### 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.2.2: 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.3.3: 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.1.4: 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

### 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.1.5: 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.2.4: 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: 9/24/2019

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