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

#### Unit Overview

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(x – c), 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.

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 B1, C1, D1, D2; III. Geometric Reasoning C1; VI. Statistical Reasoning B2, C3; VII. Functions B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

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

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

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

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

Transformations 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 and composition of functions create new functions.

• What relationships and characteristics exist between a function and its inverse?
• What is the purpose of composition of functions?
• Why would a function need to be decomposed into two or more functions?

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?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Expressions
• Multiple Representations
• Patterns/Rules
• Solve

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

Exponential functions have key attributes, including domain, range, zeros, asymptotes, intervals over which the function is increasing or decreasing, and end behavior.

• What are the key attributes of exponential functions?
• How can the key attributes of an exponential function be determined from multiple representations of the function?
• How can knowledge of the key attributes of an exponential function be used to sketch the graph of the function?
• How can the domain and range of an exponential function be determined and described?
• How can the zero of an exponential function be determined and described?
• How can the asymptotes of an exponential function be determined and described?
• How can the intervals where the exponential function is increasing and decreasing be determined and described?
• How can the end behavior of an exponential function be determined and described using infinity notation?

Logarithmic functions have key attributes, including domain, range, zeros, asymptotes, intervals over which the function is increasing or decreasing, and end behavior.

• What are the key attributes of logarithmic functions?
• How can the key attributes of a logarithmic function be determined from multiple representations of the function?
• How can knowledge of the key attributes of a logarithmic function be used to sketch the graph of the function?
• How can the domain and range of a logarithmic function be determined and described?
• How can the zero of a logarithmic function be determined and described?
• How can the asymptotes of a logarithmic function be determined and described?
• How can the intervals where the logarithmic function is increasing and decreasing be determined and described?
• How can the end behavior of a logarithmic function be determined and described using infinity notation?

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?

Transformations of exponential and logarithmic functions can be used to determine graphs and equations of representative functions in problem situations.

• What are the effects of changes on the graph of exponential and logarithmic functions when f(x) is replaced by af(x), for specific values of a?
• What are the effects of changes on the graph of exponential and logarithmic functions when f(x) is replaced by f(bx), for specific values of b?
• What are the effects of changes on the graph of exponential and logarithmic functions when f(x) is replaced by f(x - c) for specific values of c?
• What are the effects of changes on the graph of exponential and logarithmic functions when f(x) is replaced by f(x) + d, for specific values of d?
 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
• Patterns/Rules
• 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 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?
• How does the domain of a logarithmic function affect the solution of a logarithmic equation?
• What causes extraneous solutions in logarithmic equations?
• How can extraneous solutions be identified in graphs, tables, and algebraic calculations?

Exponential and 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.

• How are properties of logarithms used to evaluate or transform logarithmic expressions and/or equations?
• What representations can be used to display exponential and logarithmic function models?
• What key attributes identify an exponential or 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 an exponential or logarithmic function model and the real-world problem situation?
• How can exponential or 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 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(a – b) 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 Creator if your district has granted access to that tool.

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
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)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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:
• X. Connections
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:
• IX. Communication and Representation
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 g-1(x) = .
• If the restricted domain of g(x) is x ≤ 0, then the inverse function is g-1(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:
• II Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VI Statistical Reasoning
• B2 – Select and apply appropriate visual representations of data.
• C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 AND LOGARITHMIC, FUNCTIONS INCLUDING af(x), f(x) + d, f(x – c), f(bx) FOR SPECIFIC VALUES OF abc, 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) = exf(x) = 10x
• Logarithmic functions: f(x) = log2(x), f(x) = ln(x), f(x) = logx
• Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters abc, 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(x – c)
• 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:
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 ≥ 0}
• {y| 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  , –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:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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: logbx = y  by = x
• Common logarithms (base 10): log x = y  10y = x
• Natural logarithms (base e): ln x = y  ey = x
• Logarithms of a product: logb(xy) = logbx + logby
• Logarithms of a quotient: logb = logbx – logby
• Power rule of logarithms: logb(xr) = r logbx
• Change of base property: logbx =

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:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 abc, 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. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections