1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations. Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions Composition of and the Transformation of Functions Ways it could potentially be assessed Sketch the graph of a function that is the sum, difference, product or quotient of two functions, given their graphs. Write the equation of a function that is the sum, difference, product or quotient of two or more functions, given their equations. Determine the domain and range of a function that is the sum, difference, product or quotient of two functions. Write a function h as the sum, difference, product or quotient of two or more functions. Determine the value of the composition of functions when evaluated at a point, including: f g a f f a g f a f g ha Determine, given the equations of two functions restrictions. f f f g g f f g h Sketch, given the equations of two functions Write a function Write a function f and g, the equation of the composite function and eplain any f and g, the graph of the composite function: f f f g g f h as the composition of two or more functions. h by combining two or more functions through operations on and compositions of functions. Compare the graphs of a set of functions of the form inductive reasoning, a rule about the effect of k. y k f to the graph of Compare the graphs of a set of functions of the form y f h to the graph of, and generalize, using inductive reasoning, a rule about the effect of h. Compare the graphs of a set of functions of the form y k f h to the graph of, and generalize, using inductive reasoning, a rule about the effects of h and k. Sketch the graph of y k f, y f h or y k f h of the function, where the equation of is not given., and generalize, using for given values of h and k, given a sketch Write the equation of a function whose graph is a vertical and/or horizontal translation of the graph of the function. Compare the graphs of a set of functions of the form reasoning, a rule about the effect of a. y af to the graph Compare the graphs of a set of functions of the form y f b to the graph of, and generalize, using inductive, and generalize, using Level of Understanding
4 5 6 and their related equations. inductive reasoning, a rule about the effect of b. Apply translations and stretches to the graphs and equations of functions. Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the: -ais y-ais line y Demonstrate an understanding of inverses of functions and relations. Compare the graphs of a set of functions of the form y af b to the graph of inductive reasoning, a rule about the effects of a and b. Sketch the graph of y af ), y f b or y af b, where the equation of is not given., and generalize, using for given values of a and b, given a sketch of the function Write the equation of a function, given its graph which is a vertical and/or horizontal stretch of the graph of the function. Sketch the graph of the function for given values of a, b, h and k, given the graph of the function: y k af b h where the equation of is not given. Write the equation of a function, given its graph which is a translation and/or stretch of the graph of the function., Generalize the relationship between the coordinates of an ordered pair and the coordinates of the corresponding ordered pair that results from a reflection through the -ais, the y-ais or the line y y f, where the equation of Sketch the reflection of the graph of a function of the function through the -ais, the y-ais or the line y, given the graph is not given. Generalize, using inductive reasoning, and eplain rules for the reflection of the graph of the function the -ais, the y-ais or the line y. Sketch the graphs of the functions y f, y f and f y, where the equation of is not given., given the graph of the function Write the equation of a function, given its graph which is a reflection of the graph of the function ais, the y-ais or the line y. Eplain how the graph of the line y can be used to sketch the inverse of a relation. Eplain how the transformation, y y, can be used to sketch the inverse of a relation. through through the - Sketch the graph of the inverse relation, given the graph of a relation. Determine if a relation and its inverse are functions. Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation. Eplain the relationship between the domains and ranges of a relation and its inverse. Determine, algebraically or graphically, if two functions are inverses of each other.
How well do I need to understand these concepts Acceptable Standard: The Student can sketch the graph of a function that is the sum, difference, product, quotient, or composition of two functions, with a calculator. determine the domain and range of a function that is the sum, difference, product, or quotient of two functions, with a calculator write the function, h, as: the sum or difference of two functions the product or quotient of two functions a single composition of functions: e: h f f, h f g determine the value of the composition of functions at a point: e: perform, analyze and describe: f f a, f g a a horizontal and/or vertical translation a horizontal and/or vertical stretch a vertical stretch and translation(s) a horizontal stretch and translation(s) where the parameter b is removed through factoring. reflection in the -ais and/or in the y-ais graphically or algebraically, given the function in equation or graphical form or mapping notation determine the equation of a transformed function which involves a combination of transformations perform, analyze, sketch, and/or describe a reflection in the line y, given the function or relation in graphical form determine the equation of the inverse of a linear or quadratic function and analyze its graph Useful equations and formulas to keep in mind Standard of Ecellence: The Student also can given their graphs, sketch the graph of a function that is the sum, difference, product, or quotient of two functions, without using a calculator. write the function, h, as: the product or quotient of three functions the composition of two function, h f and g the composition of functions involving two compositions: e: and eplain any restrictions on h f g g, h f g h h, by combining two or more functions through operations on, and/or compositions of, functions, limited to operations. E: h g f g h f g k write the function,, perform, analyze, and describe: a horizontal stretch and a vertical stretch a horizontal stretch and translation where the parameter b is not removed through factoring a horizontal stretch and a vertical stretch and/or translation(s) a combination of transformations involving at least one stretch and one reflection graphically or algebraically, given the function in equation or graphical form or mapping notation determine the equation of a transformed function which involves a combination of transformations including at least one reflection and one stretch on different aes determine restrictions on the domain of a function in order for its inverse to be a function, given the graph or equation
1 3 4 Specific Outcome Demonstrate an understanding of factoring polynomials of degree greater than (limited to polynomials of degree 5 with integral coefficients). Graph and analyze polynomial functions (limited to polynomial functions of degree 5 ). Graph and analyze radical functions (limited to functions involving one radical). Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials or trinomials). Polynomial, Radical and Rational Functions, Equations and Graphs Ways it could potentially be assessed Eplain how long division of a polynomial epression by a binomial epression of the form a, a I, is related to synthetic division. Divide a polynomial epression by a binomial epression of the form a, a I, using long division or synthetic division. Eplain the relationship between the linear factors of a polynomial epression and the zeros of the corresponding polynomial function. Eplain the relationship between the remainder when a polynomial epression is divided by a, a I, and the value of the polynomial epression at a (remainder theorem). Eplain and apply the factor theorem to epress a polynomial epression as a product of factors. Identify the polynomial functions in a set of functions, and eplain the reasoning. Eplain the role of the constant term and leading coefficient in the equation of a polynomial function with respect to the graph of the function. Generalize rules for graphing polynomial functions of odd or even degree. Eplain the relationship between: the zeros of a polynomial function the roots of the corresponding polynomial equation the -intercepts of the graph of the polynomial function. Eplain how the multiplicity of a zero of a polynomial function affects the graph. Sketch, with or without technology, the graph of a polynomial function. Solve a problem by modeling a given situation with a polynomial function and analyzing the graph of the function. Sketch the graph of the function y, using a table of values, and state the domain and range. Sketch the graph of the function y k a b h by applying transformations to the graph of the function y, and state the domain and range. Sketch the graph of the function: Compare the domain and range of the function:, given the graph of the function, and eplain the strategies used., to the domain and range of the function eplain why the domains and ranges may differ. Describe the relationship between the roots of a radical equation and the intercepts of the graph of the corresponding radical function. Determine, graphically, an approimate solution of a radical equation. Graph, algebraically OR with a calculator, a rational function. Analyze the graphs of a set of rational functions to identify common characteristics. Eplain the behavior of the graph of a rational function for values of the variable near a non-permissible value. Determine if the graph of a rational function will have an asymptote or a hole for a non-permissible value. Match a set of rational functions to their graphs, and eplain the reasoning. Describe the relationship between the roots of a rational equation and the -intercepts of the graph of the corresponding rational function. Determine graphically an approimate solution of a rational equation., and Level of Understanding
How well do I need to understand these concepts Acceptable Standard: The Student can identify if a binomial is a factor of a given polynomial factor a polynomial of degree 3 completely partially factor a polynomial of degree 4 or 5 completely factor a polynomial of degree 4 or 5 identify and eplain if a given function is a polynomial function find the zeros of a polynomial function and eplain their relationship to the -intercepts of the graph and the roots of an equation sketch and analyze (multiplicities, y-intercept, domain and range, etc.) a polynomial function having only rational and integral zeros determine the equation of a polynomial function in factored form, given its graph and/or key characteristics sketch and analyze (domain, range, invariant points, - and y-intercepts): the graph or equation of given find the zeros of a radical function graphically and eplain how they relate to the -intercepts of the graph and the roots of an equation sketch and analyze (vertical asymptotes or point of discontinuity, domain, - and y-intercepts) rational functions find the zeros of a rational function graphically and eplain their relationship to the - intercepts of the graph and the roots of an equation Standard of Ecellence: The Student also can solve a problem by modeling a given situation with a polynomial function and analyzing the graph of the function determine the equation of a horizontal asymptote and the range of a rational function find the coordinates of the point of discontinuity of a rational function Useful equations and formulas to keep in mind
1 3 4 Specific Outcome Demonstrate an understanding of logarithms. Demonstrate an understanding of the product, quotient and power laws of logarithms. Graph and analyze eponential and logarithmic functions. Solve problems that involve eponential and logarithmic equations. Logarithms and Eponential Functions, Graphs and Equations Ways it could potentially be assessed Eplain the relationship between logarithms and eponents. Epress a logarithmic epression as an eponential epression and vice versa Determine algebraically the eact value of a logarithm, such as log 8 3, log 16 4, log 1 a a Estimate the value of a logarithm, using benchmarks, and eplain the reasoning; e.g., since log 8 3, thus log 16 4 and log 9 3.1. Determine, using the laws of logarithms, an equivalent epression for a logarithmic epression. Determine, with a calculator, the approimate value of a logarithmic epression, such as log 9. Use the laws of logarithms in solving problems using numeric and algebraic quantities. Sketch, manually OR with a calculator, a graph of an eponential function of the form y a, a 0. Identify the characteristics of the graph of an eponential function of the form y a, a 0 : including the domain, range, horizontal asymptote and intercepts, and eplain the significance of the horizontal asymptote. Sketch the graph of an eponential function by applying a set of transformations to the graph of y a, a 0, and state the characteristics of the graph. Sketch, manually OR with a calculator, the graph of a logarithmic function of the form y log b, b 1. Identify the characteristics of the graph of a logarithmic function of the form y log b, b 1, including the domain, range, vertical asymptote and intercepts, and eplain the significance of the vertical asymptote Sketch the graph of a logarithmic function by applying a set of transformations to the graph of y log b, b 1, and state the characteristics of the graph. Demonstrate, graphically, that a logarithmic function and an eponential function with the same base are inverses of each other. Determine the solution of an eponential equation in which the bases are powers of one another. Determine the solution of an eponential equation in which the bases are not powers of one another, using a variety of strategies. Determine the solution of a logarithmic equation, and verify the solution. Eplain why a value obtained in solving a logarithmic equation may be etraneous. Solve a problem that involves eponential growth or decay. Solve a problem that involves the application of eponential equations to loans, mortgages and investments. Solve a problem that involves logarithmic scales, such as the Richter scale and the ph scale. Solve a problem by modeling a situation with an eponential or a logarithmic equation. Level of Understanding
How well do I need to understand these concepts Acceptable Standard: The Student can determine, without technology, the eact values of simple logarithmic epressions estimate the value of a logarithmic epression using benchmarks convert between y b and log b y simplify and/or epand logarithmic epressions using a law of logarithms sketch and analyze (domain, range, intercepts, asymptote) the graphs of eponential or logarithmic functions and their transformations solve eponential equations that: can be simplified to a common base cannot be simplified to a common base and the eponents are monomials solve logarithmic equations but cannot recognize when a solution is etraneous solve eponential and logarithmic function problems solve for a value, such as earthquake intensities, ph and decibels in comparison problems Standard of Ecellence: The Student also can convert between eponential and logarithmic forms involving multiple steps simplify and/or epand logarithmic epressions using a combination of laws of logarithms solve eponential equations that cannot be simplified to a common base, where the eponents are not monomials, or where there is a numerical coefficient solve logarithmic equations and recognize when a solution is etraneous solve for an eponent in comparison problems
1 Specific Outcome Demonstrate an understanding of angles in standard position, epressed in degrees and radians. Develop and apply the unit circle. 3 Solve problems, using the si trigonometric ratios for angles epressed in radians and degrees. Ways it could potentially be assessed Trigonometric Functions Sketch, in standard position, an angle (positive or negative) when the measure is given in degrees. Describe the relationship among different systems of angle measurement, with emphasis on radians and degrees. Sketch, in standard position, an angle with a measure epressed in the form k radians, where k is a rational number. Epress the measure of an angle in radians (eact value or decimal approimation), given its measure in degrees Epress the measure of an angle in degrees, given its measure in radians (eact value or decimal approimation). Determine the measures, in degrees or radians, of all angles in a given domain that are co-terminal with a given angle in standard position. Determine the general form of the measures, in degrees or radians, of all angles that are co-terminal with a given angle in standard position. Eplain the relationship between the radian measure of an angle in standard position and the length of the arc cut on a circle of radius r, and solve problems based upon that relationship. Describe the si trigonometric ratios, using a point P (, y) that is the intersection of the terminal arm of an angle and the unit circle. Determine with a calculator, the approimate value of a trigonometric ratio for any angle with a measure epressed in either degrees or radians. Determine, using a unit circle, the eact value of a trigonometric ratio for angles epressed in degrees that are multiples of 0º, 30º, 45º, 60º or 90º, or for angles epressed in radians that are multiples of 0, 6, 4, 3 or and eplain the strategy. Determine, algebraically OR with a calculator, the measures, in degrees or radians, of the angles in a specified domain, given the value of a trigonometric ratio. Eplain how to determine the eact values of the si trigonometric ratios, given the coordinates of a point on the terminal arm of an angle in standard position. Determine the measures of the angles in a specified domain in degrees or radians, given a point on the terminal arm of an angle in standard position. Determine the eact values of the other trigonometric ratios, given the value of one trigonometric ratio in a specified domain. Sketch a diagram to represent a problem that involves trigonometric ratios. Solve a problem, using trigonometric ratios. Level of Understanding How well do I need to understand these concepts Acceptable Standard: The Student can demonstrate an understanding of the radian measure of an angle as a ratio of the subtended arc to the radius of a circle convert from radians to degrees and vice versa solve problems involving arc length, radius, and angle measure in either radians or degrees determine the measures, in degrees or radians, of all angles that are co-terminal with a given angle in standard position, within a specified domain determine the missing coordinate of a point P, y that lies on the unit circle using your Standard of Ecellence: The Student also can solve more difficult multi-step problems based on the relationship a r primary trigonometric ratios if you are given an angle in radians or degrees. find the eact or approimate values of trigonometric ratios of special angles,, where find the eact values of trigonometric ratios of special angles,, where
0 or 0 360 (the general solution) determine the eact values of all the trigonometric ratios, given the value of one trigonometric ratio in a restricted domain or the coordinates of a point on the terminal arm of an angle in standard position determine the measures of the angles,, in degrees or radians, given the value of a trigonometric ratio, where 0 or 0 360 OR given a point on the terminal arm of an angle in standard position Useful equations and formulas to keep in mind
4 Specific Outcome Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems. Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems. Ways it could potentially be assessed Trigonometric Graphs Sketch, manually OR with a calculator, the graphs of y sin, y cos or y tan. Determine the characteristics (amplitude, asymptotes, domain, period, range and zeros) of the graph of y sin, y cos or y tan. Know how varying the value of a affects the graphs of y asin and y acos Know how varying the value of d affects the graphs of y sin d and y cos d Know how varying the value of c affects the graphs of y sin c and y cos c Know how varying the value of b affects the graphs of y sin b and y cos b Sketch,, manually OR with a calculator, graphs of the form y a sinb c d or y a cosb c d, using transformations, and eplain the strategies Determine the characteristics (amplitude, asymptotes, domain, period, phase shift, range and zeros) of the graph of a trigonometric function of the form y a sinb c d or y a cosb c d. Determine the values of a, b, c and d for functions of the form y a sinb c d or y a cosb c d that correspond to a given graph, and write the equation of the function. Know how to set up a trigonometric function that models a situation in order to solve a problem. Eplain how the characteristics of the graph of a trigonometric function relate to the conditions in a problem situation. Solve a problem by analyzing the graph of a trigonometric function. Level of Understanding
How well do I need to understand these concepts Acceptable Standard: The Student can sketch the graphs of y sin, y cos or y tan and analyze the characteristics of those graphs. describe the characteristics of sinusoidal functions of the form y a sinb c d or y a cosb c d and sketch the graph give partial eplanations of the relationships between equation parameters and transformations of sinusoidal functions determine a partial equation for a sinusoidal curve given the graph, the characteristics, or a real-world situation provide a partial eplanation of how the characteristics of the graph of a trigonometric function relate to the conditions in a contetual situation Standard of Ecellence: The Student also can describe the characteristics of sinusoidal functions where the parameter b must be factored, and sketch the graph give full eplanations of the relationships between equation parameters and transformations of sinusoidal functions determine a complete equation for a sinusoidal curve given the graph, the characteristics, or a real-world situation provide a complete eplanation of how the characteristics of the graph of a trigonometric function relate to the conditions in a contetual situation Useful equations and formulas to keep in mind
5 6 Specific Outcome Solve, algebraically and graphically, first and second degree trigonometric equations with the domain epressed in degrees and radians. Prove trigonometric identities, using: reciprocal identities quotient identities Pythagorean identities sum or difference identities (restricted to sine, cosine and tangent) double-angle identities (restricted to sine, cosine and tangent). Ways it could potentially be assessed Trigonometric Identities Verify, algebraically OR with a calculator, that a given value is a solution to a trigonometric equation. Determine, algebraically, the solution of a trigonometric equation, stating the solution in eact form when possible. Determine, using a calculator, the approimate solution of a trigonometric equation in a restricted domain. Relate the general solution of a trigonometric equation to the zeros of the corresponding trigonometric function (restricted to sine and cosine functions). Determine, using a calculator, the general solution of a given trigonometric equation. Identify and correct errors in a solution for a trigonometric equation. Eplain the difference between a trigonometric identity and a trigonometric equation. Verify a trigonometric identity numerically for a given value in either degrees or radians. Eplain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid. Determine, graphically, the potential validity of a trigonometric identity, using technology. Determine the non-permissible values of a trigonometric identity. Prove, algebraically, that a trigonometric identity is valid. Determine, using the sum, difference and double-angle identities, the eact value of a trigonometric ratio. Level of Understanding
How well do I need to understand these concepts Acceptable Standard: The Student can Standard of Ecellence: The Student also can identify restrictions on the variable in the domain 0 or 0 360 identify restrictions on the variable in the domain where (the general solution) determine, in a restricted domain, the graphical solution for any trigonometric equation algebraically determine, in a restricted domain, the solution set of: first-degree trigonometric equations second-degree binomial trigonometric equations second-degree trinomial trigonometric equations where the coefficient in front of the squared term is equal to 1. epressed as eact values (using the unit circle) or decimal approimations algebraically determine, in a restricted domain, the solution set of: second-degree trinomial trigonometric where the coefficient in front of the squared term is not equal to 1. trigonometric equations involving trigonometric identity substitutions epressed as eact values (using the unit circle) or decimal approimations determine the general solution of a given trigonometric equation eplain the difference between a trigonometric identity and a trigonometric equation determine the non-permissible values of a trigonometric identity eplain the difference between verifying for a given value and proving an identity for all permissible values verify a trigonometric identity graphically or numerically for a given value algebraically simplify and prove simple identities, and recognize that there may be nonpermissible values algebraically simplify and prove more difficult identities which include sum and difference identities, double-angle identities, conjugates, or the etensive use of rational operations determine the eact value of a trigonometric ratio using the sum, difference, and double-angle identities of sine and cosine Useful equations and formulas to keep in mind determine the eact value of a trigonometric ratio using the sum, difference, and double-angle identities of a tangent
1 3 4 Specific Outcome Apply the fundamental counting principle to solve problems. Determine the number of permutations of n elements taken r at a time to solve problems. Determine the number of combinations of n different elements taken r at a time to solve problems. Epand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to eponents that are natural numbers). Permutations and Combinations Ways it could potentially be assessed Count the total number of possible choices that can be made, using graphic organizers such as lists and tree diagrams. Eplain, using eamples, why the total number of possible choices is found by multiplying rather than adding the number of ways the individual choices can be made. Solve a simple counting problem by applying the fundamental counting principle. Count, using graphic organizers such as lists and tree diagrams, the number of ways of arranging the elements of a set in a row. Determine, in factorial notation, the number of permutations of n different elements taken n at a time to solve a problem. Determine, using a variety of strategies, the number of permutations of n different elements taken r at a time to solve a problem. Eplain why n must be greater than or equal to r in the notation n P notation, such as 30 Solve an equation that involves P n r n. Eplain, using eamples, the effect on the total number of permutations when two or more elements are identical. Eplain, using eamples, the difference between a permutation and a combination. Determine the number of ways that a subset of k elements can be selected from a set of n different elements. Determine the number of combinations of n different elements taken r at a time to solve a problem. n Eplain why n must be greater than or equal to r in the notation n C r or r n n Eplain, using eamples, why n C r n C nr or r n r Solve an equation that involves n C r or P r n notation, such as 15 r Eplain the patterns found in the epanded form of ( y) n, 4 C n or n 15 Eplain how to determine the subsequent row in Pascal s triangle, given any row. Relate the coefficients of the terms in the epansion ( n and n N, by multiplying n factors of ( y) y) n to the n 1 Eplain, using eamples, how the coefficients of the terms in the epansion of ( row in Pascal s triangle. y) n are determined by combinations. Level of Understanding Epand, using the binomial theorem, ( y) n. Determine a specific term in the epansion of ( y) n.
How well do I need to understand these concepts Acceptable Standard: The Student can apply the fundamental counting principle to various problems involving a single case or constraint recognize and address problems using the terms and or or understand and use factorial notation solve problems involving permutations or combinations solve problems involving permutations when two or more elements are identical (repetitions) solve for n in equations involving one occurrence of n P r or n C r given r, where r 3 and identify any etraneous solutions (ie: negative values are etraneous) obtain solutions to problems involving a single case or constraint demonstrate an understanding of patterns that eist in the binomial epansion epand ( terms y ) n or determine a specified term in the epansion of a binomial with linear Standard of Ecellence: The Student also can apply the fundamental counting principle to various problems involving two or more cases or constraints recognize and address problems using the terms at least or at most solve problems involving both permutations and combinations solve problems involving permutations when two or more elements are identical (repetitions), with constraints obtain solutions to problems involving two or more cases or constraints or determine a specified term in the epansion of a binomial with non- epand ( linear terms y ) n determine an unknown value in ( y ) n given a specified term in its epansion Other useful equations and formulas to keep in mind Arrangements with repetition: n! abc!!! The number of diagonals in an n-sided shape is determined by C n where n is the number of terms, and a, b, c are the repetitions. n The number of ways two objects must be separated is determined by: (the total number of ways they can be arranged) (the ways they can be together) The number of possible of routes through a pathway can be determined by EITHER treating the path as a word with repetitions OR using combinations where you choose to go in a direction so many times. Pascal s Triangle, used for n combinations C0... n n n y starts off at the ZEROTH POWER (n) and the FIRST ROW: The sum of the coefficients is n and this is the sum of the sequence of C. The FIRST term starts at k 0, and the value of the th c term in the th d row of Pascal s triangle is found using c1 d 1 C