A: Quadratic Functions

A.1: expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;

A.1.1: pose problems involving quadratic relations arising from real-world applications and represented by tables of values and graphs, and solve these and other such problems (e.g., "From the graph of the height of a ball versus time, can you tell me how high the ball was thrown and the time when it hit the ground?")

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

A.1.3: factor quadratic expressions in one variable, including those for which a is not equal to 1 (e.g., 3x² + 13x - 10), differences of squares (e.g., 4x² - 25), and perfect square trinomials (e.g., 9x² + 24x + 16), by selecting and applying an appropriate strategy

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

A.1.4: solve quadratic equations by selecting and applying a factoring strategy

Factoring Special Products
Modeling the Factorization of x²+bx+c

A.1.5: determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the graph of the corresponding quadratic relation

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

A.1.6: explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numeric example; follow a demonstration of the algebraic development, with technology, such as computer algebra systems, or without technology [student reproduction of the development of the general case is not required]), and apply the formula to solve quadratic equations, using technology

Roots of a Quadratic

A.1.7: relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph, and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if b² - 4ac < 0)

Polynomials and Linear Factors
Roots of a Quadratic

A.1.8: determine the real roots of a variety of quadratic equations (e.g., 100x² = 115x + 35), and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula)

Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Roots of a Quadratic

A.2: demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;

A.2.1: 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., 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.2.3: explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of linear and quadratic functions, and 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)

Functions Involving Square Roots

A.2.5: determine, through investigation using technology, the roles of a, h, and k in quadratic functions of the form f(x) = a(x - h)² + k, and describe these roles in terms of transformations on the graph of f(x) = x² (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

Reflections of a Quadratic Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A
Translations

A.2.6: sketch graphs of g(x) = a(x - h)² + k by applying one or more transformations to the graph of f(x) = x²

Translating and Scaling Functions

A.2.9: sketch graphs of quadratic functions in the factored form f(x) = a(x - r)(x - s) by using the x-intercepts to determine the vertex

Polynomials and Linear Factors

A.2.10: describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form f(x) = ax² + bx + c, the vertex form f(x) = a(x - h)² + k, and the factored form f(x) = a(x - r)(x - s) of a quadratic function

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

A.2.11: sketch the graph of a quadratic function whose equation is given in the standard form f(x) = ax² + bx + c by using a suitable strategy (e.g., completing the square and finding the vertex; factoring, if possible, to locate the x-intercepts), and identify the key features of the graph (e.g., the vertex, the x- and y-intercepts, the equation of the axis of symmetry, the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing)

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

A.3: solve problems involving quadratic functions, including problems arising from real-world applications.

A.3.2: determine, through investigation using a variety of strategies (e.g., applying properties of quadratic functions such as the x-intercepts and the vertex; using transformations), the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology (e.g., graphing software, graphing calculator)

Translating and Scaling Functions

B: Exponential Functions

B.1: simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;

B.1.1: 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(m/n), where x > 0 and m and n are integers)

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

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
Using Tables, Rules and Graphs

B.1.5: determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numeric expressions involving exponents [e.g., (½)³ x (½)²], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., (5³)²], and use the rules to simplify numerical expressions containing integer exponents [e.g., (2³)(2 to the 5th power) = 2 to the 8th power]

Multiplying Exponential Expressions

B.1.6: 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), within the same context when possible (e.g., simple interest and compound interest, population growth)

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

B.2: identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;

B.2.1: 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

B.2.2: 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)

Exponential Functions - Activity A
Exponential Growth and Decay - Activity B
Functions Involving Square Roots

B.2.3: 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

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

B.3: demonstrate an understanding of compound interest and annuities, and solve related problems.

B.3.1: compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time

Linear Functions
Simple and Compound Interest

B.3.2: solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), and the principal, P (also referred to as present value, PV), 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

B.3.3: determine, through investigation (e.g., using spreadsheets and graphs), that compound interest is an example of exponential growth [e.g., the formulas for compound interest, A = P((1 + i) to the n power), and present value, PV = A((1 + i) to the -n power), are exponential functions, where the number of compounding periods, n, varies]

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

B.3.4: solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in the compound interest formula A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]

Simple and Compound Interest

B.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 (i.e., annuities in which payments are made at the end of each period, and the compounding period and the payment period are the same) (e.g., long-term savings plans, loans)

Simple and Compound Interest

B.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

C: Trigonometric Functions

C.1: solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications;

C.1.1: solve problems, including those that arise from real-world applications (e.g., surveying, navigation), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios

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

C.2: demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;

C.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 numeric or graphical representation

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

C.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

C.2.3: make connections between the sine ratio and the sine function by graphing the relationship between angles from 0º to 360º and the corresponding sine 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, and explaining why the relationship is a function

Cosine Function
Linear Functions
Sine Function
Sine, Cosine and Tangent
Tangent Function
Unit Circle

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

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

C.2.5: make connections, through investigation with technology, between changes in a real-world situation that can be modelled using a periodic function and transformations of the corresponding graph (e.g., investigate the connection between variables for a swimmer swimming lengths of a pool and transformations of the graph of distance from the starting point versus time)

Cosine Function
Distance-Time Graphs
Sine Function
Tangent Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A

C.2.6: determine, through investigation using technology, the roles of the parameters a, c, and d in functions in the form f(x) = a sinx, f(x) = sinx + c, and f(x) = sin(x - d), and describe these roles in terms of transformations on the graph of f(x) = sinx with angles expressed in degrees (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

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

C.2.7: sketch graphs of f(x) = a sinx, f(x) = sinx + c, and f(x) = sin(x - d) by applying transformations to the graph of f(x) = sinx, and state the domain and range of the transformed functions

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

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

C.3.1: collect data that can be modelled as a sine 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

Sine Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions - Activity A
Unit Circle

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