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Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st Nineweeks 1. Enduring Understanding - A linear model can be used to represent relationships between data. 1a. Essential Question - How can a linear relationship be determined from a set of data? DAP.6.AII.1 A. Find the equation and graph the regression *draw a scatter plot regression line Find regression line for scatter plot, using line for a scatterplot using appropriate *use appropriate technology to calculate and graph the scatter plot appropriate technology, and interpret the technology. regression line correlation coefficient correlation coefficient B. Interpret the correlation coefficient. *use appropriate technology to calculate the correlation coefficient and determine its significance DAP.6.AII.2 Interpret and use the correlation coefficient to *find the correlation coefficient using appropriate correlation coefficient Interpret and use the correlation coefficient to assess the strength of the linear relationship technology linear relationship assess the strength of the linear relationship between two variables. *interpret the correlation coefficient as it moves from -1 between two variables to 1 DAP.6.AII.4 Identify strengths and weaknesses of using *use real-world data sets over a small interval to extrapolate Identify strengths and weaknesses of using regression equations to approximate data. calculate regression equation interval regression equations to approximate data *approximate data using the regression equation regression equation *identify strengths and weaknesses of using the Note: This SLE will be revisited later. regression to extrapolate (approximate) data RF.1.AII.1 A. Determine, with and without appropriate A. domain Determine, with or without technology, the technology, the domain and range of a relation *know and use function notation range of function domain and range of a relation defined by a defined by a graph, a table of values, or a *use a variety of mathematical notation to state the relation graph, a table of values, or a symbolic symbolic equation, including those with domain (i.e.: set-builder, interval, roster, sets, etc.) function equation including those with restricted restricted domains. *evaluate functions for given values in their domain set-builder notation domains and whether a relation is a function B. Determine whether a relation is a function. *determine domain and range of a relation defined by a interval notation table (mapping) of values with and without technology roster (set notation) *determine domain and range of a relation defined by a mapping graph with and without technology function notation *determine domain and range of a relation defined by a evaluate symbolic equation with and without technology B. *identify functions from mappings, tables or equations *apply the vertical line test to graphs rev. 5-2009 1 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) RF.1.AII.9 Communicate real-world problems involving *identify key information in a real-world problem function Apply the concepts of functions to real-world 1st Nineweeks (graphically, algebraically, functions, graphically, algebraically, *determine the format reasonableness situations numerically and verbally. numerically or verbally) to represent the problem and its level of precision solution Note: This SLE will be revisited. *evaluate the reasonableness of the solution *communicate real-world problems graphically, algebraically, numerically, and verbally Note: Achieve includes solving literal equations and solving problems that can be modeled using absolute value, step, and other piece-wise defined functions. 1b. Essential Question - What are the effects of changing parameters on linear functions? RF.1.AII.4 A. Recognize parent function (i.e.. y=x , y=x n A. translation Analyze and report, with and without where n is a positive integer). *identify different parent functions reflection appropriate technology, the effect of changing B. Analyze and describe, with and without B. coefficient coefficients, exponents, and other parameters appropriate technology, the effect of changing *describe changes in slope and y-intercept from the exponent on functions and their graphs (linear, coefficients, and other parameters on linear parent graph parameter quadratic, and higher-degree polynomial) functions and their graphs. (y=mx+b ) *describe changes in slope and y-intercept between two parent function C. Analyze and describe, with and without linear functions slope Note: Only be concerned about linear appropriate technology, the effect of changing C. y-intercept functions at this point in the curriculum. coefficients, and other parameters on *describe changes using the words translate, reflect, translate quadratic functions and their graphs. (y=a(x- stretch or compress (shrink) including direction and reflect h) 2 + k ) units from the parent graph stretch D. Analyze and describe, with and without *describe changes using the words translate, reflect, compress appropriate technology, the effect of changing stretch or compress (shrink) including direction and transformation coefficients, and other parameters on higher- units between two quadratic functions linear functions degree polynomial functions and their graphs. D. quadratic functions Note: Only be concerned about Parts A and *describe changes using the words translate, reflect, cubic functions stretch or compress (shrink) including direction and quartic functions B with linear functions at this point in the units from the parent graph higher-degree curriculum. *describe changes using the words translate, reflect, polynomial functions stretch or compress (shrink) including direction and units between two higher-degree polynomial functions Note: Achieve includes comparing to the parent graph (linear, quadratic, square root, absolute value) as the fundamental concept in this objective. 1c. Essential Question - How can the inverse of a linear function be determined? rev. 5-2009 2 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) RF.1.AII.3 A. Determine the inverse of a function. A. inverse of a function Determine the inverse of a function (Graph, B. Graph, with and without appropriate 1st Nineweeks of an inverse and its notation f *know the definition -1 f -1 notation with and without appropriate technology, technology, functions and their inverses. *determine the inverse of a function given a set of exponential function functions and their inverses) ordered pairs (Achieve) *determine the inverse of a function algebraically identity function B. reflection *know the identity function y=x and its graph Horizontal Line Test *determine the inverse of a function graphically without one-to-one and with technology Note: Achieve includes -inverses which may not be functions (not one-to-one) -explain why an inverse function would be only either the positive or the negative part of the graph -explain why the graphs of a function and its inverse are reflections of each other over the line y=x -show that when the inverse of a function is a function f - 1 (f(x))=x and f(f -1(x))=x -inverses of exponential functions may be required graphically but not in algebraic form rev. 5-2009 3 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st Nineweeks 2. Enduring Understanding - A function represents a unique relationship between a domain and range. 2a. Essential Question - How can special functions be represented graphically? LEI.2.AII.1 A. Solve without and with appropriate A. absolute value Solve, with and without appropriate technology, absolute value equations and *solve without appropriate technology absolute value absolute value equation technology, absolute value equations and inequalities written in one variable and graph equations written in one variable and graph solutions inequalities inequalities written in one or two variables, and solutions. *solve with appropriate technology absolute value absolute value graph solutions B. Solve (graph), without and with appropriate equations written in one variable and graph solutions inequalities technology, absolute value equations and *solve without appropriate technology absolute value inequalities written in two variables. inequalities written in one variable and graph the solution *solve with appropriate technology absolute value inequalities written in one variable and graph the solution B. *solve (graph) without appropriate technology, absolute value equations written in two variables *solve (graph) with appropriate technology, absolute value equations written in two variables *solve (graph) without appropriate technology, absolute value inequalities written in two variables *solve (graph) with appropriate technology, absolute value inequalities written in two variables rev. 5-2009 4 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) RF.1.AII.5 A. Graph, without and with technology, A. piece-wise function Graph, with and without appropriate functions defined as piece-wise. 1st Nineweeks technology, graph functions with *without appropriate piece-wise function technology, functions defined as piece-wise B. Graph, without and with technology, domain restrictions using open and closed circles notation and step functions defined as step. *with appropriate technology, enter domain restrictions step function on the calculator constant function B. greatest integer function *without appropriate technology, graph constant greatest integer notation functions *without appropriate technology, evaluate expressions using "greatest integer" notation *without appropriate technology, create an appropriate table of values and use the table to graph the step function *with appropriate technology, enter greatest integer notation on the calculator Note: Achieve includes writing an algebraic representation for a given piece-wise defined function. 3. Enduring Understanding - Systems of linear equations and inequalities can be solved using multiple methods. 3a. Essential Question - What methods can be used to solve systems of linear equations and inequalities? LEI.2.AII.2 Solve, without and with appropriate *solve systems of linear equations in two variables system of linear Solve, with and without appropriate technology, systems of linear equations with graphically without appropriate technology equations technology, systems of linear equations with two variables graphically and algebraically. *solve systems of linear equations in two variables solution set: Ø , (x, y) , two variables and graph the solution set graphically with appropriate technology or {(x,y)| y=mx+b} *solve systems of linear equations in two variables empty set algebraically (i.e.. linear combination (elimination), substitution, matrices) without appropriate technology *determine the number of solutions rev. 5-2009 5 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) LEI.2.AII.5 A, B, C: A. Apply linear equations and inequalities to linear programming Apply, with or without technology, the model real-world situations. 1st Nineweeksinformation *identify the key reasonableness concepts of linear and absolute value B. Apply absolute value equations and *write the equations or inequalities to represent the equations and inequalities and systems of inequalities to model real-world situations. situation algebraically and solve linear equations and inequalities to model real- C. Apply systems of linear equations and *evaluate the reasonableness of the solution world situations including linear programming inequalities to model real-world situations, Note: Achieve includes time/rate/distance, percentage including linear programming. increase/decrease, ratio and proportion, mixture problems, and break-even problems. LEI.2.AII.3 A. Apply basic operations to matrices with A. With and without appropriate technology: dimensions Develop and apply, with and without and without appropriate technology. *add two matrices scalar appropriate technology, the basic operations B. Find the inverse of a matrix with and *subtract two matrices associative and properties of matrices (associative, without appropriate technology. *multiply a matrix by a scalar commutative commutative, identity, and inverse) C. Investigate the properties (associative, *multiply two matrices identity commutative, identity, and inverse) in B. inverse of a matrix relationship to matrices. *find the determinant of a matrix matrix *find the inverse of a 2x2 matrix without appropriate determinant technology *find the inverse of a 3x3 matrix with appropriate technology C. *investigate with and without appropriate technology the properties (associative, commutative, identity, and inverse) in relationship to matrices LEI.2.AII.4 A. Solve, systems of linear equations with A. elimination Solve, with and without appropriate three variables using algebraic methods. *solve systems of linear equations with three variables substitution technology, systems of linear equations with B. Solve, with appropriate technology, using algebraic methods (i.e.. elimination, substitution) systems of equations three variables using algebraic methods, systems of linear equations with three B. including matrices variables using matrices. *using appropriate technology, solve systems of linear equations with three variables using matrices. Note: Achieve suggests using systems of linear equations limited to those with integer solutions and small integral coefficients. 2nd Nineweeks 1. Enduring Understanding - Radical expressions and rational exponents represent a root of a quantity. 1a. Essential Question - How are radical expressions simplified? rev. 5-2009 6 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) QEF.3.AII.1 Perform computations with radicals: Perform computations with radicals Note: Be sure that 1st Nineweeks all computations are done with • simplify radicals with different indices different indices. radicals simplify radicals • simplify radicals with different indices • add, subtract, multiply and divide radicals Perform computations with radicals: indices • add, subtract, multiply and divide radicals • rationalize denominators *find conjugates of radical expressions radical expressions • rationalize denominators • solve equations that contain radicals or *simplify radicals with different indices rationalize • solve equations that contain radicals or radical expressions (check for extraneous *add, subtract, multiply and divide radicals denominators radical expressions solutions) *rationalize denominators extraneous *solve equations that contain radicals or radical conjugate expressions (check for extraneous solutions) Note: Achieve includes the example: x 2 6x 9 x 32 x3 1b. Essential Question - How can radical expressions be written and simplified using rational exponents? PRF.4.AII.7 Convert between and among radical and *convert the radical form of an algebraic expression to exponential form Establish the relationship between radical exponential forms of algebraic expressions. its exponential form (rational exponents) expressions and expressions containing *convert the exponential form of an algebraic expression radical form rational exponents to its radical form PRF.4.AII.8 Simplify variable expressions containing *review laws of exponents variable expression Simplify variable expressions containing rational exponents using the laws of *simplify variable expressions containing rational laws of exponents rational exponents using the laws of exponents. exponents using the laws of exponents exponents 2. Enduring Understanding - A quadratic function is a second-degree polynomial represented graphically as a parabola. 2a. Essential Question - How are real, imaginary, and complex numbers related? QEF.3.AII.2 Extend the number system to include the *define the set of complex numbers conjugate Extend the number system to include the complex numbers. *find conjugates of complex numbers complex number complex numbers • define the set of complex numbers *add, subtract, multiply, and divide complex numbers pure imaginary numbers • define the set of complex numbers • add, subtract, multiply, and divide complex *rationalize denominators "i" • add, subtract, multiply, and divide complex numbers Note: Achieve Core includes "simplify powers of pure rationalize numbers • rationalize denominators imaginary numbers (i.e. i5= i)" denominators • rationalize denominators rev. 5-2009 7 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) QEF.3.AII.5 A. Develop and analyze, with and without A. discriminate Develop and analyze, with and without 1st Nineweeks with and without appropriate *develop and analyze, appropriate technology, quadratic relations by maximum value appropriate technology, quadratic relations graphing a parabolic relationship when given technology, quadratic relations by graphing a quadratic minimum value • graph a parabolic relationship when given its its equation. relationship when given its equation (Note: this could axis of symmetry equation B. Develop and analyze, with and without include equations such as x=y2.) vertex • write an equation when given its roots (zeros appropriate technology, quadratic functions by B. quadratic relations or solutions) or graph writing an equation when given its roots (zeros *develop and analyze, with and without appropriate quadratic function • determine the nature of the solutions or solutions) or graph. technology, quadratic functions by writing an equation roots graphically and by evaluating the discriminate C. Develop and analyze, with and without when given its roots (zeros or solutions) or graph zeros • determine the maximum or minimum values appropriate technology, quadratic functions by C. solutions and the axis of symmetry both graphically and determining the nature of the solutions *develop and analyze, with and without appropriate nature of solutions algebraically graphically and by evaluating the discriminate. technology, quadratic functions by determining the parabolic relationship D. Develop and analyze, with and without nature (number and type) of the solutions graphically appropriate technology, quadratic functions by *know the discriminate as part of the quadratic formula determining the maximum or minimum values *develop and analyze, with and without appropriate and the axis of symmetry both graphically and technology, quadratic functions by determining the algebraically. nature (number and type) of the solutions by evaluating the discriminate D. *develop and analyze, with and without appropriate technology, quadratic functions by determining the maximum or minimum values and the axis of symmetry graphically *develop and analyze, with and without appropriate technology, quadratic functions by determining the maximum or minimum values and the axis of symmetry algebraically Note: Achieve includes graphing the solution set of a two-variable quadratic inequalities, and graphing horizontal parabolas. rev. 5-2009 8 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) PRF.4.AII.2 With technology: Analyze and sketch, with and without A. Use appropriate to: 1st Nineweeks technologyof a given polynomial A. Analyze and sketch the graph of a given *analyze and sketch the graph polynomial function end behavior appropriate technology, the graph of a given polynomial function. function multiplicity polynomial function, determining the B. Determine the characteristics of domain *determine domain and range y-intercept characteristics of domain and range, and range, maximum and minimum points, *determine maximum and minimum points symmetry maximum and minimum points, end behavior, end behavior, zeros, multiplicity of zeros, y- *determine maximum and minimum values synthetic division zeros, multiplicity of zeros, y-intercept, and intercept, and symmetry. *distinguish between local (relative) and global maximum point symmetry Without technology: (absolute) maximums and minimums maximum value C. Determine domain, end behavior, zeros *determine end behavior maximum point Note: This SLE will be revisited. and multiplicity of zeros, y-intercept, and *determine zeros and multiplicity of zeros minimum value symmetry. *determine y-intercept absolute/relative *determine different types of symmetry maximum and B. Without appropriate technology: minimums *determine domain. domain *determine end behavior by using the leading coefficient range and the degree of the polynomial. zeros *determine zeros and multiplicity of zeros when factorable to linear or quadratic factors. *determine y-intercept. *determine different types of symmetry. RF.1.AII.5 A. Graph without and with technology, A. piece-wise function Graph, with and without appropriate functions defined as piecewise. *without appropriate technology, graph functions with piece-wise function technology, functions defined as piece-wise B. Graph without and with technology, domain restrictions using open and closed circles notation and step functions defined as step. *with appropriate technology, enter domain restrictions step function Note: At this point the emphasis is on Part on the calculator constant function Note: At this point the emphasis is on A. B. greatest integer function piece-wise functions. *without appropriate technology, graph constant greatest integer notation functions *without appropriate technology, evaluate expressions using "greatest integer" notation *without appropriate technology, create an appropriate table of values and use the table to graph the step function *with appropriate technology, enter greatest integer notation on the calculator Note: Achieve includes writing an algebraic representation for a given piece-wise defined function. rev. 5-2009 9 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) RF.1.AII.4 A. Recognize parent function (i.e.. y=x , y=x n A. 1st Nineweeks parent functions translation Analyze and report, with and without where n is a positive integer). *identify different reflection appropriate technology, the effect of changing B. Analyze and describe, with and without B. coefficient coefficients, exponents, and other parameters appropriate technology, the effect of changing *describe changes in slope and y-intercept from the exponent on functions and their graphs (linear, coefficients, and other parameters on linear parent graph parameter quadratic, and higher degree polynomial) functions and their graphs. (y=mx+b ) *describe changes in slope and y-intercept between two parent function C. Analyze and describe, with and without linear functions slope Note: At this point the emphasis is on appropriate technology, the effect of changing C. y-intercept quadratics. coefficients, and other parameters on *describe changes using the words translate, reflect, translate quadratic functions and their graphs. (y=a(x- stretch or compress (shrink) including direction and reflect h) 2 + k ) units from the parent graph stretch D. Analyze and describe, with and without *describe changes using the words translate, reflect, compress appropriate technology, the effect of changing stretch or compress (shrink) including direction and transformation coefficients, and other parameters on higher- units between two quadratic functions linear functions degree polynomial functions and their graphs. D. quadratic functions Note: At this point the emphasis is on Parts *describe changes using the words translate, reflect, cubic functions stretch or compress (shrink) including direction and quartic functions A and C with quadratic functions. units from the parent graph higher-degree *describe changes using the words translate, reflect, polynomial functions stretch or compress (shrink) including direction and units between two higher degree polynomial functions Note: Achieve includes comparing to the parent graph (linear, quadratic, square root, absolute value) is the fundamental concept in this objective. rev. 5-2009 10 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) RF.1.AII.3 A. Determine the inverse of a function. A. inverse of a function Determine the inverse of a function (Graph, B. Graph, with and without appropriate 1st Nineweeks of an inverse and its notation f *know the definition -1 f -1 notation with and without appropriate technology, technology, functions and their inverses. *determine the inverse of a function given a set of exponential function functions and their inverses) ordered pairs (Achieve) *determine the inverse of a function algebraically identity function Note: At this point the emphasis is on the B. reflection inverse of a function. This SLE will be *know the identity function y=x and its graph Horizontal Line Test revisited in the 4th Nine Weeks. *determine the inverse of a function graphically without one-to-one and with appropriate technology Note: Achieve includes -Inverses which may not be functions (not one-to-one) -Explain why an inverse function would be only either the positive or the negative part of the graph -Explain why the graphs of a function and its inverse are reflections of each other over the line y=x -Show that when the inverse of a function is a function f -1 (f(x))=x and f(f -1(x))=x -Inverses of exponential functions may be required graphically but not in algebraic form DAP.6.AII.3 Find the quadratic curve of best fit using *draw a scatter plot quadratic regression Find the quadratic curve of best fit using appropriate technology. *use appropriate technology to calculate the curve of curve of best fit appropriate technology best fit scatter plot DAP.6.AII.4 Identify strengths and weaknesses of using *use real-world data sets over a small interval to extrapolate Identify strengths and weaknesses of using regression equations to approximate data. calculate regression equation interval regression equations to approximate data *approximate data using the regression equation regression equation *identify strengths and weaknesses of using the regression to extrapolate (approximate) data rev. 5-2009 11 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st Nineweeks 2b. Essential Question - How can various methods be used to solve quadratic equations? QEF.3.AII.3 Analyze and solve quadratic equations with *solve quadratic equations without appropriate extracting the square Analyze and solve quadratic equations, with and without appropriate technology by technology by graphing root and without appropriate technology, by • factoring *solve quadratic equations with appropriate technology completing the square • factoring • graphing by graphing Quadratic Formula • graphing • extracting the square root (i.e.. The Square *solve quadratic equations without appropriate The Square Root • extracting the square root Root Property) technology by: Property • completing the square • completing the square - factoring parabola • using the quadratic formula • using the quadratic formula - extracting square roots - completing the square - using the quadratic formula Note: Achieve includes quadratic equations involving absolute value and quadratic inequalities. QEF.3.AII.4 A. Derive the quadratic formula. A. derive Derive the quadratic formula and use it to B. Use the quadratic formula to solve *complete the square quadratic formula solve equations equations. *derive the quadratic formula B. *use the quadratic formula to solve equations 2c. Essential Question - How can quadratic functions be used to solve real-world problems? QEF.3.AII.6 A. Apply the concepts of quadratic equations *identify key information quadratic equation Apply the concepts of quadratic equations and and functions to model real-world situations by *write an equation to represent the situation reasonableness functions to model real-world situations by using appropriate technology when needed. algebraically using appropriate technology when needed B. Communicate real-world problems *use the equation to find the unknown value involving functions, graphically, algebraically, *determine the reasonableness of the solution numerically and verbally. rev. 5-2009 12 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st 3rd Nineweeks 1. Enduring Understanding - A relationship exists between a polynomial's factors, zeros, roots, and x-intercepts. 1a. Essential Question - How are a polynomial function and its factors related? PRF.4.AII.1 Determine the factors of polynomials by *review algebra I factoring techniques sum or difference of two Determine the factors of polynomials by • using factoring techniques including grouping *factor polynomials by grouping cubes • using factoring techniques including grouping and the sum or difference of two cubes *factor the sum or difference of two cubes synthetic division and the sum or difference of two cubes • using long division *apply the Rational Root (Zero) Theorem and analyze polynomial • using long division • using synthetic division results using technology factor • using synthetic division *find factors using long division and synthetic division Rational Root Theorem Note: Achieve includes solving higher-order polynomial equations PRF.4.AII.3 Write the equation of a polynomial function *determine a factor given a zero zero Write the equation of a polynomial function given its zeros. *determine a zero given a factor polynomial function given its roots (zeros) *write the equation of a polynomial function given its root zeros factor Note: include zeros of the form a+bi 1b. Essential Question - How can a graph be used to analyze a polynomial function? PRF.4.AII.2 With technology: A. Use appropriate technology to: polynomial function Analyze and sketch, with and without A. Analyze and sketch the graph of a given *analyze and sketch the graph of a given polynomial end behavior appropriate technology, the graph of a given polynomial function. function multiplicity polynomial function, determining the B. Determine the characteristics of domain *determine domain and range y-intercept characteristics of domain and range, and range, maximum and minimum points, *determine maximum and minimum points symmetry maximum and minimum points, end behavior, end behavior, zeros, multiplicity of zeros, y- *determine maximum and minimum values synthetic division zeros, multiplicity of zeros, y-intercept, and intercept, and symmetry. *distinguish between local (relative) and global maximum point symmetry Without technology: (absolute) maximums and minimums maximum value C. Determine domain, end behavior, zeros *determine end behavior maximum point and multiplicity of zeros, y-intercept, and *determine zeros and multiplicity of zeros minimum value symmetry. *determine y-intercept absolute/relative *determine different types of symmetry maximum and B. Without appropriate technology: minimums *determine domain domain *determine end behavior by using the leading coefficient range and the degree of the polynomial zeros *determine zeros and multiplicity of zeros when factorable to linear or quadratic factors *determine y-intercept *determine different types of symmetry rev. 5-2009 13 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) PRF.4.AII.4 Identify from a list of possible equations the Identify the equation of a polynomial function *use finite (first, 1st Nineweekssecond, third, etc.) differences to equation of a polynomial function given its identify linear, quadratic, or higher-order polynomial finite differences first differences given its graph or table graph or table. functions from a table second differences *use a graph to determine an equation of a polynomial polynomial function function linear quadratic RF.1.AII.4 A. Recognize parent function (i.e.. y=x, A. translation Analyze and report, with and without y=x n where n is a positive integer). *identify different parent functions reflection appropriate technology, the effect of changing B. Analyze and describe, with and without B. coefficient coefficients, exponents, and other parametersappropriate technology, the effect of changing *describe changes in slope and y-intercept from the exponent on functions and their graphs (linear, coefficients, and other parameters on linear parent graph parameter quadratic, and higher degree polynomial) functions and their graphs. (y=mx+b ) *describe changes in slope and y-intercept between two parent function C. Analyze and describe, with and without linear functions slope Note: The emphasis is on higher-degree appropriate technology, the effect of changing C. y-intercept polynomials at this point in the curriculum. coefficients, and other parameters on *describe changes using the words translate, reflect, translate quadratic functions and their graphs. (y=a(x- stretch or compress (shrink) including direction and reflect units from the parent graph stretch h) 2 + k ) *describe changes using the words translate, reflect, compress D. Analyze and describe, with and without stretch or compress (shrink) including direction and transformation appropriate technology, the effect of units between two quadratic functions linear functions changing coefficients, and other D. quadratic functions parameters on higher degree polynomial *describe changes using the words translate, reflect, cubic functions functions and their graphs. stretch or compress (shrink) including direction and quartic functions Note: The emphasis is on parts A and D at units from the parent graph higher-degree this point with higher-degree polynomial *describe changes using the words translate, reflect, polynomial functions functions. stretch or compress (shrink) including direction and units between two higher-degree polynomial functions Note: Achieve includes comparing to the parent graph (linear, quadratic, square root, absolute value) is the fundamental concept in this objective. rev. 5-2009 14 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st Nineweeks 1c. Essential Question - How can polynomial function be used to solve real-world problems? RF.1.AII.9 Communicate real world problems involving *identify key information in a real-world problem function Apply the concepts of functions to real-world functions, graphically, algebraically, *determine the format (graphically, algebraically, reasonableness situations numerically and verbally. numerically or verbally) to represent the problem and its level of precision solution *evaluate the reasonableness of the solution *communicate real-world problems graphically, algebraically, numerically, and verbally Note: Achieve includes solving literal equations and solving problems that can be modeled using absolute value, step, and other piece-wise defined functions. 2. Enduring Understanding - Mathematical operations can be applied to functions. 2a. Essential Question - How can mathematical operations be applied to functions? RF.1.AII.2 A. Evaluate functions. A. domain restrictions Evaluate, add, subtract, multiply, and divide B. Add, subtract, multiply and divide functions *know and use function notation range restrictions functions and give appropriate domain and and give appropriate domain and range B. function range restrictions restrictions. *combine functions by adding, subtracting, or function notation multiplying (i.e. polynomials) composition *express division of two polynomials as a quotient and include domain restrictions when appropriate Note: Achieve includes composition of two functions. rev. 5-2009 15 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st of polynomials. 3. Enduring Understanding - A rational function is a ratio Nineweeks 3a. Essential Question - How do you perform operations on rational expressions? PRF.4.AII.6 A. Simplify rational expressions A. rational expression Simplify, add, subtract, multiply, and divide B. Add, subtract, multiply, and divide rational *simplify rational expressions complex fractions with rational expressions expressions B. *add and subtract rational expressions with common denominators *add and subtract rational expressions with unlike denominators *multiply and divide rational expressions (including complex fractions) Note: Achieve includes solving rational equations, and solving problems that can be modeled with rational functions. 3b. Essential Question - What are the characteristics of a rational function? PRF.4.AII.5 Identify the characteristics of graphs of power *graph power function power function Identify the characteristics of graphs of power functions of the form f(x) = ax n , for negative *examine end behavior of a power function integral values functions of the form f(x) = ax n , for negative integral values of n , including domain, range, *examine behavior of the function as x approaches zero end behavior integral values of n, including domain, range, end behavior, and behavior at x = 0, and from the left and right sides of the graph domain end behavior, and behavior at x = 0, and compare these characteristics to the graphs of *determine vertical and horizontal asymptotes range compare these characteristics to the graphs of related positive integral power functions. Note: Achieve includes rational functions with linear, asymptote related positive integral power functions quadratic, or monomial denominators and also graph rational functions. rev. 5-2009 16 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st 4th Nineweeks 1. Enduring Understanding - Exponential and logarithmic functions model extreme growth or decay. 1a. Essential Question - How do exponential relationships model rapid change? ELF.5.AII.1 Recognize the graphs of exponential functions *identify exponential growth graphs exponential functions Recognize the graphs of exponential functions distinguishing between growth and decay. *identify exponential decay graphs growth distinguishing between growth and decay decay ELF.5.AII.2 A. Graph exponential functions without and A. domain Graph exponential functions and identify key with technology. *graph exponential functions without appropriate intercepts characteristics: domain, range, intercepts, B. Identify key characteristics of exponential technology asymptotes asymptotes, and end behavior functions, including domain, range, intercepts, *graph exponential functions with appropriate end behavior asymptotes, and end behavior, without and technology range of functions with technology. B. *identify key characteristics of exponential functions, including domain, range, intercepts, asymptotes, and end behavior, without appropriate technology *identify key characteristics of exponential functions, including domain, range, intercepts, asymptotes, and end behavior, with appropriate technology ELF.5.AII.3 A. Identify the effect that changes in the A. parameters Identify the effect that changes in the parameters of the base have on the graph of *identify an exponential function exponential function parameters of the base have on the graph of the exponential function without technology. *identify the effect that changes in the parameters of the base the exponential function B. Identify the effect that changes in the base have on the graph of the exponential function parameters of the base have on the graph of without appropriate technology the exponential function with technology. B. *identify the effect that changes in the parameters of the base have on the graph of the exponential function with appropriate technology Note: Achieve includes explain or illustrate the effect that changes in a parameter (a or c) or the base (b) have on the graph of the exponential function f(x) = abx + c. rev. 5-2009 17 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) ELF.5.AII.4 A. Recognize problems that can be modeled A. Change-of-Base Recognize and solve problems that can be using exponential functions. 1st Nineweeks that can be modeled using *recognize problems Formula modeled using exponential functions B. Solve problems that can be modeled using exponential functions logarithm exponential functions. B. exponential function C. Use technology to approximate solutions to *solve exponential equations using logarithms "e" exponential equations. *apply the Change-of-Base Formula to approximate solutions to exponential equations *represent a real world problem using an exponential equation and solve C. *use technology to approximate solutions to exponential equations Note: Achieve includes base e. rev. 5-2009 18 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) 1st Nineweeks 1b. Essential Question - How can logarithmic functions be used to determine an unknown exponent? RF.1.AII.3 A. Determine the inverse of a function. A. inverse of a function Determine the inverse of a function (Graph, B. Graph, with and without appropriate *know the definition of an inverse and its notation f -1 f -1 notation with and without appropriate technology, technology, functions and their inverses. *determine the inverse of a function given a set of exponential function functions and their inverses) ordered pairs (Achieve) *determine the inverse of a function algebraically identity function Note: This SLE was introduced in the 2nd B. reflection Nine Weeks. *know the identity function y=x and its graph Horizontal Line Test *determine the inverse of a function graphically with and one-to-one without appropriate technology Note: Achieve includes -inverses which may not be functions (not one-to-one) -explain why an inverse function would be only either the positive or the negative part of the graph -explain why the graphs of a function and its inverse are reflections of each other over the line y=x -show that when the inverse of a function is a function f - 1 (f(x))=x and f(f -1(x))=x -inverses of exponential functions may be required graphically but not in algebraic form ELF.5.AII.5 Establish the relationship between exponential *establish the relationship between exponential and logarithmic functions Establish the relationship between exponential and logarithmic functions without and with logarithmic functions with and without appropriate exponential functions and logarithmic functions technology. technology *use the definition of a logarithm to convert between exponential and logarithmic form *find the inverse function of a given exponential or logarithmic function ELF.5.AII.6 Evaluate simple logarithms using the definition *evaluate simple logarithms using the definition (i.e. log logarithm Evaluate simple logarithms using the definition (i.e. log 3 81 ) 3 81 ) natural logarithm (Ex. log 3 81 ) (Achieve) ELF.5.AII.7 A. Use properties of logarithms to expand *use properties of logarithms to expand logarithmic expand Use properties of logarithms to manipulate logarithmic expressions. expressions condense logarithmic expressions B. Use properties to condense logarithmic *use properties to condense logarithmic expressions expressions. rev. 5-2009 19 of 20 Arkansas Algebra II Arch Ford/Northwest Arkansas Instructional Alignment Algebra II AR Department of Education Essential Vocabulary Materials/ CONTENT STANDARD/ Objective Task Analysis *teacher word Resources Student Learning Expectations (SLE) DAP.6.AII.5 A. Compute measures of spread (range, A. range of data Compute and explain measures of spread 1st Nineweeks of spread (range, percentiles, percentiles, variance, and standard deviation). *compute measures σ percentiles σ (range, percentiles, variance, standard B. Explain measures of spread (range, variance, and standard deviation) variance 2, s2 deviation) percentiles, variance, and standard deviation). B. standard deviation , s *explain measures of spread (range, percentiles, mean µ , X- bar variance, and standard deviation) measures of spread DAP.6.AII.6 Describe the characteristics of a Gaussian *sketch a Gaussian normal distribution Gaussian normal Describe the characteristics of a Gaussian normal distribution. *show the symmetry of the distribution distribution normal distribution *locate the mean of the distribution standard deviation *find the percent of data within one, two and three distribution standard deviations from the mean symmetry mean 2. Enduring Understanding - Periodic functions can be used to describe natural relations. 2a. Essential Question - How can sine or cosine model specific phenomena such as frequency or amplitude? RF.1.AII.6 Recognize periodic phenomena (sine or *given a graph determine if the graph exhibits periodic periodic phenomena Recognize periodic phenomena (sine or cosine functions such as sound waves, length behavior circular motion cosine functions such as sound waves, length of daylight, circular motion) *given real-world situations (i.e.. sound waves, length of sine function of daylight, circular motion) daylight, circular motion) determine if periodic behavior cosine function is displayed RF.1.AII.7 Investigate and identify key characteristics of *investigate periodic functions period Investigate and identify key characteristics of periodic functions such as period, amplitude, *know and identify period, amplitude, maximum, and amplitude period functions and their graphs (period, maximum and minimum from a graph or table. minimum from a graph or table of a periodic function maximum amplitude, maximum, and minimum) minimum periodic function RF.1.AII.8 Apply frequency and amplitude to solve *know and identify frequency and amplitude from a table frequency Use basic properties of frequency and problems that model real-world phenomena of values or graph amplitude amplitude to solve problems (i.e. involving sound waves, tides, *apply frequency and amplitude to solve problems temperature, length of daylight, etc.) rev. 5-2009 20 of 20