additionally, students formulated and solved exponential and logarithmic equations for problem situations. In Precalculus Unit 01, students analyzed the key attributes of various function types, including exponential and logarithmic functions.

During this Unit
Students determine inverse functions for exponential and logarithmic functions and represent these inverse functions using multiple representations, including graphs, tables, and algebraic methods. Students graph exponential and logarithmic functions and their transformations, including af(x), f(x) + d, f(xc), and f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems. Students determine and analyze the key attributes of exponential and logarithmic functions (including domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, intervals over which the function is increasing or decreasing, end behavior, and discontinuities) in mathematical and real-world problems. Students analyze and describe end behavior using infinity notation based on an analysis of the function type and constants used. Students use the properties of logarithms (including product, quotient, power rule, and change of base) to evaluate or transform logarithmic expressions. Students generate and solve logarithmic equations in mathematical and real-world problems using algebraic, tabular, and graphical methods. Students generate and solve exponential equations in mathematical and real-world problems using algebraic, tabular, and graphical methods. Logarithmic and exponential equations are used to model and solve mathematical and real-world problem situations.

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

After this Unit
In subsequent courses in mathematics, students will apply concepts of exponential and logarithmic functions, inverses, and equations as they arise in problem situations.

Function analysis serves as the foundation for college readiness. Focusing on real world function analysis and representation is emphasized in the 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 a 2007 report published by the Mathematical Association of America, a “fundamental experience” for students entering college mathematics courses is to address problems in real-world situations by creating and interpreting mathematical models. Functions provide a way to quantitatively study the relationships and change in numerous real-world phenomena; this, coupled with the applicability of functions to many mathematical topics, make functions one of the most important topics in high school mathematics (Cooney, Beckmann, & Lloyd, 2010). In Texas, the importance of these skills is emphasized in the Texas College and Career Readiness Standards (2009), which call for students to be able to understand and analyze features of a function to model real-world situations. Algebraic models allow us to efficiently visualize and analyze the vast amount of interconnected information that is contained in a functional relationship; these tools are particularly helpful as the mathematical models become increasingly complex (National Research Council, 2005). Additionally, research argues that students need both a strong conceptual understanding of functions, as well as procedural fluency; as such, good instruction must include “a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions” (NRC, 2005, p. 353). Lastly, students need to be involved in metacognitive engagement in mathematics as they problem solve and reflect on their solutions and strategies; this is particularly important as students transition into more abstract mathematics where fewer “clues” may exist warning students of a mathematical misstep (NRC, 2005). In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) notes the necessity for high school students to understand and compare the properties and classes of functions, including exponential and logarithmic functions. In the AP Calculus Course Description, the College Board (2012) states that mathematics designed for college-bound students should involve analysis and understanding of elementary functions, including exponential and logarithmic functions. Specifically, students must be familiar with the properties, algebra, graphs, and language of these functions.

Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Katz, Victor J. (Ed.). (2007). Algebra: Gateway to a technological future. Mathematical Association of America.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2005). How students learn: Mathematics in the classroom. M.S. Donovan & J.D. Bransford (Eds.). Washington, DC: The National Academies Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical 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 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?
• Functions can be represented in various ways (including algebraically, graphically, verbally, and numerically) 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
• Symmetries
• Increasing or decreasing
• End behavior
• Functions
• Exponential
• Logarithmic
• Inverse
• Patterns, Operations, and Properties
• Relations and Generalizations
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 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 (including algebraically, graphically, verbally, and numerically) 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?
• The ability to represent quantities in various forms develops the understanding of equivalence and allows for working flexibly with algebraic expressions in order to communicate and reason about quantities.
• How can expressions be used to represent situations?
• What mathematical conventions are used when representing expressions? Why?
• How can it be determined if two expressions are equivalent?
• How are properties and operational understandings used to generate equivalent expressions?
• Why can it be useful to transform expressions?
• Why can it be useful to factor expressions?
• How does the structure of the expression influence the selection of an efficient method for transforming logarithmic expressions?
• 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
• Functions
• Exponential
• Logarithmic
• Patterns, Operations, and Properties
• Relations and Generalizations
• Algebraic Reasoning
• Expressions and Equations
• Logarithmic
• Exponential
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that logarithmic expressions can be simplified in ways that do not follow the properties of logarithms, such as log(a) + log(b) = log(a + b) or log(a) – log(b) = log(ab) rather than log(a) + log(b) = log(ab) or log(a) – log(b) = log().

Underdeveloped Concepts:

• Some students may not realize that geometric sequences can be thought of as restrictions of exponential functions to the natural numbers.
• Some students may not realize that e represents an irrational number (e = 2.71828182…) rather than a variable.
• Some students may incorrectly sum the effects of percent changes over time, such as believing that if a population increased by 50% each year, then in two years the population will increase by 100%.
• Some students may miss solutions to logarithmic equations based on how they transform the logarithmic expressions involved. For example, if students solve the logarithmic equation log(x²) = 2 by incorrectly applying the power rule, they might determine that x = 10 is a solution to the equation but miss that x = –10 is also a solution.

#### Unit Vocabulary

• Covariation – pattern of related change between two variables in a function
• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and f –1(f(x)) = x.

Related Vocabulary:

 Base Change of base Common logarithm Compound interest Composition Compression Decay Decreasing Domain End behavior Equation Exponent Exponential function Expression Extraneous solution Function Growth Horizontal asymptote Increasing Inequality notation Infinity notation Interval notation Inverse function notation Logarithm Logarithm of a product Logarithm of a quotient Logarithmic function Natural logarithm Power rule Range Reflection Root Set notation Solution Stretch Transformation Translation Undefined Vertical asymptote x-intercept y-intercept Zero
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 Precalculus 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.
• 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
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.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.
P.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.
P.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.
P.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.
P.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.
P.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.
P.1G Display, explain, and 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.
P.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
P.2E Determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations.

Determine

AN INVERSE FUNCTION, WHEN IT EXISTS, FOR A GIVEN FUNCTION OVER ITS DOMAIN OR A SUBSET OF ITS DOMAIN

Represent

THE INVERSE OF A FUNCTION USING MULTIPLE REPRESENTATIONS

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.
• Characteristics of inverse functions
• Domain of the function becomes an appropriate range of the inverse function.
• Range of the function becomes an appropriate domain of the inverse function.
• Composed as f(f –1(x)) = x and f –1(f(x)) = x
• Multiple representations
• Inverse function notation
• When a function f(x) has an inverse that is also a function, the inverse can be written with f –1(x).
• For the function f(x) = x + 4, the inverse function is f –1(x) = x – 4.
• For the function g(x) = x2:
• If the restricted domain of g(x) is x ≥ 0, then the inverse function is g1(x) = .
• If the restricted domain of g(x) is x ≤ 0, then the inverse function is g1(x) = –.
• Algebraic
• The inverse of a function can be found algebraically by:
• Writing the original function in “y = ” form
• Interchanging the x and y variables
• Solving for y
• A function’s inverse can be confirmed algebraically if both of the following are true: f(f –1(x)) = x and f –1(f(x)) = x.
• Tabular
• From the table of values for a given function, the tabular values of the inverse function can be found by switching the x- and y-values of each ordered pair.
• Graphical
• The graphs of a function and its inverse are reflections over the line y = x.
• Verbal description of the relationships between the domain and range of a function and its inverse
• Restrictions on the domain of the original function to maintain functionality
• Inverse functions over a subset of the domain of the original function

Note(s):

• Algebra II analyzed the relationship between functions and inverses, such as quadratic and square root, or logarithmic and exponential, including necessary restrictions on the domain.
• Precalculus extends the analysis of inverses to include other types of functions, such as trigonometric and others.
• 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.
P.2F

Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Graph

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Including, but not limited to:

• Graphs of the parent functions
• Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations
• Various methods for graphing
• Curve sketching
• Plotting points from a table of values
• Transformations of parent functions (parameter changes abc, and d)
• Using graphing technology

Note(s):

• Algebra II graphed various types of functions, including square root, cube root, absolute value, and rational functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• 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.
• 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.
P.2G

Graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems.

Graph

FUNCTIONS, INCLUDING EXPONENTIAL LOGARITHMIC, AND LOGARITHMIC FUNCTIONS INCLUDING af(x), f(x) + d, f(xc), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• General form of parent function
• Exponential functions: f(x) = 2x, f(x) = ex, f(x) = 10x
• Logarithmic functions: f(x) = log2(x), f(x) = ln(x), f(x) = log(x)
• Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters a, b, c, and d on graphs
• Effects of a on f(x) in af(x)
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the horizontal axis (x-axis)
• Effects of d on f(x) in f(x) + d
• d = 0, no vertical shift
• Translation, vertical shift up or down by |d| units
• Effects of c on f(x) in f(xc)
• c = 0, no horizontal shift
• Translation, horizontal shift left or right by |c| units
• Effects of b on f(x) in f(bx)
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the vertical axis or y-axis
• Combined transformations of parent functions
• Transforming a portion of a graph
• Illustrating the results of transformations of the stated functions in mathematical problems using a variety of representations
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra II graphed transformations of various types of functions, including square root, cube, cube root, absolute value, rational, exponential, and logarithmic functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• 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.
P.2I

Determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing.

Determine, Analyze

THE KEY FEATURES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS SUCH AS DOMAIN, RANGE, ZEROS, ASYMPTOTES, AND INTERVALS OVER WHICH THE FUNCTION IS INCREASING OR DECREASING

Including, but not limited to:

• Covariation – pattern of related change between two variables in a function
• Multiplicative patterns
• Exponential functions
• Logarithmic functions
• Domain and range
• Represented as a set of values
• {0, 1, 2, 3, 4}
• Represented verbally
• All real numbers greater than or equal to zero
• All real numbers less than one
• Represented with inequality notation
• x ≥ 0
• y < 1
• Represented with set notation
• {x|x ∈ ℜ, x ≥ 0}
• {y|y ∈ ℜ, y < 1}
• Represented with interval notation
• [0, ∞)
• (–∞, 1)
• Zeros
• Roots/solutions
• x-intercepts
• Asymptotes
• Vertical asymptotes (x = h)
• Horizontal asymptotes (y = k)
• Slant asymptotes (y = mx + b)
• Intervals where the function is increasing or decreasing
• Represented with inequality notation, –1 <  ≤ 3
• Represented with set notation, {x|x ∈ ℜ, –1 < x ≤ 3}
• Represented with interval notation, (–1, 3]
• Connections among multiple representations of key features
• Graphs
• Tables
• Algebraic
• Verbal

Note(s):

• Algebra II analyzed functions according to key attributes, such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum values over an interval.
• Precalculus extends the analysis of key attributes of functions to include zeros and intervals where the function is increasing or decreasing.
• 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.
• 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.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.
P.2J

Analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems.

Analyze, Describe

END BEHAVIOR OF FUNCTIONS, INCLUDING EXPONENTIAL AND LOGARITHMIC FUNCTIONS, USING INFINITY NOTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Describing end behavior with infinity notation
• Right end behavior
• As x → ∞ (or as x approaches infinity) the function becomes infinitely large; f(x) → ∞.
• As x → ∞ (or as x approaches infinity) the function becomes infinitely small; f(x) → –∞.
• As x → ∞ (or as x approaches infinity) the function approaches a constant value, cf(x) → c.
• Left end behavior
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely large; f(x) → ∞.
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely small; f(x) → –∞.
• As x → –∞ (or as x approaches negative infinity) the function approaches a constant value, cf(x) → c.
• Determining end behavior from multiple representations
• Tables: evaluating the function for extreme negative (left end) and positive (right end) values of x
• Graphs: analyzing behavior on the left and right sides of the graph
• Determining end behavior from analysis of the function type and the constants used
• Exponential: f(x) = abx
• Ex: When a > 0 and b > 1, as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → 0.
• Ex: When a > 0 and 0 < b < 1, as x → ∞ (on the right), f(x) → 0, and as x → –∞ (on the left), f(x) → ∞.
• Logarithmic: f(x) = alogb(x)
• Ex: When a > 0 and b > 1, as x → ∞ (on the right), f(x) → ∞.
• Ex: When a > 0 and b > 1, as x → 0 (on the left), f(x) → –∞.
• Interpreting end behavior in real-world situations

Note(s):

• Algebra II analyzed the domains and ranges of quadratic, square root, exponential, logarithmic, and rational functions.
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends analysis of domain, range, and asymptotic restrictions to determine the end behavior of functions and describes this behavior using infinity notation.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• 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.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.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.2N

Analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems.

Analyze, To Solve

SITUATIONS MODELED BY FUNCTIONS, INCLUDING EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Including, but not limited to:

• Models that represent problem situations
• Understanding the meaning of the variables (both independent and dependent)
• Evaluating the function when independent quantities (x-values) are given
• Solving equations when dependent quantities (y-values) are given
• Appropriateness of given models for a situation
• Analyzing the attributes of a problem situation
• Determining which type of function models the situation
• Determining a function to model the situation
• Using transformations
• Using attributes of functions
• Using technology
• Describing the reasonable domain and range values
• Comparing the behavior of the function and the real-world relationship
• Exponential functions
• Exponential growth (e.g., accrued interest, population growth, etc.)
• Exponential decay (e.g., half-life, cooling rate, etc.)
• Logarithmic functions (e.g., pH, sound (decibel measures), earthquakes (Richter scale), etc.)

Note(s):

• Algebra II analyzed situations involving exponential, logarithmic, and rational functions.
• Precalculus extends function analysis to include polynomial and power functions and expects students to solve real-world problems and interpret solutions to those problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VI.C.2. Develop a function to model a situation.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• 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.
• 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.
P.5 Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:
P.5G Use the properties of logarithms to evaluate or transform logarithmic expressions.

Use

THE PROPERTIES OF LOGARITHMS

Including, but not limited to:

• Connection of logarithms to exponents: logb(x) = yby = x
• Common logarithms (base 10): log(x) = y ↔ 10y = x
• Natural logarithms (base e): ln(x) = yey = x
• Logarithms of a product: logb(xy) = logb(x) + logb(y)
• Logarithms of a quotient: logb = logb(x) – logb(y)
• Power rule of logarithms: logb(xr) = r • logb (x)
• Change of base property:

To Transform, To Evaluate

LOGARITHMIC EXPRESSIONS

Including, but not limited to:

• Evaluating logarithmic expressions
• Changing to exponential notation
• With technology
• Transforming logarithmic expressions
• Numerical expressions
• Algebraic expressions

Note(s):

• Algebra I simplified expressions using the laws (properties) of exponents, including integral and rational exponents.
• Algebra II rewrote exponential equations to logarithmic equations and vice versa.
• Algebra II formulated and solved exponential and logarithmic equations.
• Precalculus applies the properties of logarithms to transform expressions.
• 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.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.
P.5H Generate and solve logarithmic equations in mathematical and real-world problems.

Generate, Solve

LOGARITHMIC EQUATIONS IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Solution strategies
• Solving logarithmic equations algebraically
• Simplifying expressions on both sides of an equation by writing them as single logarithms
• Rewriting logarithmic equations in exponential form
• Extraneous solutions
• Solving logarithmic equations with technology
• Graphs
• Tables
• Various situations
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra II rewrote exponential equations to logarithmic equations and vice versa.
• Algebra II formulated and solved exponential and logarithmic equations.
• Algebra II determined the resonableness of a solution to a logarithmic equation.
• Precalculus applies the properties of logarithms to simplify expressions and 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.
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• 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.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.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.
• 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.
P.5I Generate and solve exponential equations in mathematical and real-world problems.

Generate, Solve

EXPONENTIAL EQUATIONS IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Various solution strategies
• Solving exponential equations algebraically
• Simplifying expressions on both sides of an equation
• Rewriting exponential equations in logarithmic form
• Solving exponential equations with technology
• Graphs
• Tables
• Various situations
• Mathematical and real-world problem situations
• Exponential growth
• Exponential decay
• Other exponential behavior

Note(s):

• Algebra I analyzed and investigated quadratic and exponential functions and their applications.
• Algebra II analyzed and investigated logarithmic, exponential, absolute value, rational, square root, cube root, and cubic functions.
• Algebra II formulated and solved exponential and logarithmic equations.
• Algebra I and Algebra II analyzed and described the effects of transformations on the parent functions with changes in a, b, c, and d parameters.
• Precalculus extends these skills to generate and solve exponential equations in mathematical and real-world situations.
• 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.
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• 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.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.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.
• 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.