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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 09: Investigation and Application of Exponential Functions SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address graphing and identifying key attributes of exponential functions and writing exponential functions in the form f(x) = abx. 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 Unit 02, students explored various representations of functions such as graphs, tables, verbal descriptions, and algebraic generalizations. Students investigated graphs and key features of functions, including domain and range written in inequality notation.

During this Unit
Students graph exponential functions that model growth and decay. Students identify key features, including y-intercept and asymptote, and determine the domain and range of exponential functions in the form f(x) = abx, representing the domain and range using inequality notation and verbal descriptions. Students interpret the effects of the values of a and b in exponential functions in the form f(x) = abx and write exponential functions in the form f(x) = abx (where b is a rational number greater than 0) to describe problems arising from mathematical and real-world situations, including growth and decay. Students use technology to write exponential functions that provide a reasonable fit to data to estimate solutions, make predictions, and justify solutions in terms of the problem situation for real-world problems and data collection activities.

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

After this Unit
In Unit 10, students will compare exponential functions to geometric sequences. In Unit 11, students will review exponential functions and their applications. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving exponential functions and equations.

In Algebra I, graphing and identifying key attributes of exponential functions are identified as STAAR Readiness Standard A.9D. Writing exponential functions to describe mathematic and real-world problem situations is identified as STAAR Readiness Standard A.9C. These STAAR Readiness Standards are subsumed under STAAR Reporting Category 5: Exponential Functions and Equations. Identifying domain and range of exponential functions is identified as STAAR Supporting Standard A.9A. Writing exponential functions and interpreting the meaning of a and b for exponential functions in problem situations are identified as STAAR Supporting Standards A.9B and A.9E. These STAAR Supporting Standards are subsumed under STAAR Reporting Category 5: Exponential Functions and Equations. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VI. Functions B1, C1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

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

Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6 – 10. Portsmouth, VA: Heinemann.
National Council of Teachers of Mathematics. (1995). Curriculum and evaluation standards for school mathematics: Algebra in a technological world. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Functions, Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (1998). High school mathematics at work: Essays and examples for the education of all students. Washington, DC: National Academy Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationship?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• 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.
• What kinds of mathematical and real-world situations can be modeled by exponential functions?
• What graphs, key attributes, and characteristics are unique to exponential functions?
• What patterns of covariation are associated with exponential functions?
• How can the key attributes of exponential functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of an exponential function?
• What are the real-world meanings of the key attributes of an exponential function model?
• How can the key attributes of an exponential function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
• How 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 equations?
• How does the given information and/or representation influence the selection of an efficient method for writing exponential equations?
• Functions, Equations, and Inequalities
• Attributes of functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Asymptotes
• Functions and Equations
• Exponential
• Relations and Generalizations
• Statistical Relationships
• Regression methods
• 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 the variable is the base and the numeric base is the exponent.
• Some students may think that the graph of the exponential parent function will at some point cross the x-axis rather than being asymptotic and only approaching the x-axis.

#### Unit Vocabulary

• Asymptote – a line that is approached and may or may not be crossed
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Domain – set of input values for the independent variable over which the function is defined
• Exponential decay –an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Range – set of output values for the dependent variable over which the function is defined
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)

Related Vocabulary:

 a value b value Base Decreasing Exponent Increasing Initial value Rational number
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

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

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway –Resources Aligned to Algebra I Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

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

Note(s):

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

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

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

Note(s):

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

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

Note(s):

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

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
A.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
A.9 Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:
A.9A Determine the domain and range of exponential functions of the form f(x) = abx and represent the domain and range using inequalities.
Supporting Standard

Determine, Represent

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

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• Domain and range of exponential functions in mathematical problem situations
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Domain and range values in real-world problem situations
• Reasonable domain and range for the real-world problem situation
• Comparison of domain and range of a function model to appropriate domain and range for real-world problem situation
• Inequality representations
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6
• Ex: y ≥ 0, yΖ

Note(s):

• Grade 6 identified independent and dependent quantities.
• Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I introduces exponential functions.
• Algebra I introduces the concept of domain and range of a function.
• Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
• Algebra II will extend the concept of domain and range.
• Algebra II will introduce representing domain and range using interval and set notation.
• Algebra II will continue to investigate exponential functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
A.9B Interpret the meaning of the values of a and b in exponential functions of the form f(x) = abx in real-world problems.
Supporting Standard

Interpret

THE MEANING OF THE VALUES OF a AND b IN EXPONENTIAL FUNCTIONS OF THE FORM f(x) = abx IN REAL-WORLD PROBLEMS

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• Exponential functions in real-world problem situations
• Representative exponential function for the real-world problem situation
• Meaning of a and b in terms of the real-world problem situation
• Growth rate, r, is b – 1
• b = 1 + r, where r is in decimal form
• Rate of decay, r, is 1 – b
• b = 1 – r, where r is in decimal form

Note(s):

• Algebra I introduces exponential functions.
• Algebra II will continue to investigate exponential functions and equations.
• 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.
• 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.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
A.9C Write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay.

Write

EXPONENTIAL FUNCTIONS IN THE FORM f(x) = abx (WHERE b IS A RATIONAL NUMBER) TO DESCRIBE PROBLEMS ARISING FROM MATHEMATICAL AND REAL-WORLD SITUATIONS, INCLUDING GROWTH AND DECAY

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• The b value is a rational number greater than 0.
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Exponential functions in real-world problem situations.
• Representative exponential function for the real-world problem situation
• Meaning of a and b in terms of the real-world problem situation
• Growth rate, r, is b – 1
• b = 1 + r, where r is in decimal form
• Rate of decay, r, is 1 – b
• b = 1 – r, where r is in decimal form
A.9D Graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems.

Graph

EXPONENTIAL FUNCTIONS THAT MODEL GROWTH AND DECAY

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• The b value is a rational number greater than 0.
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Mathematical problem situations
• Real-world problem situations

Identify

KEY FEATURES, INCLUDING y-INTERCEPT AND ASYMPTOTE, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
•  Key attributes
• y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• y-intercept in an exponential function: (0, a) where a is the a value in f(x) = abx
• Asymptote – a line that is approached and may or may not be crossed
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra I introduces exponential functions.
• Algebra II will continue to investigate exponential functions, including continuous growth and decay.
• 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.
• 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.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
A.9E Write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems.
Supporting Standard

Write

EXPONENTIAL FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• The b value is a rational number greater than 0.
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Concrete models
• Data representations
• Real-world problem situations
• Data collections using technology
• Technology to determine a function model using exponential regression

Make

PREDICTIONS FOR REAL-WORLD PROBLEMS

Including, but not limited to:

• Real-world problem situations represented by exponential functions
• Exponential functions in the form f(x) = abx
• Predictions in terms of the real-world problem situation
• Reasonableness of predictions in terms of the real-world problem situation

Note(s):

• Algebra I introduces exponential functions.
• Algebra I predicts solutions for exponential functions.
• Algebra II will continue to investigate exponential functions, including continuous growth and decay.
• Aglebra II will formulate and solve exponential functions and equations.
• Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• 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.
• 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.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.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.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.