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)

Linear Functions
Point-Slope Form of a Line - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic
Using Tables, Rules and Graphs

A.1.2: represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions [e.g., evaluate f(½), given f(x) = 2x² + 3x - 1]

Exponential Functions - Activity A
Translating and Scaling Functions

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

Functions Involving Square Roots
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Radical Functions
Roots of a Quadratic

A.1.5: determine the numeric or graphical representation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the graph of a function and the graph of its inverse (e.g., the graph of the inverse is the reflection of the graph of the function in the line y = x)

Cosine Function
Fourth-Degree Polynomials - Activity A
Linear Functions
Logarithmic Functions - Activity A
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A

A.1.6: determine, through investigation, the relationship between the domain and range of a function and the domain and range of the inverse relation, and determine whether or not the inverse relation is a function

Functions Involving Square Roots
Quadratics in Factored Form
Radical Functions
Roots of a Quadratic

A.1.7: determine, using function notation when appropriate, the algebraic representation of the inverse of a linear or quadratic function, given the algebraic representation of the function [e.g., f (x) = (x - 2)² - 5], and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the algebraic representations of a function and its inverse (e.g., the inverse of a linear function involves applying the inverse operations in the reverse order)

Translating and Scaling Functions

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)

Functions Involving Square Roots
Linear Functions
Quadratics in Factored Form
Radical Functions
Reflections of a Linear Function
Roots of a Quadratic
Translating and Scaling Functions

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

Functions Involving Square Roots
Quadratics in Factored Form
Radical Functions
Roots of a Quadratic
Translating and Scaling 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)

Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Polynomials and Linear Factors
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Roots of a Quadratic

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)

Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Polynomials and Linear Factors

A.2.3: solve problems involving quadratic functions arising from real-world applications and represented using function notation

Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

A.2.4: determine, through investigation, the transformational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding quadratic equation and a point on the function

Roots of a Quadratic
Translating and Scaling Functions

A.2.5: solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determining the time when two identical cylindrical water tanks contain equal volumes of water, if one tank is being filled at a constant rate and the other is being emptied through a hole in the bottom)

Prisms and Cylinders - Activity A
Solving Equations By Graphing Each Side

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

A.3.1: simplify polynomial expressions by adding, subtracting, and multiplying

Addition of Polynomials - Activity A

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)]

Operations with Radical Expressions
Simplifying Radicals - Activity A
Square Roots

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.1.2: determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with a rational exponent (i.e., x to the (m/n) power, where x > 0 and m and n are integers)

General Form of a Rational Function
Rational Functions

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]

Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
General Form of a Rational Function
Rational Functions
Using Tables, Rules and Graphs

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

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)

Exponential Functions - Activity A
Reflections of a Quadratic Function
Translating and Scaling Functions

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

Functions Involving Square Roots
Logarithmic Functions: Translating and Scaling
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

B.2.4: determine, through investigation using technology, that the equation of a given exponential function can be expressed using different bases [e.g., f(x) = 9 to the x power can be expressed as f(x) = 3 to the 2x power], and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws)

Exponential Functions - Activity A

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

Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Translating and Scaling Functions

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

B.3.1: collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g.,websites such as Statistics Canada, E-STAT), and graph the data

Exponential Functions - Activity A
Half-life
Triple Beam Balance

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)

Drug Dosage
Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Rational Functions
Simple and Compound Interest
Sine Function
Tangent Function

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

Drug Dosage
Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Half-life
Simple and Compound Interest

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.1.1: make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function [e.g., f(x) = 2x, where the domain is the set of natural numbers, is a discrete linear function, and its graph is a set of equally spaced points; f(x) = 2x, where the domain is the set of real numbers, is a continuous linear function and its graph is a straight line]

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences
Linear Functions

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)

Arithmetic Sequences
Arithmetic and Geometric Sequences

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

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

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)

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

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

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

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

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

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

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

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

C.3.1: make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology (e.g., use a spreadsheet or graphing calculator to make simple interest calculations, determine first differences in the amounts over time, and graph amount versus time)

Arithmetic Sequences
Arithmetic and Geometric Sequences
Linear Functions
Simple and Compound Interest
Slope-Intercept Form of a Line - Activity A

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)

Arithmetic and Geometric Sequences
Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Geometric Sequences
Half-life
Simple and Compound Interest

C.3.3: solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), the principal, P (also referred to as present value, PV), or the interest rate per compounding period, i, using the compound interest formula in the form A = P ((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]

Simple and Compound Interest

C.3.4: determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula in the form A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]; describe strategies (e.g., guessing and checking; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems

Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Simple and Compound Interest

C.3.5: explain the meaning of the term annuity, and determine the relationships between ordinary simple annuities (i.e., annuities in which payments are made at the end of each period, and compounding and payment periods are the same), geometric series, and exponential growth, through investigation with technology (e.g., use a spreadsheet to determine and graph the future value of an ordinary simple annuity for varying numbers of compounding periods; investigate how the contributions of each payment to the future value of an ordinary simple annuity are related to the terms of a geometric series)

Arithmetic and Geometric Sequences
Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Geometric Sequences
Half-life

C.3.6: determine, through investigation using technology (e.g., the TVM Solver on a graphing calculator, online tools), the effects of changing the conditions (i.e., the payments, the frequency of the payments, the interest rate, the compounding period) of ordinary simple annuities (e.g., long-term savings plans, loans)

Simple and Compound Interest

C.3.7: solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan)

Simple and Compound Interest

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

Cosine Function
Sine Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Function
Tangent Ratio
Unit Circle

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)

Cosine Function
Sine Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Function
Tangent Ratio
Unit Circle

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

Cosine Function
Sine Function
Tangent Function

D.1.4: define the secant, cosecant, and cotangent ratios for angles in a right triangle in terms of the sides of the triangle (e.g., sec A = hypotenuse/adjacent), and relate these ratios to the cosine, sine, and tangent ratios (e.g., sec A = 1/cos A)

Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Ratio

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

Simplifying Trigonometric Expressions

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)

Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent

D.1.7: pose problems involving right triangles and oblique triangles in three-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law

Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent

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.2.1: describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numerical or graphical representation

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions - Activity A

D.2.2: predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements; predicting natural gas consumption in Ontario from previous consumption)

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions - Activity A

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

Linear Functions
Sine Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Function
Tangent Ratio

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)

Cosine Function
Sine Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

D.2.5: 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, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

Sine Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

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

Cosine Function
Sine Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

D.2.7: sketch graphs of y = af(k(x - d)) + c by applying one or more transformations to the graphs of f(x) = sinx and f (x) = cosx, and state the domain and range of the transformed functions

Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

D.2.8: represent a sinusoidal function with an equation, given its graph or its properties

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

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

D.3.1: collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data

Cosine Function
Sine Function
Tangent Function
Unit Circle

D.3.2: identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions - Activity A

D.3.3: determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions - Activity A

D.3.4: predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied (e.g., varying the conditions, such as speed and direction, when walking in a circle in front of a motion sensor)

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Functions

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

Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A