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INSTRUCTIONAL FOCUS DOCUMENTAlgebra II

TITLE : Unit 10: Exponential and Logarithmic Functions and Equations SUGGESTED DURATION : 10 days

UNIT OVERVIEW

This unit bundles student expectations that address the inverse relationships of exponential functions and logarithmic functions, transformations of logarithmic functions, andkey attributes of logarithmic functions. Student expectations also address connections between exponential and logarithmic equations and formulating, solving, and justifyingsolutions to exponential and logarithmic equations for problem situations.

Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards includingapplication, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skillsso that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Algebra I Unit 06, Unit 09, and Unit 11, students analyzed exponential functions and applied laws of exponents in mathematical and real-world problemsituations. In Algebra II Unit 04, students applied laws of exponents. In Unit 09, students analyzed transformations and key attributes of exponential functions. Students alsosolved and applied exponential equations in mathematical and real-world problem situations.

During this unit, students describe and analyze the inverse relationship between the exponential and logarithmic functions, including the restriction(s) on domain/range, and

graph and write the inverse functions using notation such as f -1 (x). Students graph the function f(x) = logb x where b is 2, 10, and e, and analyze the key attributes such as

domain, range, intercepts, and asymptotic behavior. Students determine the effects on the key attributes on the graph of f(x) = logb x where b is 2, 10, and e when f(x) is

replaced by af(x), f(x) + d, and f(x – c) for specific positive and negative real values of a, c, and d, and investigate parameter changes and key attributes in terms of real-worldproblem situations. Students make connections between exponential and logarithmic equations by rewriting exponential equations as their corresponding logarithmic equationsand logarithmic equations as their corresponding exponential equations. Students use this understanding to solve single logarithmic equations having real solutions and justifythe reasonableness of the solutions. Students formulate exponential and logarithmic equations that model real-world situations, solve the equations, and determine thereasonableness of the solution in terms of the problem situation.

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

In Algebra II, analyzing the inverse relationship between exponential and logarithmic functions and graphing, transforming, and analyzing key attributes of logarithmic functionsare identified in STAAR Readiness Standards 2A.2A, 2A.2C, and 2A.5A. Solving exponential equations and single logarithmic equations are identified in STAAR ReadinessStandard 2A.5D. These Readiness Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses and STAAR ReportingCategory 5: Exponential and Logarithmic Functions and Equations. Graphing and writing inverse functions using notation are identified in STAAR Supporting Standard 2A.2Band subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Writing exponential equations as logarithmic equations and writinglogarithmic equations as exponential equations are identified in STAAR Supporting Standard 2A.5C. Formulating exponential and logarithmic equations to model problemsituations and determining the reasonableness of solutions are identified in STAAR Supporting Standards 2A.5B and 2A.5E. These Supporting Standards are subsumed under

Last Updated 08/18/2016 Page 1 of 34Print Date 08/08/2017 Printed By Christina James, NOCONA H S

STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. This unit is supporting the development of Texas College and Career ReadinessStandards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, C1, D1, D2; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2, C2; VIII. ProblemSolving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics, students should develop an understanding of thealgebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12, “High schoolstudents continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility withrepresentation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages anddisadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation,choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions” (NCTM, 2002, p. 3). Research found in National Council ofTeachers of Mathematics also states, “Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplishedby working with symbolic representations alone” (as cited by NCTM, 2009, p. 41). This unit places particular emphasis on multiple representations. State and nationalmathematics standards support such an approach. The Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions, such asalgebraic, tabular, graphical, and verbal descriptions. This skill is mirrored in the Principles and Standards for School Mathematics (1989). Specifically, this work calls forinstructional programs that enable all students to understand relations and functions and select, convert flexibly among, and use various representations for them. Morerecently, according to National Council of Teachers of Mathematics (2007), the importance of multiple representations has been highlighted in Curriculum Focal Points forPrekindergarten through Grade 8 Mathematics. According to this resource, students should be able to translate among verbal, tabular, graphical, and algebraic representationsof functions and describe how aspects of a function appear in different representations as early as Grade 8. Also, in research summaries such as Classroom Instruction ThatWorks: Research-Based Strategies for Increasing Student Achievement, such concept development is even cited among strategies that increase student achievement.Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences has been statistically shown toimprove student performance on standardized measures of progress (Marzano, Pickering & Pollock, 2001).

Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA:Association for Supervision and Curriculum Development.National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.National Council of Teachers of Mathematics. (2007). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers ofMathematics, Inc.National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers ofMathematics, Inc.Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved fromhttp://www.thecb.state.tx.us/collegereadiness/crs.pdf




http://www.thecb.state.tx.us/collegereadiness/crs.pdf

OVERARCHING UNDERSTANDINGS AND QUESTIONS

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

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

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

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

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

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

Inverses of functions create new functions.




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

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

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

PERFORMANCE ASSESSMENT(S)OVERARCHING CONCEPTS

UNIT CONCEPTSUNIT UNDERSTANDINGS

Algebra II Unit 10 PA 01

Click on the PA title to view related rubric.

Teacher Note:

Graphing calculator required. Provide graph paper.

Student Note:

For each task:

Algebraic Reasoning

Multiple Representations

Functions

Attributes of Functions

Inverses of Functions

Non-Linear Functions

Logarithmic functions have unique graphs and attributes.

What representations can be used to represent a logarithmic

function?

How do the different bases of 2, 10, and e, affect the

representations of the logarithmic function?

What are the key attributes of a logarithmic function and how

can they be determined?




/module/content/search/item/689556/viewdetail.ashx



Analyze the situation(s).

Organize and record your work.

Use precise mathematical language to justify

and explain each mathematical process.

1. Given the function f(x) = 3(2)x, analyze f –1(x), using

tabular, graphical, and symbolic representations.

Describe and justify restrictions on the domain and

range of the inverse function in comparison to the

domain and range of the given function.

2. Given the functions f(x) = log(x) and

g(x) = log(x – 3) – 5

a. Graph and label f(x) and g(x) on a coordinate

plane.

b. Determine the effects of the parameter changes

on the graph of f(x) = log(x) when replaced by

g(x) = log(x – 3) – 5.

c. Identify and analyze the key attributes of f(x) and

g(x), including domain, range, intercepts, and

asymptotic behavior.

3. Given the functions f(x) = ln x and

g(x) = 3ln(x + 2) + 4.

Geometric Reasoning

Transformations

Associated Mathematical Processes

Application

Tools and Techniques

Problem Solving Model

Communication

Representations

Relationships

Justification

Transformations of the logarithmic parent function, f(x) = bx, can be

used to determine graphs and equations of representative logarithmic

functions in problem situations.

What are the effects of changes on the graph of f(x) = bx when

f(x) is replaced by af(x), for specific positive and negative values

of a?


f(x) is replaced by f(x) + d,for specific positive and negative

values of d?


f(x) is replaced by f(x – c) for specific positive and negative

values of c?

The inverse of a function can be determined from multiple

representations.

How can the inverse of a function be determined from the graph

of the function?

How can the inverse of a function be determined from a table of

coordinate points of the function?

How can the inverse of a function be determined from the

equation of the function?

How are a function and its inverse distinguished symbolically?

How are function compositions related to inverse functions?






a. Graph and label f(x) and g(x) on a coordinate

plane.

b. Determine the effects of the parameter changes

on the graph of f(x) = ln x when replaced by

g(x) = 3ln(x + 2) + 4.

c. Identify and analyze the key attributes of f(x) and

g(x), including domain, range, intercepts, and

asymptotic behavior.

Standard(s): 2A.1B , 2A.1C , 2A.1D , 2A.1E ,

2A.1F , 2A.1G , 2A.2A , 2A.2B , 2A.2C , 2A.5A

ELPS.c.1C , ELPS.c.1E , ELPS.c.2D , ELPS.c.4G

, ELPS.c.4K , ELPS.c.5F , ELPS.c.5G

How do the attributes of inverse functions compare to the

attributes of original functions?

The domain and range of the inverse of a function may need to be

restricted in order for the inverse to also be a function.

When must the domain of an inverse function be restricted?

How does the relationship between a function and its inverse,

including the restriction(s) on the domain, affect the

restriction(s) on its range?

Algebra II Unit 10 PA 02

Click on the PA title to view related rubric.

Teacher Note:

Graphing calculator required. Provide graph paper.

Student Note:

Algebraic Reasoning

Equations

Evaluate

Expressions

Multiple Representations

Solution Representations

Solve

Equations can be used to model and solve mathematical and real-

world problem situations.

How are real-world problem situations identified as ones that

can be modeled by exponential and logarithmic equations?

How are exponential and logarithmic equations used to model

problem situations?

What methods can be used to solve exponential and




/module/standards/0/183020/standard.ashx




















For each task:

Analyze the situation(s).

Organize and record your work.

Use precise mathematical language to justify

and explain each mathematical process.

1. Write the exponential equations as single

logarithmic equations.

a. e10 = x

b. (0.8)x = 240

c. x2 • x3 = 1024

2. Convert the logarithmic equations to their

corresponding exponential equations and solve for x.

a. log3 243 = x

b. log5 x – log5 2 = 3

c. logx 8 + logx 16 = 7

3. Josephina is preparing Indian pudding for a party.

She is cooking the pudding in a water-tight

container submerged in boiling water. When the

pudding is done, Josephina places the container of

pudding out in the snow to cool down. When

Functions

Attributes of Functions

Non-Linear Functions

Associated Mathematical Processes

Application

Tools and Techniques

Problem Solving Model

Communication

Representations

Relationships

Justification

logarithmic equations?

What are the advantages and disadvantages of various methods

used to solve exponential and logarithmic equations?

What methods can be used to justify the reasonableness of

solutions to exponential and logarithmic equations?

What causes extraneous solutions in exponential and

logarithmic equations?

How can extraneous solutions be identified in graphs, tables,

and algebraic calculations?

Exponential functions can be used to model real-world problem

situations by analyzing collected data, key attributes, and various

representations in order to interpret and make predictions and critical

judgments.

What representations can be used to display exponential

function models?

What key attributes identify an exponential function model?

How does the domain and range of the function compare to the

domain and range of the problem situation?

What are the connections between the key attributes of an

exponential function model and the real-world problem

situation?

How can exponential function representations be used to

interpret and make predictions and critical judgments in terms

of the problem situation?






Josephina places the Indian pudding in the snow,

the temperature of the pudding is 100°C. After 1

minute, the temperature of the pudding drops to

95°C.

a. Assuming the temperature is dependent on the

minutes the pudding remains in the snow,

formulate an exponential function to represent

the problem situation. Explain your process.

b. Graph the exponential function over an

appropriate domain and range. Identify the key

attributes including domain, range, intercept(s),

and asymptotic behavior and describe their

meaning in the problem situation.

c. Josephina wants to serve the pudding when it is

at a temperature of 10°C. How long will she need

to leave the Indian pudding cooling in the snow

for it to reach the desired temperature? Solve the

problem algebraically, and round the answer to

the nearest minute.

d. Use the graph of the function to justify your

response.

4. A titration of Solution A had already begun when

Brian began taking measurements of the

concentration values in moles. Solution B was being

Logarithmic functions can be used to model real-world problem

situations by analyzing collected data, key attributes, and various

representations in order to interpret and make predictions and critical

judgments.

What representations can be used to display logarithmic

function models?

What key attributes identify a logarithmic function model?

How does the domain and range of the function compare to the

domain and range of the problem situation?

What are the connections between the key attributes of a

logarithmic function model and the real-world problem situation?

How can logarithmic function representations be used to

interpret and make predictions and critical judgments in terms

of the problem situation?






used as a control and was not being titrated. The

concentration of Solution A was increasing and

represented by A(x) = 2log2 + log(x + 1), where x

represents time in hours since the beginning of the

titration. The concentration of Solution B was

constant and represented by B(x) = log16.

a. How long after Brian began taking concentration

measurements was the concentration of

Solution A equal to the concentration of Solution

B? What was the concentration amount? Solve

the problem algebraically.

b. How long was this from the beginning of the

experiment?

c. Justify the reasonableness of your response.

Standard(s): 2A.1A , 2A.1B , 2A.1C , 2A.1D ,

2A.1E , 2A.1F , 2A.1G , 2A.2A , 2A.5B , 2A.5C ,

2A.5D , 2A.5E ELPS.c.1E , ELPS.c.2D ,

ELPS.c.4D , ELPS.c.4G , ELPS.c.4K , ELPS.c.5F

, ELPS.c.5G

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

Some students, when rewriting an exponential equation to its logarithmic equation form, may confuse the position of the argument and exponent. Students may rewrite























the equation y = bx logb x = y rather than y = bx logb y = x.

UNIT VOCABULARY

Asymptotic behavior –behavior such that as x approaches infinity, f(x) approaches a given valueContinuous function –function whose values are continuous or unbroken over the specified domainDiscrete 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 definedExponential decay –an exponential function where 0 < b < 1 and as x increases, y decreases exponentiallyExponential growth –an exponential function where b > 1 and as x increases, y increases exponentiallyHorizontal asymptote –horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by thecurve.Inequality notation –notation in which the solution is represented by an inequality statementInterval notation –notation in which the solution is represented by a continuous interval

Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and f –1(f(x)) = x.Range –set of output values for the dependent variable over which the function is definedSet notation –notation in which the solution is represented by a set of valuesVertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

DecreasingDependentEvaluateExponent

Horizontal shifthorizontal stretchIncreasingIndependent

Parent functionreflectionVertical compressionVertical shift




Exponential regressionFunction notationHorizontal compression

Logarithmic regressionParameter change

Vertical stretchy = x line

UNIT ASSESSMENT ITEMS SYSTEM RESOURCES OTHER RESOURCES

Unit Assessment Items that have been published byyour district may be accessed through Search AllComponents in the District Resources tab.Assessment items may also be found using theAssessment Creator if your district has granted accessto that tool.

Mathematics Algebra II STAAR EnhancedBlueprint

Mathematics Algebra II TEKS Introduction withAligned Standards

Mathematics Concepts Tree

Texas Education Agency Mathematics Algebra IITEKS Supporting Information (with TEKSResource System comments)

Texas Higher Education Coordinating Board – TexasCollege and Career Readiness Standards (selectCCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR MathematicsResources

Texas Education Agency – Revised MathematicsTEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response toCurriculum Focal Points for K-8 MathematicsRevised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS:Supporting Information

Texas Education Agency – Interactive MathematicsGlossary




/module/content/search/tcmpcdistrict/





http://www.thecb.state.tx.us/index.cfm?objectid=BCA1DEF2-02B0-B3FB-5A72BD7F7FB2448E

http://texasgateway.org/resource/interactive-math-glossary

TEKS UNIT LEVEL SPECIFICITY

Bold black text in italics: Knowledge andSkills Statement (TEKS)Bold black text: Student Expectation (TEKS)Bold red text in italics: Student Expectationidentified by TEA as a Readiness Standard forSTAARBold green text in italics: Student Expectationidentified by TEA as a Supporting Standard forSTAARStrike-through: Indicates portions of the StudentExpectation that are not included in this unit butare taught in previous or future unit(s)

Blue text: Supporting information / Clarifications from TCMPC (Specificity)Blue text in italics: Unit-specific clarificationBlack text: Texas Education Agency (TEA); Texas College and Career Readiness Standards(TxCCRS)

2A.1 Mathematical process standards. The student usesmathematical processes to acquire anddemonstrate mathematical understanding. Thestudent is expected to:

2A.1A Apply mathematics to problems arising ineveryday 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 disciplinesEveryday lifeSocietyWorkplace

TEKS#

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/module/standards/Tools/Browse?StandardId=183015


TEKS UNIT LEVEL SPECIFICITYTEKS#

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Note(s):

The mathematical process standards may be applied to all content standards as appropriate.TxCCRS:

X. Connections

2A.1B Use a problem-solving model that incorporatesanalyzing given information, formulating a planor strategy, determining a solution, justifying thesolution, and evaluating the problem-solvingprocess 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 THESOLUTION


Problem-solving modelAnalyze given informationFormulate a plan or strategyDetermine a solutionJustify the solutionEvaluate the problem-solving process and the reasonableness of the solution

Note(s):


VIII. Problem Solving and Reasoning







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2A.1C Select tools, including real objects,manipulatives, paper and pencil, and technologyas appropriate, and techniques, including mentalmath, estimation, and number sense asappropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGYAS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBERSENSE AS APPROPRIATE, TO SOLVE PROBLEMS


Appropriate selection of tool(s) and techniques to apply in order to solve problemsTools

Real objectsManipulativesPaper and pencilTechnology

TechniquesMental mathEstimationNumber sense

Note(s):


VIII. Problem Solving and Reasoning

2A.1D Communicate mathematical ideas, reasoning,and their implications using multiple Communicate







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representations, including symbols, diagrams,graphs, and language as appropriate. MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE

REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE ASAPPROPRIATE


Mathematical ideas, reasoning, and their implicationsMultiple representations, as appropriate

SymbolsDiagramsGraphsLanguage

Note(s):


IX. Communication and Representation

2A.1E Create and use representations to organize,record, and communicate mathematical ideas. Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS


Representations of mathematical ideasOrganizeRecordCommunicate






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Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas beingcommunicatedAppropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):



2A.1F Analyze mathematical relationships to connectand communicate mathematical ideas. Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS


Mathematical relationshipsConnect and communicate mathematical ideas

Conjectures and generalizations from sets of examples and non-examples,patterns, etc.Current knowledge to new learning

Note(s):


X. Connections







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2A.1G Display, explain, or justify mathematical ideasand arguments using precise mathematicallanguage in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE INWRITTEN OR ORAL COMMUNICATION


Mathematical ideas and argumentsValidation of conclusions

Displays to make work visible to othersDiagrams, visual aids, written work, etc.

Explanations and justificationsPrecise mathematical language in written or oral communication

Note(s):



2A.2 Attributes of functions and their inverses. Thestudent applies mathematical processes tounderstand that functions have distinct keyattributes and understand the relationship betweena function and its inverse. The student is expectedto:

2A.2A Graph the functions f(x)= , f(x)=1/x, f(x)=x3, f(x)=

, f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2,Graph








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10, and e, and, when applicable, analyze the keyattributes such as domain, range, intercepts,symmetries, asymptotic behavior.

Readiness Standard

THE FUNCTIONS f(x) = , f(x)= , f(x) = x3, f(x) = , f(x) = bx, f(x) = |x|, AND f(x) = logb (x)

WHERE b IS 2, 10, AND e


Representations of functions, including graphs, tables, and algebraic generalizations

Exponential, f(x) = bx, where b is 2, 10, and Connections between representations of families of functionsComparison of similarities and differences of families of functions

Analyze

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


Domain and range of the functionDomain – set of input values for the independent variable over which the function isdefined

Continuous function – function whose values are continuous or unbroken over thespecified domainDiscrete function – function whose values are distinct and separate and notconnected; values are not continuous. Discrete functions are defined by theirdomain.

Range – set of output values for the dependent variable over which the function is definedRepresentation for domain and range

Verbal descriptionEx: x is all real numbers less than five.





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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 inequalitystatement

Ex: x < 5, x Ex: x Ex: –3 < y ≤ 6, x Ex: y ≥ 0, y

Set notation – notation in which the solution is represented by a set of valuesBraces are used to enclose the set.Solution is read as “The set of x such that x is an element of …”Ex: x|x , x < 5Ex: x|x Ex: y|y , –3 < y ≤ 6Ex: y|y , y ≥ 0

Interval notation – notation in which the solution is represented by a continuousinterval

Parentheses indicate that the endpoints are open, meaning the endpointsare excluded from the interval.Brackets indicate that the endpoints are closed, meaning the endpointsare included in the interval.Ex: (– , 5)Ex: (– , )Ex: (–3, 6]

Domain and range of the function versus domain and range of the contextual situationKey attributes of functions

Intercepts/Zerosx-intercept(s) – x coordinate of a point at which the relation crosses the x-axis,





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meaning the y coordinate equals zero, (x, 0)Zeros – the value(s) of x such that the y value of the relation equals zeroy-intercept(s) – y coordinate of a point at which the relation crosses the y-axis,meaning the x coordinate equals zero, (0, y)

Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches agiven value

Vertical asymptote – vertical line approached by the curve as the functionapproaches positive or negative infinity. Vertical asymptotes are never crossed bythe curve.Horizontal asymptote – horizontal line approached by the curve as the functionapproaches positive or negative infinity. Horizontal asymptotes may be crossedby the curve.

Use key attributes to recognize and sketch graphsApplication of key attributes to real-world problem situations

Note(s):

Grade Level(s):The notation represents the set of real numbers, and the notation represents theset of integers.

Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = bx and their keyattributes.Precalculus will study polynomial, power, trigonometric, inverse trigonometric, andpiecewise defined functions, including step functions.Various mathematical process standards will be applied to this student expectation asappropriate.

TxCCRS:III. Geometric Reasoning

B1 – Identify and apply transformations to figures.C1 – Make connections between geometry and algebra.





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VII. FunctionsA1 – Recognize whether a relation is a function.A2 – Recognize and distinguish between different types of functions.B1 – Understand and analyze features of a function.

VIII. Problem Solving and ReasoningIX. Communication and RepresentationX. Connections

2A.2B Graph and write the inverse of a function usingnotation such as f -1(x).

Supporting Standard

Graph, Write

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


Inverse of a function – function that undoes the original function. When composed f(f –1(x))

= x and f –1(f(x)) = x.Inverse functions

Exponential and logarithmicInverses of functions on graphs

Symmetric to y = x Inverses of functions in tables

Interchange independent (x) and dependent (y) coordinates in ordered pairsInverses of functions in equation notation

Interchange independent (x) and dependent (y) variables in the equation, then solve for yInverses of functions in function notation

f –1(x) represents the inverse of the function f(x).

Note(s):






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Grade Level(s):Algebra II introduces inverse of a function.Various mathematical process standards will be applied to this student expectation asappropriate.


C1 – Make connections between geometry and algebra.VII. Functions

A1 – Recognize whether a relation is a function.A2 – Recognize and distinguish between different types of functions.B1 – Understand and analyze features of a function.B2 – Algebraically construct and analyze new functions.


2A.2C Describe and analyze the relationship between afunction and its inverse (quadratic and squareroot, logarithmic and exponential), including therestriction(s) on domain, which will restrict itsrange.

Readiness Standard

Describe, Analyze

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


Relationships between functions and their inversesAll inverses of functions are relations.Inverses of one-to-one functions are functions.Inverses of functions that are not one-to-one can be made functions by restricting the






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domain of the original function, f(x).Characteristics of inverse relations

Interchange of independent (x) and dependent (y) coordinates in ordered pairsReflection over y = x

Domain and range of the function versus domain and range of the inverse of the given functionFunctionality of the inverse of the given function

Exponential function and logarithmic function, f(x) = bx and g(x) = logb (x) where b is 2, 10, and

e

Note(s):

Grade Level(s):Algebra I determined if relations represented a function.Algebra II introduces inverse of a function and restricting domain to maintain functionality.Various mathematical process standards will be applied to this student expectation asappropriate.


C1 – Make connections between geometry and algebra.VII. Functions

A1 – Recognize whether a relation is a function.A2 – Recognize and distinguish between different types of functions.B1 – Understand and analyze features of a function.B2 – Algebraically construct and analyze new functions.


2A.5 Exponential and logarithmic functions and






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equations. The student applies mathematicalprocesses to understand that exponential andlogarithmic functions can be used to modelsituations and solve problems. The student isexpected to:

2A.5A Determine the effects on the key attributes on thegraphs of f(x) = bx and f(x) = logb

(x) where b is 2,

10, and e when f(x) is replaced by af(x), f(x) + d,and f(x - c) for specific positive and negative realvalues of a, c, and d. Readiness Standard

Determine

THE EFFECTS ON THE KEY ATTRIBUTES ON THE GRAPHS

OF f(x) = bx AND f(x) = logb(x) WHERE b IS 2, 10, AND e WHEN f(x) IS REPLACED BY af(x), f(x) +

d, and f(x – c) FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, c, AND d


General form of the power function

Exponential functions, f(x) = bx, where b is 2, 10, and e

f(x) = 2x; f(x) = 10x; f(x) = ex

Logarithmic functions, y = logbx, where b is 2, 10, and e

f(x) = log2x; f(x) = log10x or f(x) = log(x); f(x) = logex or f(x) = ln(x)

Representations with and without technologyGraphsTablesVerbal descriptionsAlgebraic generalizations

Effects on the graphs of f(x) = bx and y = logbx when parameters a, b, c, and d are changed in

f(x) = a • b(x – c) + d and f(x) = a • logb(x – c) + d






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Effects on the graphs of f(x) = 2x and f(x) = log2x, when f(x) is replaced by af(x) with and

without technologya ≠ 0|a| > 1, the graph stretches vertically0 < |a| < 1, the graph compresses verticallyOpposite of a reflects vertically over the x-axis

Effects on the graphs of f(x) = 10x and f(x) = logx, when f(x) is replaced by f(x – c) withand without technology

c = 0, no horizontal shiftHorizontal shift left for values of c < 0 by |c| units

For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left twounits.

Horizontal shift right for values of c > 0 by |c| unitsFor f(x – 2), c = 2, and the function moves to the right two units.

Effects on the graphs of f(x) = ex and f(x) = ln(x), when f(x) is replaced by f(x) + d withand without technology

d = 0, no vertical shiftVertical shift down for values of d < 0 by |d| unitsVertical shift up for values of d > 0 by |d| units

Connections between the critical attributes of transformed functions and f(x) = bx and y = logbx

Determination of parameter changes given a graphical or algebraic representationDetermination of a graphical representation given the algebraic representation orparameter changesDetermination of an algebraic representation given the graphical representation orparameter changes

Descriptions of the effects on the domain and range by the parameter changesEffects of multiple parameter changes

Mathematical problem situation





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Effects of parameter changes in real-world problem situations

Note(s):

Grade Level(s):

Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x2

when f(x) is replaced by af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d.Algebra II continues to investigate the exponential parent function and introduceslogarithmic parent function and transformations of both functions.Various mathematical process standards will be applied to this student expectation asappropriate.


B1 – Identify and apply transformations to figures.C1 – Make connections between geometry and algebra.

VII. FunctionsA1 – Recognize whether a relation is a function.A2 – Recognize and distinguish between different types of functions.B1 – Understand and analyze features of a function.


2A.5B Formulate exponential and logarithmic equationsthat model real-world situations, includingexponential relationships written in recursivenotation.

Formulate

EXPONENTIAL AND LOGARITHMIC EQUATIONS THAT MODEL REAL-WORLD SITUATIONS






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Supporting StandardIncluding, but not limited to:

Data collection activities with and without technologyData modeled by exponential functions

Exponential growth – an exponential function where b > 1 andas x increases, y increases exponentiallyExponential decay – an exponential function where 0 < b < 1 andas x increases, y decreases exponentially

Data modeled by logarithmic functionsReal-world problem situations

Real-world problem situations modeled by exponential functionsExponential growthExponential decay

Real-world problem situations modeled by logarithmic functionsRepresentations of exponential and logarithmic equations

Tables/graphsVerbal descriptions

Technology methods

Transformations of f(x) = bx and y = logbx

Exponential regressionLogarithmic regression

Note(s):

Grade Level(s):Algebra II introduces formulating exponential and logarithmic equations.Various mathematical process standards will be applied to this student expectation asappropriate.

TxCCRS:VII. Functions





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B1 – Understand and analyze features of a function.B2 – Algebraically construct and analyze new functions.C2 – Develop a function to model a situation.


2A.5C Rewrite exponential equations as theircorresponding logarithmic equations andlogarithmic equations as their correspondingexponential equations. Supporting Standard

Rewrite

EXPONENTIAL EQUATIONS AS THEIR CORRESPONDING LOGARITHMIC EQUATIONS ANDLOGARITHMIC EQUATIONS AS THEIR CORRESPONDING EXPONENTIAL EQUATIONS


Laws (properties) of exponents

Product of powers (multiplication when bases are the same): am • an = am+n

Quotient of powers (division when bases are the same): = am-n

Power to a power: (am)n = amn

Negative exponents: a-n =

Zero exponent: a0 = 1Laws (properties) of logarithms

Product: logb(p • q) = logbp + logbq; ln(p • q) = lnp + lnq

Quotient: logb = logbp – logbq; ln = lnp – lnq

Power: logbpq = qlogbp; lnp

q = qlnp






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Reciprocal: logb = –logbp; ln = –lnp

Log of base: logbb = 1; lne = 1

Log of 1: logb1 = 0; ln1 = 0

Exponential equations to corresponding logarithmic equations

bx = A → logb(A) = x

Logarithmic equations to corresponding exponential equations

logb(A) = x → bx = A

Note(s):

Grade Level(s):Algebra I applied exponential functions to problem situations using tables, graphs, andthe algebraic generalization,

f(x) = a • bx.Algebra II introduces logarithms.Algebra II connects between exponential equations and logarithmic equations.Various mathematical process standards will be applied to this student expectation asappropriate.

TxCCRS:





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I. Numeric ReasoningB1 – Perform computations with real and complex numbers.

II. Algebraic ReasoningA1 – Explain and differentiate between expressions and equations using wordssuch as “solve,” “evaluate,” and “simplify.”C1 – Recognize and use algebraic (field) properties, concepts, procedures, andalgorithms 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 ReasoningC1 – Make connections between geometry and algebra.


2A.5D Solve exponential equations of the form y = abx

where a is a nonzero real number and b isgreater than zero and not equal to one and singlelogarithmic equations having real solutions. Readiness Standard

Solve

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


Exponential equation, y = abx

a – initial value at x = 0b – common ratio

Solving exponential equationsApplication of laws (properties) of exponentsApplication of logarithms as necessary






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Real-world problem situations modeled by exponential functionsExponential growth

f(x) = abx, where b > 1

f(x) = aekx, where k > 0Exponential decay

f(x) = abx, where 0 < b < 1

f(x) = aekx, where k < 0

Solve

SINGLE LOGARITHMIC EQUATIONS HAVING REAL SOLUTIONS


Single logarithmic equation, y = logbx

x – argumentb – basey – exponent

Solving logarithmic equationsTransformation to exponential form as necessary

Real-world problem situations modeled by logarithmic functions

Note(s):

Grade Level(s):Algebra I applied exponential functions to problem situations using tables, graphs, andthe algebraic generalization,

f(x) = a • bx.Algebra II solves exponential equations algebraically.





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Algebra II introduces logarithms and solving logarithmic equations.Various mathematical process standards will be applied to this student expectation asappropriate.

TxCCRS:I. Numeric Reasoning

B1 – Perform computations with real and complex numbers.II. Algebraic Reasoning

A1 – Explain and differentiate between expressions and equations using wordssuch as “solve,” “evaluate,” and “simplify.”C1 – Recognize and use algebraic (field) properties, concepts, procedures, andalgorithms 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.



2A.5E Determine the reasonableness of a solution to alogarithmic equation. Supporting Standard

Determine

THE REASONABLENESS OF A SOLUTION TO A LOGARITHMIC EQUATION


Justification of solutions to logarithmic equations with and without technologyVerbal descriptionTablesGraphs






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Substitution of solutions into original functionsJustification of reasonableness of solutions in terms of mathematical and real-world problemsituations or data collections

Note(s):

Grade Level(s):Algebra II introduces logarithms and solving logarithmic equations.Various mathematical process standards will be applied to this student expectation asappropriate.

TxCCRS:I. Numeric Reasoning

B1 – Perform computations with real and complex numbers.II. Algebraic Reasoning

A1 – Explain and differentiate between expressions and equations using wordssuch as “solve,” “evaluate,” and “simplify.”C1 – Recognize and use algebraic (field) properties, concepts, procedures, andalgorithms 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.






The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English languageproficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part ofeach subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodatedcommensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.

School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of theELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4

Choose appropriate ELPS to support instruction.

ELPS# SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.

Last Updated 08/18/2016




http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4

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TEKS Resource System · 2017-08-08 · 29(5$5&+,1*81'(567$1',1*6$1'48(67,216 (txdwlrqvfdqprghosureohpvlwxdwlrqvdqgehvroyhgxvlqjydulrxvphwkrgv :k\duhhtxdwlrqvxvhgwrprghosureohpvlwxdwlrqv

TEKS Resource System · 2017-08-08 · 29(5$5&+,181'(567$1',16$1'48(67,216 (txdwlrqvfdqprghosureohpvlwxdwlrqvdqgehvroyhgxvlqjydulrxvphwkrgv :k\duhhtxdwlrqvxvhgwrprghosureohpvlwxdwlrqv