### 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

A.1.3: determine, through investigation using graphing technology, the point of intersection of the graphs of two exponential functions (e.g., y = 4 to the -x power and y = 8 to the (x + 3) power), recognize the x-coordinate of this point to be the solution to the corresponding exponential equation (e.g., 4 to the -x power = 8 to the (x + 3) power), and solve exponential equations graphically (e.g., solve 2 to the (x + 2) power = (2 to the x power) + 12 by using the intersection of the graphs of y = 2 to the (x + 2) power and y = (2 to the x power) + 12)

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

#### 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))

A.2.6: pose problems based on real-world applications that can be modelled with given exponential equations, and solve these and other such problems algebraically by rewriting them in logarithmic form

### 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.1.1: recognize a polynomial expression (i.e., a series of terms where each term is the product of a constant and a power of x with a nonnegative integral exponent, such as x³ - 5x² + 2x - 1); recognize the equation of a polynomial function and give reasons why it is a function, and identify linear and quadratic functions as examples 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)

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)

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

B.1.5: substitute into and evaluate polynomial functions expressed in function notation, including functions arising from real-world applications

B.1.6: pose problems based on real-world applications that can be modelled with polynomial functions, 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

B.1.7: recognize, using graphs, the limitations of modelling a real-world relationship using a polynomial function, and identify and explain any restrictions on the domain and range (e.g., restrictions on the height and time for a polynomial function that models the relationship between height above the ground and time for a falling object)

#### 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)

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)

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]

#### 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)

B.3.2: solve problems algebraically that involve polynomial functions and equations of degree no higher than four, including those arising from real-world applications

B.3.4: expand and simplify polynomial expressions involving more than one variable [e.g., simplify -2xy(3x²y³ - 5x³y²)], including expressions arising from real-world applications

B.3.7: make connections between formulas and linear, quadratic, and exponential functions [e.g., recognize that the compound interest formula, A = P((1 + i) to the n power), is an example of an exponential function A(n) when P and i are constant, and of a linear function A(P) when i and n are constant], using a variety of tools and strategies (e.g., comparing the graphs generated with technology when different variables in a formula are set as constants)

### 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

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)

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)

C.1.4: solve multi-step problems in two and three dimensions, 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

#### 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

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)

C.2.3: determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = sin (x - d) + c and y = cos (x - d) + c, and describe these roles in terms of transformations on the graphs of f(x) = sin x and f(x) = cos x with angles expressed in degrees (i.e., vertical and horizontal translations)

C.2.4: determine, through investigation using technology, the roles of the parameters a and k in functions of the form y = a sin kx and y = a cos kx, and describe these roles in terms of transformations on the graphs of f(x) = sin x and f(x) = cos x with angles expressed in degrees (i.e., reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

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

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

#### C.3: demonstrate an understanding that sinusoidal functions can be used to model some periodic phenomena, and solve related problems, including those arising from real-world applications.

C.3.1: collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, pressure in sound waves, the height of a tack on a bicycle wheel that is rotating at a fixed speed), 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

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

C.3.3: 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, in degree mode, from a table of values or from its equation

### 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)

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

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)

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

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

#### 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

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

#### 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)

D.3.2: determine the length of an arc and the area of a sector or segment of a circle, and solve related problems

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)

Correlation last revised: 8/18/2015

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