INSTRUCTIONAL FOCUS DOCUMENT Algebra II

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1 UNIT OVERVIEW This unit bundles student expectations that address connections between functions and their characteristics, as well as connections between parent functions and their transformations. The concept of the inverse of a function is also introduced and bundled within this unit. Prior to this unit, in Algebra 1 Units 01, 04, 05, 09, 11, and 12, students experienced an in-depth study of linear and quadratic functions. Inverse (rational) and exponential functions were also investigated in Algebra 1 Unit 1. In Geometry Units 05 and 07, linear and quadratic functions were applied in various pattern situations. During this unit, students analyze functions by investigating their characteristics and representing them using various models. Students determine the domain (continuous or discrete) and range as well as independent and dependent variables. Students identify functions as increasing or decreasing. Transformations are identified, described, and applied to various parent functions. In addition, inverses of functions are explored by comparing and contrasting the function and its inverse using tables, graphs, and algebraic representations. After this unit, in Algebra 2 Units 03 10, the study of families of functions will be extended using data collection and analysis in problem situations. Each family of functions will be studied in-depth by describing and applying the parent function, transformational effects of parameter changes, and characteristics of the function. 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 as well as exponential and logarithmic functions. Analysis of functions is foundational to high school and college mathematics. Focusing on the properties, attributes, and applications of functions using multiple representations is emphasized in the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning C1; II. Algebraic Reasoning A1, C1, D1, D2; VI. Statistical Reasoning B2; VII. Functions A1, B1, B2, C2; IX. Communication and Representation B1, B2, C1; and STAAR Readiness Standards: 2A.1A, 2A.1B, and 2A.4B. Domain, range, inverses of functions, parent functions, and effects of parameter changes are addressed in STAAR Supporting Standards: 2A.4A, 2A.4C, 2A.7B, 2A.9A, 2A.9G, 2A.10A, and 2A.11B. 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) lists the key elements of reasoning and sense making with functions as being using multiple representations of functions, modeling by using families of functions, and analyzing the effects of parameters (NCTM, p. 41). page 1 of 25

2 National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 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. (2009). Texas college and career readiness standards. Austin, TX: Author. PERFORMANCE ASSESSMENT(S) Unit 02 PA 01 Analyze a function in a problem situation such as the following: OVERARCHING CONCEPTS UNIT CONCEPTS Foundations for Functions Representations; Properties and Attributes UNIT UNDERSTANDINGS Data relationships in problem situations can be interpreted through the analysis of properties and attributes of functions and modeled using multiple representations to make predictions and critical judgments in terms of the problem situations. The following data, comparing the mean height of children in relation to their age, was collected in Kalama, Egypt. Predict the mean height of a 21.5 month-old baby. Could you predict the height of a 21 year-old adult? Explain your reasoning. page 2 of 25

3 PERFORMANCE ASSESSMENT(S) OVERARCHING CONCEPTS UNIT CONCEPTS UNIT UNDERSTANDINGS Data from: United States Agency for International Development (USAID) Create a graphic organizer that includes a table, graph, verbal description, and algebraic generalization. Write an analysis of the relationships between the representations and a description of the attributes (function/non-function, representative parent function, domain/range, continuous/discrete, increasing/decreasing, function notation, etc.). Use the representations to make predictions and critical judgments in the problem situation. Standard(s): 2A.1A, 2A.1B ELPS.c.1C, ELPS.c.4J, ELPS.c.5B, ELPS.c.5G page 3 of 25

4 PERFORMANCE ASSESSMENT(S) Unit 02 PA 02 Create a chart comparing effects of transformations. The chart should include the name and equation of the parent function, horizontal and vertical shifts, horizontal and vertical stretch or compression, reflection, and a composite graph of the parent function and translated function. Formulate a sample graph of a quadratic function that includes changes on all four parameters, determine an algebraic generalization to represent the quadratic function, and justify your reasoning. Standard(s): 2A.4A, 2A.4B, 2A.7B, 2A.9A, 2A.10A, 2A.11B ELPS.c.1C, ELPS.c.5B, ELPS.c.5F OVERARCHING CONCEPTS UNIT CONCEPTS Algebra and Geometry Parent Functions Quadratic and Square Root Functions Effects of Parameter Changes Rational Functions Effects of Parameter Changes Exponential and Logarithmic Functions Effects of Parameter Changes UNIT UNDERSTANDINGS Changes in the parameters of parent functions have specific effects on the equation of the function and the graph in the coordinate plane. Unit 02 PA 03 Compare and contrast a function and its inverse given various representations such as the following: graph Algebra and Geometry Inverses of Functions Quadratic and Square Root Functions Inverses of Functions The graph of a function and its inverse reflect across the y = x line, and the elements of the ordered pairs are reversed; the x and y variables are interchanged in the algebraic generalizations. page 4 of 25

5 PERFORMANCE ASSESSMENT(S) OVERARCHING CONCEPTS UNIT CONCEPTS UNIT UNDERSTANDINGS table equation Create a graphic organizer that includes a table of values, a graph, and an algebraic generalization for both the function and its inverse. Write a summary comparing the domain and range, tables, graphs, and algebraic generalizations of the function and its inverse. Standard(s): 2A.4C, 2A.9G ELPS.c.1C, ELPS.c.5B, ELPS.c.5G page 5 of 25

6 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. Some students may think that the inverse of a function means to take the opposite sign or reciprocal of the function instead of 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. UNIT VOCABULARY Functions relations in which each element of the input (domain) is paired with exactly one element of the output (range) Parent functions set of basic functions from which related functions are derived by transformations Relation a set of ordered pairs (pairing of a set of inputs (domain) with a set of outputs (range) Transformations general term for ways to manipulate the shape or position of a graph; types of transformation include translation (shifts), reflection, and dilation (compression and stretch); specific parameter changes on parent functions cause certain types of transformations on the graph of the function. page 6 of 25

7 SYSTEM RESOURCES Transition Alignment Guide (TAG) Tool OTHER RESOURCES Algebra 2 End of Course Success: Vocabulary Reporting Category 1 On the Move, On Your Own: Putting It All Together, Keep It Moving 1, Keep It Moving 2 MTR Grades 9 12: Tab 4, : Understanding Functions TEA STAAR Mathematics Resources: TEA STAAR Released Test Questions: TEXTEAMS: Part I: /Precalculus Institute: I Foundations of Functions; 1. Foundations, 1.1 Transformations, Act. 1 (Explore Transformations), Act. 2 (Domains), Act. 3 (Transformations with Technology), Act. 4 (Summarize Your Findings); 2. Applying Transformations, 2.1 Applications, Act. 1 (Good Viewing Windows), Act. 2 (Friendly Viewing Windows) TEKS UNIT LEVEL SPECIFICITY Legend: Bold black text in italics: Knowledge and Skills Statement (TEKS) Bold black text: Student Expectation (TEKS) Bolded red text in italics: Student Expectation identified by TEA as a Readiness Standard for STAAR Legend: 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) Bolded green text in italics: Student page 7 of 25

8 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) 2A.1 Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to: 2A.1A Identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations. Readiness Standard Use, Apply, Identify, Determine REASONABLE MATHEMATICAL DOMAINS AND RANGES Including, but not limited to: Comparison of discrete and continuous domain Representations of domain and range with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Connections among the representations Notation of domain and range including set builder notation Lists of domain and range in sets (including f(x) for y) Verbal Ex: Domain is all real numbers; domain is all real numbers greater than five. page 8 of 25

9 Ex: Range is all real numbers less than zero; range is all real numbers. Symbolic Ex: domain (e.g., {x x > 0}; {x -3 < x < 4}; {x x R}; Ø) Ex: range (e.g., {y y 0}; {y -7 < y 4}; { f(x) -5 < f(x) 2}; {y y R}; Ø) Determination of scales for graphs and windows on graphing calculators using domain and range Contextual domain and range of real-world problem situation Specific range values at given domain values from various representations (e.g., tables, graphs, or algebraic generalizations) Specific domain values at given range values from various representations (e.g., tables, graphs, or algebraic generalizations) Domain and range of the problem situation versus the domain and range of the mathematical function Note: In Algebra 1, students determine domain and range using various representations of functions. 2A.1B Collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. Readiness Standard TxCCRS Note: VII. Functions B1 Understand and analyze features of a function. Collect, Organize DATA Including, but not limited to: Data sets and real-world problem situations Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing page 9 of 25

10 calculators, and computers) Ex: measurement activities, experiments, simulations, Internet, etc. Representations of data with and without technology Tables Graphs including scatterplots Verbal descriptions Algebraic generalizations (including equation and function notation) Use, Make, Interpret, Fit, Model, Apply SCATTERPLOTS AND REPRESENTATIVE FUNCTIONS FOR DATA Including, but not limited to: Data sets and real-world problem situations General trends in data Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing calculators, and computers) Ex: measurement activities, experiments, simulations, Internet, etc. Representations of data with and without technology Tables Graphs, including scatterplots Verbal descriptions Algebraic generalizations (including equation and function notation) Parent functions as representations of scatterplots of data with and without technology Linear Quadratic Square root Rational Exponential page 10 of 25

11 Logarithmic Absolute value Linear or nonlinear scatterplots using graphing technology Model, Predict, Make DECISIONS AND CRITICAL JUDGMENTS IN PROBLEM SITUATIONS Including, but not limited to: Data sets and real-world problem situations General trends in the data with and without technology Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing calculators, and computers) Ex: measurement activities, experiments, simulations, Internet, etc. Representations of data with and without technology Tables Graphs including scatterplots Verbal descriptions Algebraic generalizations (including equation and function notation) Parent functions as representations of scatterplots of data with and without technology Linear Quadratic Square root Rational Exponential Logarithmic Absolute value Linear or nonlinear scatterplots using graphing technology Predictions and critical judgments in terms of the scatterplots and representative functions page 11 of 25

12 Note: In Algebra 1, students collect and analyze data using various representations, which are then used to make predictions and critical judgments. TxCCRS Note: II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. III. Geometric Reasoning C2 Make connections between geometry, statistics, and probability. VI. Statistical Reasoning A1 Plan a study. VI. Statistical Reasoning B1 Determine types of data. VI. Statistical Reasoning B2 Select and apply appropriate visual representations of data. VI. Statistical Reasoning B4 Describe patterns and departure from patterns in a set of data. VI. Statistical Reasoning C1 Make predictions and draw inferences using summary statistics. VI. Statistical Reasoning C2 Analyze data sets using graphs and summary statistics. VI. Statistical Reasoning C3 Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software. VII. Functions A2 Recognize and distinguish between different types of functions. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C1 Apply known function models. VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning B2 Use various types of reasoning. IX. Communication and Representation B1 Model and interpret mathematical ideas and concepts using multiple representations. IX. Communication and Representation B2 Summarize and interpret mathematical information provided orally, visually, or in written form within the given context. IX. Communication and Representation C1 Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, graphs, and words. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. page 12 of 25

13 2A.4 Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to: 2A.4A Identify and sketch graphs of parent functions, including linear (f (x) = x), quadratic (f (x) = x 2 ), exponential (f (x) = a x ), and logarithmic (f (x) = log a x) functions, absolute value of x (f (x) = x ), square root of x (f (x) = x), and reciprocal of x (f (x) = 1/x). Supporting Standard Connect, Identify, Sketch PARENT FUNCTIONS AND THEIR GRAPHS Including, but not limited to: General forms of parent functions (including equation and function notation) Linear: f(x) = x Quadratic: f(x) = x² Exponential: f(x) = a x Logarithmic: f(x) = log a x Absolute value: f(x) = x Square root: f(x) = x Rational (reciprocal of x): f(x) = Representations of parent functions with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Connections between representations Characteristics of parent functions with and without technology Ex: domain and range, rate of change, increasing/decreasing, vertex points, end points, intercepts, symmetry, asymptotes, maximum/minimum, end behavior, etc. Connections between transformed functions to the respective parent function with and without page 13 of 25

14 technology Similarities and differences in the parent functions Identification of relations as functions (TxCCRS) Note: 2A.4B Extend parent functions with parameters such as a in f (x) = a/x and describe the effects of the parameter changes on the graph of parent functions. Readiness Standard In Algebra 1, students identified the characteristics of the linear and quadratic parent functions. TxCCRS Note: III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions A1 Recognize whether a relation is a function. VII. Functions A2 Recognize and distinguish between different types of functions. VII. Functions B1 Understand and analyze features of a function. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Extend, Describe PARAMETER CHANGES ON PARENT FUNCTIONS Including, but not limited to: General forms of parent functions (including equation and function notation) Linear: f(x) = x Quadratic: f(x) = x² Exponential: f(x) = a x Logarithmic: f(x) = log a x Absolute value: f(x) = x Square root: f(x) = x Rational (reciprocal of x): f(x) = page 14 of 25

15 Representations of parent functions with and without technology Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and k on the graph of the parent function: Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations page 15 of 25

16 Note: 2A.4C Describe and analyze the relationship between a function and its inverse. Supporting Standard In Algebra 1, students extended parameters on vertical shifts, stretches, and compressions for linear and quadratic functions. TxCCRS Note: III. Geometric Reasoning B1 Identify and apply transformations to figures. III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C1 Apply known function models. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Describe, Analyze RELATIONSHIPS BETWEEN FUNCTIONS AND THEIR INVERSES Including, but not limited to: Comparison of independent and dependent variables of a function and its inverse Comparison of domain and range of a function and its inverse Symmetry about y = x Identification of functions and non-functions Comparison of functions and their inverses using multiple representations with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Inverse relationships page 16 of 25

17 Quadratic and square root functions Exponential and logarithmic functions TxCCRS Note: III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions A1 Recognize whether a relation is a function. VII. Functions B1 Understand and analyze features of a function. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. 2A.7 Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is expected to: 2A.7B Use the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a (x - h) 2 + k form of a function in applied and purely mathematical situations. Supporting Standard Interpret, Use, Investigate, Describe, Predict PARAMETER CHANGES ON QUADRATIC FUNCTIONS IN APPLIED AND MATHEMATICAL SITUATIONS Including, but not limited to: Comparison of quadratic parent function y = x 2 to transformations reflected in y = a(x - h) 2 + k with and without technology Representations of parent function and related function with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and kon the graph of the parent function, y = a(x - h) 2 + k page 17 of 25

18 Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations TxCCRS Note: III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. 2A.9 Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to page 18 of 25

19 solve them, and analyzes the solutions in terms of the situation. The student is expected to: 2A.9A Use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges. Supporting Standard Use, Investigate, Describe, Predict GRAPHS AND EFFECTS OF PARAMETER CHANGES ON SQUARE ROOT FUNCTIONS Including, but not limited to: Comparison of square root parent function y= x to transformations reflected in y=a (x h) + k with and without technology Representations of parent function and related function with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and k on the graph of the parent function, y=a (x h)+ k Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation page 19 of 25

20 Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range and limitations of domain by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations 2A.9G Connect inverses of square root functions with quadratic functions. Supporting Standard TxCCRS Note: III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. Connect, Analyze SQUARE ROOTS AS INVERSE FUNCTIONS OF QUADRATICS Including, but not limited to: Inverse relationships Quadratic functions Square root functions Characteristics of inverse relationships with and without technology Domain and range switched in table Reflection in the y = x line on the graph Algebraic representations (switching x and y and solving for y) Comparison of functionality of the original function and its inverse (horizontal and vertical line page 20 of 25

21 tests) with and without technology TxCCRS Note: VII. Functions A1 Recognize whether a relation is a function. VII. Functions B1 Understand and analyze features of a function. 2A.10 Rational functions. The student formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: 2A.10A Use quotients of polynomials to describe the graphs of rational functions, predict the effects of parameter changes, describe limitations on the domains and ranges, and examine asymptotic behavior. Use, Examine, Describe, Analyze, Predict EFFECTS OF PARAMETER CHANGES ON RATIONAL FUNCTIONS Including, but not limited to: Supporting Standard Comparison of rational parent function y = to transformations reflected in Representations of parent function and related function with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and k on the graph of the parent function Effects of changes in a on the graph of the parent function with and without technology a 0 page 21 of 25

22 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range and limitations of domain by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations TxCCRS Note: III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. 2A.11 Exponential and logarithmic functions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the page 22 of 25

23 solutions in terms of the situation. The student is expected to: 2A.11B Use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior. Supporting Standard Use, Investigate, Describe, Predict, Examine, Analyze GRAPHS AND EFFECTS OF PARAMETER CHANGES ON EXPONENTIAL AND LOGARITHMIC FUNCTIONS Including, but not limited to: Comparison of exponential parent function f(x)=b x to transformations reflected in f(x)=a*b x-h +k Comparison of logarithmic parent function f(x)=log b x to transformations reflected in f(x)= alog b (x-h) +k Representations of parent function and related function with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and k on the graphs of the parent function, f(x)=a*b x-h +k Changes in parameters a, h, and kon the graphs of the parent function, f(x)= alog b (x-h) +k Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology page 23 of 25

24 k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range and limitations of domain (logarithms) by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations Note: In the exponential and logarithmic parent functions the a, used as the exponent and log base, was changed to b, so it would not be confused with the a parameter used in transformations of parent functions. TxCCRS Note: III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. VII. Functions C1 Apply known function models. page 24 of 25

25 ELPS# SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS. 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. Choose appropriate ELPS to support instruction. page 25 of 25