A: Exponential Functions

A.1: solve problems involving exponential equations graphically, including problems arising from real-world applications;

A.1.1: determine, through investigation with technology, and describe the impact of changing the base and changing the sign of the exponent on the graph of an exponential function

Exponential Functions

A.1.2: solve simple exponential equations numerically and graphically, with technology (e.g., use systematic trial with a scientific calculator to determine the solution to the equation 1.05 to the x power = 1,276), and recognize that the solutions may not be exact

Exponential Functions

A.1.4: pose problems based on real-world applications (e.g., compound interest, population growth) that can be modelled with exponential equations, 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

Compound Interest

A.2: solve problems involving exponential equations algebraically using common bases and logarithms, including problems arising from real-world applications.

A.2.1: simplify algebraic expressions containing integer and rational exponents using the laws of exponents (e.g., x³ ÷ x to the ½ power, square root of (x to the 6th power times y to the 12th power))

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions
Simplifying Algebraic Expressions II

A.2.2: solve exponential equations in one variable by determining a common base (eg., 2 to the x power = 32, 4 to the (5x - 1) power = 2 to the (2(x + 11)) power, 3 to the (5x + 8) power = 27 to the x power)

Exponential Functions

A.2.3: recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions

Logarithmic Functions

A.2.4: determine, with technology, the approximate logarithm of a number to any base, including base 10 [e.g., by recognizing that log base 10 of 0.372 can be determined using the LOG key on a calculator; by reasoning that log base 3 of 29 is between 3 and 4 and using systematic trial to determine that log base 3 of 29 is approximately 3.07]

Logarithmic Functions

A.2.5: make connections between related logarithmic and exponential equations (e.g., log base 5 of 125 = 3 can also be expressed as 5³ = 125), and solve simple exponential equations by rewriting them in logarithmic form (e.g., solving 3 to the x power = 10 by rewriting the equation as log base 3 of 10 = x)

Logarithmic Functions

B: Polynomial Functions

B.1: recognize and evaluate polynomial functions, describe key features of their graphs, and solve problems using graphs of polynomial functions;

B.1.2: compare, through investigation using graphing technology, the graphical and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., investigate the effect of the degree of a polynomial function on the shape of its graph and the maximum number of x-intercepts; investigate the effect of varying the sign of the leading coefficient on the end behaviour of the function for very large positive or negative x-values)

Graphs of Polynomial Functions
Polynomials and Linear Factors
Zap It! Game

B.1.3: describe key features of the graphs of polynomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or negative x-values)

Graphs of Polynomial Functions
Polynomials and Linear Factors

B.1.4: distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, f(x) = 2 to the x power)], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions

Compound Interest
Exponential Functions
Graphs of Polynomial Functions
Introduction to Exponential Functions
Logarithmic Functions
Polynomials and Linear Factors
Quadratics in Factored Form
Quadratics in Vertex Form
Translating and Scaling Functions

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

B.2.1: factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring)

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

B.2.2: make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g., f(x) = x(x – 1)(x + 1)] and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features (e.g., by determining intercepts and end behaviour; by locating positive and negative regions using test values between and on either side of the x-intercepts)

Polynomials and Linear Factors

B.2.3: determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), and describe the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function [e.g., the real roots of the equation (x to the 4th power) – 13x² + 36 = 0 are the x-intercepts of the graph of f(x) = (x to the 4th power) – 13x² + 36]

Polynomials and Linear Factors

B.3: solve polynomial equations by factoring, make connections between functions and formulas, and solve problems involving polynomial expressions arising from a variety of applications.

B.3.1: solve polynomial equations in one variable, of degree no higher than four (e.g., x² – 4x = 0, (x to the 4th power) – 16 = 0, 3x² + 5x + 2 = 0), by selecting and applying strategies (i.e., common factoring; difference of squares; trinomial factoring), and verify solutions using technology (e.g., using computer algebra systems to determine the roots of the equation; using graphing technology to determine the x-intercepts of the corresponding polynomial function)

Dividing Polynomials Using Synthetic Division

B.3.9: gather, interpret, and describe information about applications of mathematical modelling in occupations, and about college programs that explore these applications

Earthquakes 1 - Recording Station
Estimating Population Size

C: Trigonometric Functions

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

C.1.1: determine the exact values of the sine, cosine, and tangent of the special angles 0°, 30°, 45°, 60°, 90°, and their multiples

Cosine Function
Sine Function
Tangent Function

C.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 the special angles)

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios
Sum and Difference Identities for Sine and Cosine
Tangent Function

C.1.3: determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the same (e.g., determine one angle using a calculator and infer the other angle)

Sine Function
Tangent Function

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

C.2.1: 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) = sin x or f(x) = cos x, and explaining why the relationship is a function

Cosine Function
Sine Function

C.2.2: sketch the graphs of f(x) = sin x and f(x) = cos x 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)

Translating and Scaling Sine and Cosine Functions

C.2.5: determine the amplitude, period, and phase shift 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, and sketch graphs of y = a sin (k(x – d)) + c and y = a cos (k(x – d)) + c by applying transformations to the graphs of f(x) = sin x and f(x) = cos x

Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions

C.2.6: represent a sinusoidal function with an equation, given its graph or its properties

Translating and Scaling Functions

D: Applications of Geometry

D.1: represent vectors, add and subtract vectors, and solve problems using vector models, including those arising from real-world applications;

D.1.1: recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement; forces involved in structural design; simple animation of computer graphics; velocity determined using GPS)

Vectors

D.1.2: represent a vector as a directed line segment, with directions expressed in different ways (e.g., 320°; N 40° W), and recognize vectors with the same magnitude and direction but different positions as equal vectors

Vectors

D.1.3: resolve a vector represented as a directed line segment into its vertical and horizontal components

Adding Vectors

D.1.4: represent a vector as a directed line segment, given its vertical and horizontal components (e.g., the displacement of a ship that travels 3 km east and 4 km north can be represented by the vector with a magnitude of 5 km and a direction of N 36.9° E)

Vectors

D.1.5: determine, through investigation using a variety of tools (e.g., graph paper, technology) and strategies (i.e., head-to-tail method; parallelogram method; resolving vectors into their vertical and horizontal components), the sum (i.e., resultant) or difference of two vectors

Adding Vectors
Vectors

D.1.6: solve problems involving the addition and subtraction of vectors, including problems arising from real-world applications (e.g., surveying, statics, orienteering)

Adding Vectors
Vectors

D.2: solve problems involving two-dimensional shapes and three-dimensional figures and arising from real-world applications;

D.2.3: solve problems involving the areas of rectangles, parallelograms, trapezoids, triangles, and circles, and of related composite shapes, in situations arising from real-world applications

Area of Parallelograms
Circumference and Area of Circles
Perimeter and Area of Rectangles

D.2.4: solve problems involving the volumes and surface areas of spheres, right prisms, and cylinders, and of related composite figures, in situations arising from real-world applications

Prisms and Cylinders

D.3: determine circle properties and solve related problems, including those arising from real-world applications.

D.3.1: recognize and describe (i.e., using diagrams and words) arcs, tangents, secants, chords, segments, sectors, central angles, and inscribed angles of circles, and some of their real-world applications (e.g., construction of a medicine wheel)

Chords and Arcs
Circles
Inscribed Angles

D.3.3: determine, through investigation using a variety of tools (e.g., dynamic geometry software), properties of the circle associated with chords, central angles, inscribed angles, and tangents (e.g., equal chords or equal arcs subtend equal central angles and equal inscribed angles; a radius is perpendicular to a tangent at the point of tangency defined by the radius, and to a chord that the radius bisects)

Chords and Arcs
Circles
Inscribed Angles

D.3.4: solve problems involving properties of circles, including problems arising from real-world applications

Circles