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Algebra II
TITLE : Unit 01: Introduction to Functions in Algebra II SUGGESTED DURATION : 15 days

Unit Overview

This unit bundles student expectations that address an introductory exploration of the parent functions covered in Algebra II, attributes of functions including domain and range, and inverses of functions. 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 Grade 8, students were introduced to linear functions. In Algebra I, students experienced an in-depth study of linear and quadratic functions and their characteristics. Also in Algebra I, students investigated exponential functions. In Geometry, students analyzed pattern situations and represented them using linear and quadratic functions as appropriate.

During this unit, students explore representations of the families of functions to be covered in Algebra II, f(x) = , f(x) = , f(x) = x3, f(x) = , f(x) = bxf(x) = |x| and f(x) = logb x, where b is 2, 10, and e, including tables, graphs, verbal descriptions, and algebraic generalizations. Students analyze key attributes of functions, including domain (represented in interval notation, inequalities, set notation), range (represented in interval notation, inequalities, set notation), intercepts, symmetries, and asymptotic behaviors. Using various representations, students describe and analyze the relationships between a function and its inverse (quadratic and square root, logarithmic and exponential, cubic and cube root), including restriction(s) on domain. Students graph and write the inverses of functions in function notation using notation such as f–1(x). Students use composition of two functions, including domain restrictions, to determine if functions are inverses of one another.

After this unit, students will experience a more in-depth study of each family of functions by describing and applying the parent function, identifying the transformational effects of parameter changes on the graph of the parent function, and analyzing the characteristics of the family of functions. Representative functions will also be used to formulate and solve equations and inequalities in problem situations. Inverses will be addressed again when comparing and contrasting the quadratic and square root functions, exponential and logarithmic functions, and cubic and cube root functions. The concepts in this unit will also be applied in subsequent mathematics courses.

In Algebra II, graphing families of functions and identifying their key attributes is identified as STAAR Readiness Standard 2A.2A. Describing and analyzing inverse functions is identified as STAAR Readiness Standard 2A.2C. All STAAR Readiness Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Writing and graphing inverse functions is identified as STAAR Supporting Standard 2A.2B. Using the composition of functions to determine if functions are inverses is identified as STAAR Supporting Standard 2A.2D. Both STAAR Supporting Standard 2A.2B and 2A.2D are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Writing the domain and range of a function in inequality, set, and interval notation is STAAR Supporting Standard 2A.7I and part of STAAR Reporting Category 1: Number and Algebraic Methods. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.


According to the National Council of Teachers of Mathematics (NCTM):

The Algebra Standard emphasizes relationships among quantities and the ways in which quantities change relative to one another. To think algebraically, one must be able to understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; and analyze change in various contexts. In high school, students create and use tables, symbols, graphs, and verbal representations to generalize and analyze patterns, relations, and functions with increasing sophistication, and they convert flexibly among various representations. They compare and contrast situations modeled by different types of functions, and they develop an understanding of classes of functions, both linear and nonlinear, and their properties. (NCTM, 2002, p. 2-3)

In addition, Focus in High School Mathematics: Reasoning and Sense Making (2009) from the National Council of Teachers of Mathematics (NCTM), states “Functions are one of the most important mathematical tools for helping students make sense of the world around them, as well as preparing them for further study in mathematics” and lists the key elements of reasoning and sense making with functions as being “using multiple representations of functions, modeling by using families of functions” (p. 41). According to NCTM (2000), students need to learn to use a wide range of explicitly and recursively defined functions to model the world around them. Navigating through Data Analysis in Grades 9 – 12 (2003) states, “a fundamental goal of the mathematics curriculum: to develop critical thinking and sound judgment based on data” (NCTM, p. 1). According to Navigating through Algebra in Grades 9 – 12 (2002), “The Algebra Standard emphasizes relationships among quantities and the ways in which quantities change relative to one another. To think algebraically, one must be able to understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; and analyze change in various contexts” (NCTM, p. 2).

 

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. (2003). Navigating through data analysis in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf

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?

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?

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

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

Algebraic Reasoning

  • Inequalities
  • Multiple Representations

Functions

  • Attributes of Functions
  • Independent/Dependent Variables
  • Linear Functions
  • Non-linear Functions

Associated Mathematical Processes

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

 

Data collected from real-world situations can be matched with an appropriate function family comparing key attributes from the graph of the data with key attributes from the graph of the parent function.

  • What are the key attributes of the following parent functions: f(x) = , f(x) = , f(x) = x3, f(x) = , f(x) = bxf(x) = |x| and f(x) = logb x, where b is 2, 10, and e?
  • How can the function family be determined for a set of data?

Domain and range are key attributes of each function.

  • What are the domain and range of a function?
  • How can the domain and range of a function be determined?
  • How can domain and range be written?
Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

  • Multiple Representations

Functions

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

Associated Mathematical Processes

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

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

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

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

  • When must the domain of an inverse function be restricted?
  • How does the relationship between a function and its inverse, including the restriction(s) on the domain, affect the restriction(s) on its range?

The composition of two functions can be used to determine if the functions are inverse relationships.

  • How does taking the composition of two functions prove the functions are inverses?
  • When must the domain of an inverse function be restricted?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think that the domain and range of a function and the domain and range of the problem situation are always equivalent rather than that the domain and range of the problem situation may be restricted by constraints in the problem.
  • Some students may think that the inverse of a function means to take the opposite sign or reciprocal of the function rather than switching the independent and dependent variables.

Underdeveloped Concepts:

  • Some students may not have an understanding of the distinctions between discrete and continuous domains or how to represent discrete domains graphically and symbolically.
  • Some students may only be able to create situations representing linear and quadratic models at the beginning of Algebra II, since these are the functions covered in depth in Algebra I.

Unit Vocabulary

  • Asymptotic Behavior – value or line approached as the variable approaches infinity
  • Composition of Functions – process of substituting one function into another function to create a new function such that the range of one function becomes the domain of the other
  • 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
  • Horizontal Asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve
  • Inequality Notation – notation in which the solution is represented by an inequality statement
  • Interval Notation – notation in which the solution is represented by a continuous interval
  • Inverse of a Function – function that undoes the original function. When composed f(f--1(x)) = x and f--1(f(x)) = x
  • Range – set of output values for the dependent variable over which the function is defined
  • Reflectional Symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
  • Rotational Symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry. 
  • Set Notation – notation in which the solution is represented by a set of values
  • Vertical Asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve
  • x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
  • y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
  • Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

  • Absolute value function
  • Cube root function
  • Cubic function
  • Decreasing function
  • Dependent variable
  • Exponential function
  • Function notation
  • Linear function
  • Independent variable
  • Logarithmic function
  • Increasing function
  • Quadratic function
  • Rational function
  • Relation
  • Representations
  • Square root function
Unit Assessment Items System Resources Other Resources

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

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

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

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

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

Apply

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

Including, but not limited to:

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

Note(s):    

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

Use

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

Including, but not limited to:

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

Note(s):    

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

Select

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

Including, but not limited to:

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

Note(s):    

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

Communicate

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

Including, but not limited to:

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

Note(s):    

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):    

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

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):    

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

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • IX. Communication and Representation
2A.2 Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:
2A.2A

Graph the functions f(x)=, f(x)=1/x, f(x)=x3, f(x)=, f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.


Readiness Standard

Graph

THE FUNCTIONS f(x) = , f(x), f(x) = x3, f(x) = , f(x) = bx, f(x) = |x|, AND f(x) = logb (x) WHERE b IS 2, 10, AND e

Including, but not limited to:

  • Representations of functions, including graphs, tables, and algebraic generalizations
    • Square root, f(x) = 
    • Rational (reciprocal of x), f(x
    • Cubic, f(x) = x3
    • Cube root, f(x) = 
    • Exponential, f(x) = bx, where b is 2, 10, and e
    • Absolute value, f(x) = |x|
    • Logarithmic, f(x) = logbx, where b is 2, 10, and e
  • Connections between representations of families of functions
  • Comparison of similarities and differences of families of functions

Analyze

THE KEY ATTRIBUTES OF THE FUNCTIONS SUCH AS DOMAIN, RANGE, INTERCEPTS, SYMMETRIES, ASYMPTOTIC BEHAVIOR, WHEN APPLICABLE

Including, but not limited to:

  • Domain and range of the function
    • Domain – set of input values for the independent variable over which the function is defined
      • Continuous function – function whose values are continuous or unbroken over the specified domain
      • Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
    • Range – set of output values for the dependent variable over which the function is defined
    • Representation for domain and range
      • Verbal description
        • Ex: x is all real numbers less than five.
        • Ex: x is all real numbers.
        • Ex: y is all real numbers greater than –3 and less than or equal to 6.
        • Ex: y is all integers greater than or equal to zero.
      • Inequality notation – notation in which the solution is represented by an inequality statement
        • Ex: x < 5, x  
        • Ex: x  
        • Ex: –3 < y ≤ 6, x  
        • Ex: y ≥ 0, y  
      • Set notation – notation in which the solution is represented by a set of values
        • Braces are used to enclose the set.
        • Solution is read as “The set of x such that x is an element of …”
        • Ex: {x|x  x < 5}
        • Ex: {x|x  }
        • Ex: {y|y  , –3 < y ≤ 6}
        • Ex: {y|y  y ≥ 0}
      • Interval notation – notation in which the solution is represented by a continuous interval
        • Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
        • Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
        • Ex: (–, 5)
        • Ex: (–)
        • Ex: (–3, 6]
    • Domain and range of the function versus domain and range of the contextual situation
  • Key attributes of functions
    • Intercepts/Zeros
      • x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
      • Zeros – the value(s) of x such that the y value of the relation equals zero
      • y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
    • Symmetries
      • Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
      • Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
    • Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
      • Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
      • Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
    • Use key attributes to recognize and sketch graphs
    • Application of key attributes to real-world problem situations

Note(s):

  • Grade Level(s):
    • The notation represents the set of real numbers, and the notation represents the set of integers.
    • Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = bx and their key attributes.
    • Precalculus will study polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • B1 – Identify and apply transformations to figures.
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.2B Graph and write the inverse of a function using notation such as f -1(x).
Supporting Standard

Graph, Write

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

Including, but not limited to:

  • Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and  f –1(f(x)) = x.
  • Inverse functions
    • Quadratic and square root
    • Exponential and logarithmic
    • Cubic and cube root
  • Inverses of functions on graphs
    • Symmetric to yx 
  • Inverses of functions in tables
    • Interchange independent (x) and dependent (y) coordinates in ordered pairs
  • Inverses of functions in equation notation
    • Interchange independent (x) and dependent (y) variables in the equation, then solve for y
  • Inverses of functions in function notation
    • f –1(x) represents the inverse of the function f(x).

Note(s):

  • Grade Level(s):
    • Algebra II introduces inverse of a function.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
      • B2 – Algebraically construct and analyze new functions.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.2C Describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range.
Readiness Standard

Describe, Analyze

THE RELATIONSHIP BETWEEN A FUNCTION AND ITS INVERSE (QUADRATIC AND SQUARE ROOT, LOGARITHMIC AND EXPONENTIAL), INCLUDING THE RESTRICTION(S) ON DOMAIN, WHICH WILL RESTRICT ITS RANGE

Including, but not limited to:

  • Relationships between functions and their inverses
    • All inverses of functions are relations.
    • Inverses of one-to-one functions are functions.
    • Inverses of functions that are not one-to-one can be made functions by restricting the domain of the original function, f(x).
  • Characteristics of inverse relations
    • Interchange of independent (x) and dependent (y) coordinates in ordered pairs
    • Reflection over y = x
  • Domain and range of the function versus domain and range of the inverse of the given function
  • Functionality of the inverse of the given function
    • Quadratic function and square root function, f(x) = x2 and f(x) =  
      • Restrictions on domain when using positive square root  
      • Restrictions on domain when using negative square root 
    • Cubic function and cube root function, f(x) = x3 and g(x) =  
    • 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 as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
      • B2 – Algebraically construct and analyze new functions.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.2D Use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.
Supporting Standard

Use

THE COMPOSITION OF TWO FUNCTIONS

Including, but not limited to:

  • Composition of functions – process of substituting one function into another function to create a new function such that the range of one function becomes the domain of the other
  • Composition notation given f(x) and g(x)
    • Verbal
      • f composed with g of x; g composed with f of x
      • f of g of xg of f of x
    • Symbolic
      • f(g(x)); g(f(x))
      • (f  g)(x); (g  f)(x)

To Determine

IF TWO FUNCTIONS ARE INVERSES OF EACH OTHER, INCLUDING THE NECESSARY RESTRICTIONS ON THE DOMAIN

Including, but not limited to:

  • Characteristics of inverse relations
    • Interchange of independent (x) and dependent (y) coordinates in ordered pairs
    • Interchange of independent (x) and dependent (y) coordinates in an equation and resolving for y
    • Reflection over the f(x) = x line
    • 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
  • Domain and range of the function versus domain and range of the inverse of the given function

Note(s):

  • Grade Level(s):
    • Algebra II introduces inverse of a function.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
      • B2 – Algebraically construct and analyze new functions.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.7 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:
2A.7I Write the domain and range of a function in interval notation, inequalities, and set notation.
Supporting Standard

Write

THE DOMAIN AND RANGE OF A FUNCTION IN INTERVAL NOTATION, INEQUALITIES, AND SET NOTATION

Including, but not limited to:

  • Domain and range of the function
    • Domain – set of input values for the independent variable over which the function is defined
      • Continuous function – function whose values are continuous or unbroken over the specified domain
      • Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
    • Range – set of output values for the dependent variable over which the function is defined
    • Reasonable domain and range values in problem situations
    • Domain and range of the function versus domain and range of the contextual situation
    • Representation for domain and range
      • Verbal description
        • Ex: x is all real numbers less than five.
        • Ex: x is all real numbers.
        • Ex: y is all real numbers greater than –3 and less than or equal to 6.
        • Ex: y is all integers greater than or equal to zero.
      • Inequality notation – notation in which the solution is represented by an inequality statement
        • Ex: x < 5
        • Ex: x  
        • Ex: –3 < y ≤ 6
        • Ex: y ≥ 0, y  
      • Set notation – notation in which the solution is represented by a set of values
        • Braces are used to enclose the set.
        • Solution is read as “The set of x such that x is an element of …”
        • Ex: {x|x  , x < 5}
        • Ex: {x|x  }
        • Ex: {y|y  , –3 < y ≤ 6}
        • Ex: {y|y  , y ≥ 0}
      • Interval notation – notation in which the solution is represented by a continuous interval
        • Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
        • Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
        • Ex: (–∞, 5)
        • Ex: (–∞, ∞)
        • Ex: (–3, 6]
        • Ex: [0, ∞)

Note(s):

  • Grade Level(s):
    • The notation  represents the set of real numbers, and the notation  represents the set of integers.
    • Algebra I introduced domain and range written in inequality notation.
    • Algebra II introduces domain and range written in set and interval notation.
    • 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
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each 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 accommodated commensurate 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 the ELPS 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.
Click here to collapse or expand this section.
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/16/2016
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