composition of and the transformation of...

Composition of and the Transformation of Functions Specific Outcome Ways it could potentially be assessed

Level of

Understanding

1

Demonstrate an understanding of

operations on, and compositions of,

functions.

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 x 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 f a f g a

g f a f g h a

Determine, given the equations of two functions f x and g x , the equation of the composite function and explain any

restrictions.

f f x f g x

g f x f g h x

Sketch, given the equations of two functions f x and g x , the graph of the composite function:

f f x f g x g f x

Write a function h x as the composition of two or more functions.

Write a function h x by combining two or more functions through operations on and compositions of functions.

2

Demonstrate an understanding of the

effects of horizontal and vertical

translations on the graphs of

functions and their related

equations.

Compare the graphs of a set of functions of the form y k f x to the graph of y f x , and generalize, using

inductive reasoning, a rule about the effect of k .

Compare the graphs of a set of functions of the form y f x h to the graph of

y f x , 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 x h to the graph of

y f x , and generalize, using inductive reasoning, a rule about the effects of h and k .

Sketch the graph of y k f x , y f x h or y k f x h for given values of h and k , given a sketch

of the function y f x , where the equation of y f x is not given.

Write the equation of a function whose graph is a vertical and/or horizontal translation of the graph of the function

y f x .

3


effects of horizontal and vertical

stretches on the graphs of functions

Compare the graphs of a set of functions of the form y af x to the graph y f x , and generalize, using inductive

reasoning, a rule about the effect of a .

Compare the graphs of a set of functions of the form y f bx to the graph of y f x , and generalize, using

and their related equations. inductive reasoning, a rule about the effect of b .

Compare the graphs of a set of functions of the form y af bx to the graph of y f x , and generalize, using

inductive reasoning, a rule about the effects of a and b .

Sketch the graph of y af x ) , y f bx or y af bx for given values of a and b, given a sketch of the function

y f x , where the equation of y f x is not given.

Write the equation of a function, given its graph which is a vertical and/or horizontal stretch of the graph of the function

y f x .

4

Apply translations and stretches to

the graphs and equations of

functions.

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 x h ,

where the equation of y f x 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

y f x .

5


effects of reflections on the graphs

of functions and their related

equations, including reflections

through the:

x-axis

y-axis

line y x

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 x-axis, the y-axis or the line y x

Sketch the reflection of the graph of a function y f x through the x-axis, the y-axis or the line y x , given the graph

of the function y f x , where the equation of y f x is not given.

Generalize, using inductive reasoning, and explain rules for the reflection of the graph of the function y f x through

the x-axis, the y-axis or the line y x .

Sketch the graphs of the functions y f x , y f x and x f y , given the graph of the function

y f x , where the equation of y f x is not given.

Write the equation of a function, given its graph which is a reflection of the graph of the function y f x through the x-

axis, the y-axis or the line y x .

6


inverses of functions and relations.

Explain how the graph of the line y x can be used to sketch the inverse of a relation.

Explain how the transformation , ,x y y x can be used to sketch the inverse of a relation.

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.

Explain 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… Standard of Excellence: The Student also can…

sketch the graph of a function that is the sum, difference, product, quotient, or composition of

two functions, with a calculator.

given their graphs, sketch the graph of a function that is the sum, difference, product, or

quotient of two functions, without using 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 x , as:

the sum or difference of two functions

the product or quotient of two functions

a single composition of functions: ex: , h x f f x h x f g x

write the function, h x , as:

the product or quotient of three functions

the composition of two function, f x and g x and explain any restrictions on

h x

the composition of functions involving two compositions: ex:

, h x f g g x h x f g h x

write the function, h x , by combining two or more functions through operations on,

and/or compositions of, functions, limited to 2 operations. Ex:

h x g x f g x , h x f x g x k x

determine the value of the composition of functions at a point: ex: f f a , f g a

perform, analyze and describe:

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 x-axis and/or in the y-axis

graphically or algebraically, given the function in equation or graphical form or mapping

notation

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

determine the equation of a transformed function which involves a combination of

transformations including at least one reflection and one stretch on different axes

perform, analyze, sketch, and/or describe a reflection in the line y x , given the function or

relation in graphical form

determine restrictions on the domain of a function in order for its inverse to be a function,

given the graph or equation

determine the equation of the inverse of a linear or quadratic function and analyze its graph

Useful equations and formulas to keep in mind…

Polynomial, Radical and Rational Functions, Equations and Graphs Specific Outcome Ways it could potentially be assessed

Level of

Understanding

1


factoring polynomials of degree

greater than 2 (limited to

polynomials of degree 5 with

integral coefficients).

Explain how long division of a polynomial expression by a binomial expression of the form x a , Ia , is related to

synthetic division.

Divide a polynomial expression by a binomial expression of the form x a , Ia , using long division or synthetic

division.

Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding

polynomial function.

Explain the relationship between the remainder when a polynomial expression is divided by x a , Ia , and the value of

the polynomial expression at x a (remainder theorem).

Explain and apply the factor theorem to express a polynomial expression as a product of factors.

2

Graph and analyze polynomial

functions (limited to polynomial

functions of degree 5 ).

Identify the polynomial functions in a set of functions, and explain the reasoning.

Explain 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.

Explain the relationship between:

the zeros of a polynomial function

the roots of the corresponding polynomial equation

the x-intercepts of the graph of the polynomial function.

Explain 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.

3

Graph and analyze radical functions

(limited to functions involving one

radical).

Sketch the graph of the function y x , using a table of values, and state the domain and range.

Sketch the graph of the function y k a b x h by applying transformations to the graph of the function

y x , and state the domain and range.

Sketch the graph of the function: y f x , given the graph of the function y f x , and explain the strategies used.

Compare the domain and range of the function: y f x , to the domain and range of the function y f x , and

explain why the domains and ranges may differ.

Describe the relationship between the roots of a radical equation and the x–intercepts of the graph of the corresponding

radical function.

Determine, graphically, an approximate solution of a radical equation.

4

Graph and analyze rational functions

(limited to numerators and

denominators that are monomials,

binomials or trinomials).

Graph, algebraically OR with a calculator, a rational function.

Analyze the graphs of a set of rational functions to identify common characteristics.

Explain 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 explain the reasoning.

Describe the relationship between the roots of a rational equation and the x-intercepts of the graph of the corresponding

rational function.

Determine graphically an approximate solution of a rational equation.



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 explain if a given function is a polynomial function

find the zeros of a polynomial function and explain their relationship to the x-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

solve a problem by modeling a given situation with a polynomial function and analyzing the

graph of the function

determine the equation of a polynomial function in factored form, given its graph and/or key

characteristics

sketch and analyze (domain, range, invariant points, x- and y-intercepts): y f x given

the graph or equation of y f x

find the zeros of a radical function graphically and explain how they relate to the x-intercepts

of the graph and the roots of an equation

sketch and analyze (vertical asymptotes or point of discontinuity, domain, x- and y-intercepts)

rational functions

determine the equation of a horizontal asymptote and the range of a rational function

find the zeros of a rational function graphically and explain their relationship to the x-

intercepts of the graph and the roots of an equation

find the coordinates of the point of discontinuity of a rational function


Logarithms and Exponential Functions, Graphs and Equations Specific Outcome Ways it could potentially be assessed

Level of

Understanding

1


logarithms.

Explain the relationship between logarithms and exponents.

Express a logarithmic expression as an exponential expression and vice versa

Determine algebraically the exact value of a logarithm, such as 2

log 8 3 , 2

log 16 4 , 12

loga a

Estimate the value of a logarithm, using benchmarks, and explain the reasoning; e.g., since 2

log 8 3 , thus 2

log 16 4

and 2

log 9 3.1 .

2


product, quotient and power laws of

logarithms.

Determine, using the laws of logarithms, an equivalent expression for a logarithmic expression.

Determine, with a calculator, the approximate value of a logarithmic expression, such as2

log 9 .

Use the laws of logarithms in solving problems using numeric and algebraic quantities.

3

Graph and analyze exponential and

logarithmic functions.

Sketch, manually OR with a calculator, a graph of an exponential function of the formx

y a , 0a .

Identify the characteristics of the graph of an exponential function of the form x

y a , 0a : including the domain,

range, horizontal asymptote and intercepts, and explain the significance of the horizontal asymptote.

Sketch the graph of an exponential function by applying a set of transformations to the graph of x

y a , 0a , and state

the characteristics of the graph.

Sketch, manually OR with a calculator, the graph of a logarithmic function of the form logby x , 1b .

Identify the characteristics of the graph of a logarithmic function of the form logby x , 1b , including the domain,

range, vertical asymptote and intercepts, and explain the significance of the vertical asymptote

Sketch the graph of a logarithmic function by applying a set of transformations to the graph of logby x , 1b , and

state the characteristics of the graph.

Demonstrate, graphically, that a logarithmic function and an exponential function with the same base are inverses of each

other.

4

Solve problems that involve

exponential and logarithmic

equations.

Determine the solution of an exponential equation in which the bases are powers of one another.

Determine the solution of an exponential 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.

Explain why a value obtained in solving a logarithmic equation may be extraneous.

Solve a problem that involves exponential growth or decay.

Solve a problem that involves the application of exponential 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 exponential or a logarithmic equation.



determine, without technology, the exact values of simple logarithmic expressions

estimate the value of a logarithmic expression using benchmarks

convert between x

y b and

logb

y x convert between exponential and logarithmic forms involving multiple steps

simplify and/or expand logarithmic expressions using a law of logarithms simplify and/or expand logarithmic expressions using a combination of laws of logarithms

sketch and analyze (domain, range, intercepts, asymptote) the graphs of exponential or

logarithmic functions and their transformations

solve exponential equations that:

can be simplified to a common base

cannot be simplified to a common base and the exponents are monomials

solve exponential equations that cannot be simplified to a common base, where the exponents

are not monomials, or where there is a numerical coefficient

solve logarithmic equations but cannot recognize when a solution is extraneous solve logarithmic equations and recognize when a solution is extraneous

solve exponential and logarithmic function problems

solve for a value, such as earthquake intensities, pH and decibels in comparison problems solve for an exponent in comparison problems

Trigonometric Functions Specific Outcome Ways it could potentially be assessed

Level of

Understanding

1


angles in standard position,

expressed in degrees and radians.

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 expressed in the form k radians, where k is a rational number.

Express the measure of an angle in radians (exact value or decimal approximation), given its measure in degrees

Express the measure of an angle in degrees, given its measure in radians (exact value or decimal approximation).

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.

Explain 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.

2 Develop and apply the unit circle. Describe the six trigonometric ratios, using a point P (x, y) that is the intersection of the terminal arm of an angle and the

unit circle.

3

Solve problems, using the six

trigonometric ratios for angles

expressed in radians and degrees.

Determine with a calculator, the approximate value of a trigonometric ratio for any angle with a measure expressed in either

degrees or radians.

Determine, using a unit circle, the exact value of a trigonometric ratio for angles expressed in degrees that are multiples of

0º, 30º, 45º, 60º or 90º, or for angles expressed in radians that are multiples of 6 4 3 2or0, , , and explain 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.

Explain how to determine the exact values of the six 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 exact 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.



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 solve more difficult multi-step problems based on the relationship ar

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 x y that lies on the unit circle using your

primary trigonometric ratios if you are given an angle in radians or degrees.

find the exact or approximate values of trigonometric ratios of special angles, , where find the exact values of trigonometric ratios of special angles, , where

0 2 or 0 360 (the general solution)

determine the exact 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 2 or 0 360 OR given a point on the terminal

arm of an angle in standard position


Trigonometric Graphs Specific Outcome Ways it could potentially be assessed

Level of

Understanding

4

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.

Sketch, manually OR with a calculator, the graphs of siny x , cosy x or tany x .

Determine the characteristics (amplitude, asymptotes, domain, period, range and zeros) of the graph of

siny x , cosy x or tany x .

Know how varying the value of a affects the graphs of siny a x and cosy a x

Know how varying the value of d affects the graphs of siny x d and cosy x d

Know how varying the value of c affects the graphs of siny x c and cosy x c

Know how varying the value of b affects the graphs of siny bx and cosy bx

Sketch, , manually OR with a calculator, graphs of the form siny a b x c d or cosy a b x c d , using

transformations, and explain the strategies

Determine the characteristics (amplitude, asymptotes, domain, period, phase shift, range and zeros) of the graph of a

trigonometric function of the form siny a b x c d or cosy a b x c d .

Determine the values of a, b, c and d for functions of the form siny a b x c d or cosy a b x 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.

Explain 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.



sketch the graphs of siny x , cosy x or tany x and analyze the characteristics of

those graphs.

describe the characteristics of sinusoidal functions of the form siny a b x c d or

cosy a b x c d and sketch the graph

describe the characteristics of sinusoidal functions where the parameter b must be factored,

and sketch the graph

give partial explanations of the relationships between equation parameters and

transformations of sinusoidal functions

give full explanations 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

determine a complete equation for a sinusoidal curve given the graph, the characteristics, or a

real-world situation

provide a partial explanation of how the characteristics of the graph of a trigonometric

function relate to the conditions in a contextual situation

provide a complete explanation of how the characteristics of the graph of a trigonometric

function relate to the conditions in a contextual situation


Trigonometric Identities Specific Outcome Ways it could potentially be assessed

Level of

Understanding

5

Solve, algebraically and graphically,

first and second degree

trigonometric equations with the

domain expressed in degrees and

radians.

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 exact form when possible.

Determine, using a calculator, the approximate 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.

6

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).

Explain the difference between a trigonometric identity and a trigonometric equation.

Verify a trigonometric identity numerically for a given value in either degrees or radians.

Explain 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 exact value of a trigonometric ratio.



identify restrictions on the variable in the domain 0 2 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.

expressed as exact values (using the unit circle) or decimal approximations

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

expressed as exact values (using the unit circle) or decimal approximations

determine the general solution of a given trigonometric equation

explain the difference between a trigonometric identity and a trigonometric equation determine the non-permissible values of a trigonometric identity

explain 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 non-

permissible values

algebraically simplify and prove more difficult identities which include sum and difference

identities, double-angle identities, conjugates, or the extensive use of rational operations

determine the exact value of a trigonometric ratio using the sum, difference, and double-angle

identities of sine and cosine

determine the exact value of a trigonometric ratio using the sum, difference, and double-angle

identities of a tangent


Permutations and Combinations Specific Outcome Ways it could potentially be assessed

Level of

Understanding

1

Apply the fundamental counting

principle to solve problems.

Count the total number of possible choices that can be made, using graphic organizers such as lists and tree diagrams.

Explain, using examples, 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.

2

Determine the number of

permutations of n elements taken r

at a time to solve problems.

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.

Explain why n must be greater than or equal to r in the notation n rP

Solve an equation that involves n rP notation, such as 2 30n P .

Explain, using examples, the effect on the total number of permutations when two or more elements are identical.

3

Determine the number of

combinations of n different elements

taken r at a time to solve problems.

Explain, using examples, 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.

Explain why n must be greater than or equal to r in the notation n rC or n

r

Explain, using examples, why n r n n rC C or n n

r n r

Solve an equation that involves n rC or n

r

notation, such as 2 15nC or 2

15n

4

Expand powers of a binomial in a

variety of ways, including using the

binomial theorem (restricted to

exponents that are natural numbers).

Explain the patterns found in the expanded form of ( )n

x y , 4n and n N , by multiplying n factors of ( )x y

Explain how to determine the subsequent row in Pascal’s triangle, given any row.

Relate the coefficients of the terms in the expansion ( )n

x y to the 1n row in Pascal’s triangle.

Explain, using examples, how the coefficients of the terms in the expansion of ( )n

x y are determined by combinations.

Expand, using the binomial theorem, ( )n

x y .

Determine a specific term in the expansion of ( )n

x y .



apply the fundamental counting principle to various problems involving a single case or

constraint

apply the fundamental counting principle to various problems involving two or more cases or

constraints

recognize and address problems using the terms “and” or “or” recognize and address problems using the terms “at least” or “at most”

understand and use factorial notation

solve problems involving permutations or combinations solve problems involving both permutations and combinations

solve problems involving permutations when two or more elements are identical (repetitions) solve problems involving permutations when two or more elements are identical (repetitions),

with constraints

solve for n in equations involving one occurrence of n rP or n rC given r, where

3r and identify any extraneous solutions (ie: negative values are extraneous)

obtain solutions to problems involving a single case or constraint obtain solutions to problems involving two or more cases or constraints

demonstrate an understanding of patterns that exist in the binomial expansion

expand ( )n

x y or determine a specified term in the expansion of a binomial with linear

terms

expand ( )n

x y or determine a specified term in the expansion of a binomial with non-

linear terms

determine an unknown value in ( )

nx y

given a specified term in its expansion

Other useful equations and formulas to keep in mind…

Arrangements with repetition: !

! ! !

n

a b c where n is the number of terms, and a, b, c are the repetitions.

The number of diagonals in an n-sided shape is determined by 2nC 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

x y starts off at the ZEROTH POWER (n) and the FIRST ROW: The sum of the coefficients is 2n and this is the sum of the sequence of

combinations 0 ...n n nC C . The FIRST term starts at 0k , and the value of the thc term in the

thd row of Pascal’s triangle is found using 1 1c dC

composition of and the transformation of...

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