### A: Characteristics of Functions

#### A.1: demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;

A.1.1: explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., identifying a one-to-one or many-to-one mapping; using the vertical-line test)

A.1.3: explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions f (x) = x, f (x) = x², f(x) = the square root of x, and f(x) = 1/x; describe the domain and range of a function appropriately (e.g., for y = x² +1, the domain is the set of real numbers, and the range is y is greater than or equal to 1); and explain any restrictions on the domain and range in contexts arising from real-world applications

A.1.4: relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations)

A.1.8: determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x – d)) + c, and describe these roles in terms of transformations on the graphs of f(x) = x, f(x) = x , f(x) = the square root of x, and f(x) = 1/x (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

A.1.9: sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) = x, f(x) = x², f(x) = the square root of x, and f(x) = 1/x, and state the domain and range of the transformed functions

#### A.2: determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications;

A.2.1: determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant)

A.2.2: determine the maximum or minimum value of a quadratic function whose equation is given in the form f (x) = ax2 + bx+ c, using an algebraic method (e.g., completing the square; factoring to determine the zeros and averaging the zeros)

#### A.3: demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.

A.3.2: verify, through investigation with and without technology, that (the square root of ab) = (the square root of a) x (the square root of b), a is greater than or equal to 0, b is greater than or equal to 0, and use this relationship to simplify radicals (e.g., the square root of 24) and radical expressions obtained by adding, subtracting, and multiplying [e.g., (2 + the square root of 6)(3 – the square root of 12)]

A.3.4: determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values)

### B: Exponential Functions

#### B.1: evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;

B.1.3: simplify algebraic expressions containing integer and rational exponents [e.g., (x³) ÷ (x to the ½ power), ((x to the 6th power)y³) to the 1/3 power], and evaluate numeric expressions containing integer and rational exponents and rational bases [e.g., 2 to the –3 power, (–6)³, 4 to the ½ power, 1.01 to the 120th power]

B.1.4: determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form f(x) = a to the x power (a > 0, a is not equal to 1), function machines]

#### B.2: make connections between the numeric, graphical, and algebraic representations of exponential functions;

B.2.1: distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations)

B.2.2: determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x - d)) + c, and describe these roles in terms of transformations on the graph of f(x) = a to the x power (a > 0, a is not equal to 1) (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

B.2.3: sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graph of f(x) = a to the x power (a > 0, a is not equal to 1), and state the domain and range of the transformed functions

B.2.5: represent an exponential function with an equation, given its graph or its properties

#### B.3: identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.

B.3.2: identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)

B.3.3: solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations

### C: Discrete Functions

#### C.1: demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal’s triangle;

C.1.2: determine and describe (e.g., in words; using flow charts) a recursive procedure for generating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, …), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)

C.1.3: connect the formula for the nth term of a sequence to the representation in function notation, and write terms of a sequence given one of these representations or a recursion formula

C.1.4: represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8,16, 32, 64, … as t base 1 = 2; t base n = 2(t base (n–1), as f(n) = 2 to the n power, or as t base n = 2 to the n power, or represent ½, 2/3, 3/4, 4/5, 5/6, 6/7,... as t base 1 = ½; t base n = t base (n-1) + (1/(n(n + 1)), as f(n) = n/(n+1), or as t base n = n/(n + 1), where n is a natural number], and describe the information that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms)

C.1.5: determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways (e.g., tables of values, algebraic notation)

#### C.2: demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems;

C.2.1: identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation

C.2.2: determine the formula for the general term of an arithmetic sequence [i.e., t base n = a + (n –1)d] or geometric sequence (i.e., t base n = a(r to the (n–1) power)), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a sequence

C.2.4: solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications

#### C.3: make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.

C.3.2: make and describe connections between compound interest, geometric sequences, and exponential growth, through investigation with technology (e.g., use a spreadsheet to make compound interest calculations, determine finite differences in the amounts over time, and graph amount versus time)

### D: Trigonometric Functions

#### D.1: determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;

D.1.1: determine the exact values of the sine, cosine, and tangent of the special angles: 0º, 30º, 45º, 60º, and 90º

D.1.2: determine the values of the sine, cosine, and tangent of angles from 0º to 360º, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles)

D.1.3: determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the same

D.1.5: prove simple trigonometric identities, using the Pythagorean identity sin²x + cos²x = 1; the quotient identity tanx = sinx/cosx; and the reciprocal identities secx = 1/cosx, cscx = 1/sinx, and cotx = 1/tanx

D.1.6: pose problems involving right triangles and oblique triangles in two-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)

#### D.2: demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;

D.2.3: make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function f(x) = sinx or f(x) = cosx, and explaining why the relationship is a function

D.2.4: sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals)

D.2.6: determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = a sin(k(x – d)) + c or f(x) = a cos(k(x – d)) + c

#### D.3: identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.

D.3.5: pose problems based on applications involving a sinusoidal function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation

Correlation last revised: 9/16/2020

This correlation lists the recommended Gizmos for this province's curriculum standards. Click any Gizmo title below for more information.