Grade 12. Mathematical Process

Objectives: Ontario Curriculum: Grade 12 Advanced Functions, University Preparation

12.A Exponential and Logarithmic Functions

• A.1 demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
  • A.1.1 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
    • Convert between exponential and logarithmic form: rational bases (12-F.1)
    • Evaluate logarithms (12-F.2)
    • Change of base formula (12-F.3)
    • Evaluate logarithms: mixed review (12-F.9)
  • A.1.2 determine, with technology, the approximate logarithm of a number to any base, including base 10 (e.g., 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)
    • Evaluate logarithms (12-F.2)
    • Evaluate logarithms: mixed review (12-F.9)
  • A.1.3 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 xth power = 10 by rewriting the equation as log base 3 of 10 = x)
    • Convert between exponential and logarithmic form: rational bases (12-F.1)
    • Solve exponential equations using common logarithms (12-G.5)
  • A.1.4 make connections between the laws of exponents and the laws of logarithms [e.g., use the statement 10 to the a + b power = 10 to the a power times 10 to the b power to deduce that log base 10 of x + log base 10 of y = log base 10 of (xy)], verify the laws of logarithms with or without technology (e.g., use patterning to verify the quotient law for logarithms by evaluating expressions such as log base 10 of 1000 – log base 10 of 100 and then rewriting the answer as a logarithmic term to the same base), and use the laws of logarithms to simplify and evaluate numerical expressions
    • Change of base formula (12-F.3)
    • Identify properties of logarithms (12-F.4)
    • Product property of logarithms (12-F.5)
    • Quotient property of logarithms (12-F.6)
    • Power property of logarithms (12-F.7)
    • Properties of logarithms: mixed review (12-F.8)
  • A.2 identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
    • A.2.1 determine, through investigation with technology (e.g., graphing calculator, spreadsheet) and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, increasing/decreasing behaviour) of the graphs of logarithmic functions of the form f(x) = log base b of x, and make connections between the algebraic and graphical representations of these logarithmic functions
      • Domain and range of exponential and logarithmic functions (12-G.1)
    • A.2.2 recognize the relationship between an exponential function and the corresponding logarithmic function to be that of a function and its inverse, deduce that the graph of a logarithmic function is the reflection of the graph of the corresponding exponential function in the line y = x, and verify the deduction using technology
    • A.2.3 determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = log base 10 of (x – d) + c and the roles of the parameters a and k in functions of the form y = alog base 10 of (kx), and describe these roles in terms of transformations on the graph of f(x) = log base 10 of x (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
    • A.2.4 pose problems based on real-world applications of exponential and logarithmic functions (e.g., exponential growth and decay, the Richter scale, the pH scale, the decibel scale), 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
      • Exponential growth and decay: word problems (12-G.11)
      • Compound interest: word problems (12-G.12)
      • Continuously compounded interest: word problems (12-G.13)
    • A.3 solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
      • A.3.1 recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
        • Convert between exponential and logarithmic form: rational bases (12-F.1)
        • Evaluate logarithms (12-F.2)
        • Change of base formula (12-F.3)
        • Identify properties of logarithms (12-F.4)
        • Product property of logarithms (12-F.5)
        • Quotient property of logarithms (12-F.6)
        • Power property of logarithms (12-F.7)
        • Properties of logarithms: mixed review (12-F.8)
        • Evaluate logarithms: mixed review (12-F.9)
      • A.3.2 solve exponential equations in one variable by determining a common base (e.g., solve 4 to the x power= 8 to the x + 3 power by expressing each side as a power of 2) and by using logarithms (e.g., solve 4 = 8 by taking the logarithm base 2 of both sides), recognizing that logarithms base 10 are commonly used (e.g., solving 3 to the x power = 7 by taking the logarithm base 10 of both sides)
        • Solve exponential equations by factoring (12-G.4)
        • Solve exponential equations using common logarithms (12-G.5)
      • A.3.3 solve simple logarithmic equations in one variable algebraically [e.g., log base 3 of(5x + 6) = 2, log base 10 of (x + 1) = 1]
        • Solve logarithmic equations I (12-G.6)
        • Solve logarithmic equations II (12-G.7)
      • A.3.4 solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications
        • Solve exponential equations by factoring (12-G.4)
        • Solve exponential equations using common logarithms (12-G.5)
        • Solve logarithmic equations I (12-G.6)
        • Solve logarithmic equations II (12-G.7)
        • Exponential growth and decay: word problems (12-G.11)
        • Compound interest: word problems (12-G.12)
        • Continuously compounded interest: word problems (12-G.13)

12.B Trigonometric Functions

• B.1 demonstrate an understanding of the meaning and application of radian measure;
  • B.1.1 recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure
    • Convert between radians and degrees (12-N.1)
    • Radians and arc length (12-N.2)
  • B.1.2 represent radian measure in terms of pi (e.g., pi/3 radians, 2pi radians) and as a rational number (e.g., 1.05 radians, 6.28 radians)
    • Convert between radians and degrees (12-N.1)
    • Radians and arc length (12-N.2)
  • B.1.3 determine, with technology, the primary trigonometric ratios (i.e., sine, cosine, tangent) and the reciprocal trigonometric ratios (i.e., cosecant, secant, cotangent) of angles expressed in radian measure
    • Find trigonometric ratios using reference angles (12-N.7)
  • B.1.4 determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0, pi/6, pi/4, pi/3, pi/2, and their multiples less than or equal to 2pi
    • Quadrants (12-N.3)
    • Coterminal and reference angles (12-N.4)
    • Find trigonometric ratios using right triangles (12-N.5)
    • Find trigonometric ratios using the unit circle (12-N.6)
    • Find trigonometric ratios using reference angles (12-N.7)
  • B.2 make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
    • B.2.1 sketch the graphs of f(x) = sin x and f(x) = cos x for angle measures expressed in radians, and determine and describe some key properties (e.g., period of 2pi, amplitude of 1) in terms of radians
      • Find properties of sine functions (12-O.1)
      • Graph sine functions (12-O.4)
      • Find properties of cosine functions (12-O.5)
      • Graph cosine functions (12-O.8)
      • Graph sine and cosine functions (12-O.9)
      • Symmetry and periodicity of trigonometric functions (12-P.2)
    • B.2.2 make connections between the tangent ratio and the tangent function by using technology to graph the relationship between angles in radians and their tangent ratios and defining this relationship as the function f(x) = tan x, and describe key properties of the tangent function
    • B.2.3 graph, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions (i.e., cosecant, secant, cotangent) for angle measures expressed in radians, determine and describe key properties of the reciprocal functions (e.g., state the domain, range, and period, and identify and explain the occurrence of asymptotes), and recognize notations used to represent the reciprocal functions [e.g., the reciprocal of f(x) = sin x can be represented using csc x, 1/f(x), or 1/sin x, but not using f to the -1 power (x) or sin to the -1 one power times x, which represent the inverse function]
      • Trigonometric identities I (12-P.3)
    • B.2.4 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, with angles expressed in radians
      • Find properties of sine functions (12-O.1)
      • Find properties of cosine functions (12-O.5)
    • B.2.5 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 with angles expressed in radians, and state the period, amplitude, and phase shift of the transformed functions
      • Find properties of sine functions (12-O.1)
      • Graph sine functions (12-O.4)
      • Find properties of cosine functions (12-O.5)
      • Graph cosine functions (12-O.8)
      • Graph sine and cosine functions (12-O.9)
    • B.2.6 represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
      • Write equations of sine functions from graphs (12-O.2)
      • Write equations of sine functions using properties (12-O.3)
      • Write equations of cosine functions from graphs (12-O.6)
      • Write equations of cosine functions using properties (12-O.7)
    • B.2.7 pose problems based on applications involving a trigonometric function with domain expressed in radians (e.g., seasonal changes in temperature, heights of tides, hours of daylight, displacements for oscillating springs), and solve these and other such problems by using a given graph or a graph generated with or without technology from a table of values or from its equation
  • B.3 solve problems involving trigonometric equations and prove trigonometric identities.
    • B.3.1 recognize equivalent trigonometric expressions [e.g., by using the angles in a right triangle to recognize that sin x and cos (pi/2 – x) are equivalent; by using transformations to recognize that cos (x + pi/2) and –sin x are equivalent], and verify equivalence using graphing technology
      • Complementary angle identities (12-P.1)
      • Symmetry and periodicity of trigonometric functions (12-P.2)
      • Trigonometric identities I (12-P.3)
      • Trigonometric identities II (12-P.4)
    • B.3.2 explore the algebraic development of the compound angle formulas (e.g., verify the formulas in numerical examples, using technology; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]), and use the formulas to determine exact values of trigonometric ratios [e.g., determining the exact value of sin (pi/12) by first rewriting it in terms of special angles as sin (pi/4 – pi/6)]
    • B.3.3 recognize that trigonometric identities are equations that are true for every value in the domain (i.e., a counter-example can be used to show that an equation is not an identity), prove trigonometric identities through the application of reasoning skills, using a variety of relationships (e.g., tan x = sin x / cos x; sin²x + cos²x = 1; the reciprocal identities; the compound angle formulas), and verify identities using technology
      • Trigonometric identities I (12-P.3)
      • Trigonometric identities II (12-P.4)
    • B.3.4 solve linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from 0 to 2pi, and solve related problems
      • Solve trigonometric equations (12-N.9)

12.C Polynomial and Rational Functions

• C.1 identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
  • C.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, give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions
    • Identify functions (12-A.2)
    • Characteristics of quadratic functions (12-C.1)
  • C.1.2 compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., compare finite differences in tables of values; 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)
    • Graph a linear function (12-A.4)
    • Characteristics of quadratic functions (12-C.1)
    • Find the maximum or minimum value of a quadratic function (12-C.2)
    • Graph a quadratic function (12-C.3)
    • Match quadratic functions and graphs (12-C.4)
    • Match polynomials and graphs (12-D.9)
  • C.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)
    • Characteristics of quadratic functions (12-C.1)
    • Find the maximum or minimum value of a quadratic function (12-C.2)
    • Graph a quadratic function (12-C.3)
    • Match quadratic functions and graphs (12-C.4)
    • Match polynomials and graphs (12-D.9)
  • C.1.4 distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, g(x) = 2 to the x power], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions
    • Match polynomials and graphs (12-D.9)
    • Match exponential functions and graphs (12-G.3)
    • Graph sine and cosine functions (12-O.9)
  • C.1.5 make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g., f(x) = 2(x – 3)(x + 2)(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)
    • Write a polynomial from its roots (12-D.2)
    • Find the roots of factored polynomials (12-D.3)
    • Match polynomials and graphs (12-D.9)
  • C.1.6 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) = x³ and f(x) = x to the 4th power (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
    • Translations of functions (12-B.1)
    • Reflections of functions (12-B.2)
    • Dilations of functions (12-B.3)
    • Transformations of functions (12-B.4)
    • Function transformation rules (12-B.5)
    • Describe function transformations (12-B.6)
  • C.1.7 determine an equation of a polynomial function that satisfies a given set of conditions (e.g., degree of the polynomial, intercepts, points on the function), using methods appropriate to the situation (e.g., using the x-intercepts of the function; using a trial-and-error process with a graphing calculator or graphing software; using finite differences), and recognize that there may be more than one polynomial function that can satisfy a given set of conditions (e.g., an infinite number of polynomial functions satisfy the condition that they have three given x-intercepts)
    • Write a polynomial from its roots (12-D.2)
  • C.1.8 determine the equation of the family of polynomial functions with a given set of zeros and of the member of the family that passes through another given point [e.g., a family of polynomial functions of degree 3 with zeros 5, –3, and –2 is defined by the equation f(x) = k(x – 5)(x + 3)(x + 2), where k is a real number, k does not equal 0; the member of the family that passes through (–1, 24) is f(x) = –2(x – 5)(x + 3)(x + 2)]
    • Write a polynomial from its roots (12-D.2)
  • C.1.9 determine, through investigation, and compare the properties of even and odd polynomial functions [e.g., symmetry about the y-axis or the origin; the power of each term; the number of x-intercepts; f(x) = f(– x) or f(– x) = – f(x)], and determine whether a given polynomial function is even, odd, or neither
• C.2 identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
  • C.2.1 determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions [e.g., make connections between f(x) = 1/(x² – 4) and its graph by using graphing technology and by reasoning that there are vertical asymptotes at x = 2 and x = –2 and a horizontal asymptote at y = 0 and that the function maintains the same sign as f(x) = x² – 4]
    • Rational functions: asymptotes and excluded values (12-E.1)
  • C.2.2 determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that have linear expressions in the numerator and denominator [e.g., f(x) = 2x/(x – 3), h(x)= (x – 2)/(3x + 4)], and make connections between the algebraic and graphical representations of these rational functions
    • Rational functions: asymptotes and excluded values (12-E.1)
    • Domain and range of radical functions (12-H.1)
  • C.2.3 sketch the graph of a simple rational function using its key features, given the algebraic representation of the function
• C.3 solve problems involving polynomial and simple rational equations graphically and algebraically;
  • C.3.1 make connections, through investigation using technology (e.g., computer algebra systems), between the polynomial function f(x), the divisor x – a, the remainder from the division f(x)/(x – a), and f(a) to verify the remainder theorem and the factor theorem
  • C.3.2 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 by grouping, remainder theorem, factor theorem)
    • Factor sums and differences of cubes (12-D.10)
    • Factor using a quadratic pattern (12-D.12)
  • C.3.3 determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this connection [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]
    • Match polynomials and graphs (12-D.9)
  • C.3.4 solve polynomial equations in one variable, of degree no higher than four (e.g., 2x³ – 3x² + 8x – 12 = 0), by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem), and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding polynomial function)
    • Solve equations with sums and differences of cubes (12-D.11)
    • Solve equations using a quadratic pattern (12-D.13)
  • C.3.5 determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection [e.g., the real root of the equation (x – 2)/(x – 3) = 0 is 2, which is the x-intercept of the function f(x) = (x – 2)/(x – 3); the equation 1/(x – 3) = 0 has no real roots, and the function f(x) = 1/(x – 3) does not intersect the x-axis]
  • C.3.6 solve simple rational equations in one variable algebraically, and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding rational function)
    • Solve rational equations (12-E.2)
  • C.3.7 solve problems involving applications of polynomial and simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem, such as determining the values of k for which the function f(x) = x³ + 6x² + kx – 4 gives the same remainder when divided by x – 1 and x + 2]
• C.4 demonstrate an understanding of solving polynomial and simple rational inequalities.
  • C.4.1 explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one variable, and demonstrate that given solutions satisfy an inequality (e.g., demonstrate numerically and graphically that the solution to (1/(x + 1)) less than (5 is x) less than -1 or x greater than -4/5);
  • C.4.2 determine solutions to polynomial inequalities in one variable [e.g., solve f(x) ≥ 0, where f(x) = x³ – x² + 3x – 9] and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities
  • C.4.3 solve linear inequalities and factorable polynomial inequalities in one variable (e.g., (x³ + x²) greater than 0) in a variety of ways (e.g., by determining intervals using x-intercepts and evaluating the corresponding function for a single x-value within each interval; by factoring the polynomial and identifying the conditions for which the product satisfies the inequality), and represent the solutions on a number line or algebraically (e.g., for the inequality (x to the 4th power – 5x² + 4) less than 0, the solution represented algebraically is –2 less than x less than –1 or 1 less than x less than 2)
    • Graph solutions to quadratic inequalities (12-L.1)
    • Solve quadratic inequalities (12-L.2)
    • Graph solutions to higher-degree inequalities (12-L.3)
    • Solve higher-degree inequalities (12-L.4)

12.D Characteristics of Functions

• D.1 demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
  • D.1.1 gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change (e.g., in words, numerically, graphically, algebraically)
  • D.1.2 recognize that the rate of change for a function is a comparison of changes in the dependent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or changing by examining applications, including those arising from real-world situations (e.g., rate of change of the area of a circle as the radius increases, inflation rates, the rising trend in graduation rates among Aboriginal youth, speed of a cruising aircraft, speed of a cyclist climbing a hill, infection rates)
  • D.1.3 sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology (e.g., motion sensor) when possible
  • D.1.4 calculate and interpret average rates of change of functions (e.g., linear, quadratic, exponential, sinusoidal) arising from real-world applications (e.g., in the natural, physical, and social sciences), given various representations of the functions (e.g., tables of values, graphs, equations)
  • D.1.5 recognize examples of instantaneous rates of change arising from real-world situations, and make connections between instantaneous rates of change and average rates of change (e.g., an average rate of change can be used to approximate an instantaneous rate of change)
  • D.1.6 determine, through investigation using various representations of relationships (e.g., tables of values, graphs, equations), approximate instantaneous rates of change arising from real-world applications (e.g., in the natural, physical, and social sciences) by using average rates of change and reducing the interval over which the average rate of change is determined
  • D.1.7 make connections, through investigation, between the slope of a secant on the graph of a function (e.g., quadratic, exponential, sinusoidal) and the average rate of change of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point
  • D.1.8 determine, through investigation using a variety of tools and strategies (e.g., using a table of values to calculate slopes of secants or graphing secants and measuring their slopes with technology), the approximate slope of the tangent to a given point on the graph of a function (e.g., quadratic, exponential, sinusoidal) by using the slopes of secants through the given point (e.g., investigating the slopes of secants that approach the tangent at that point more and more closely), and make connections to average and instantaneous rates of change
  • D.1.9 solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods (e.g., by using graphing technology to graph a tangent and measure its slope)
• D.2 determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
  • D.2.1 determine, through investigation using graphing technology, key features (e.g., domain, range, maximum/minimum points, number of zeros) of the graphs of functions created by adding, subtracting, multiplying, or dividing functions [e.g., f(x) = 2 to the -x power sin 4x, g(x) = x² + 2 to the x power, h(x) = (sin x)/(cos x)], and describe factors that affect these properties
    • Add, subtract, multiply and divide functions (12-A.10)
  • D.2.2 recognize real-world applications of combinations of functions (e.g., the motion of a damped pendulum can be represented by a function that is the product of a trigonometric function and an exponential function; the frequencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically
    • Composition of functions (12-A.11)
  • D.2.3 determine, through investigation, and explain some properties (i.e., odd, even, or neither; increasing/decreasing behaviours) of functions formed by adding, subtracting, multiplying, and dividing general functions [e.g., f(x) + g(x), f(x)g(x)]
  • D.2.4 determine the composition of two functions [i.e., f(g(x))] numerically (i.e., by using a table of values) and graphically, with technology, for functions represented in a variety of ways (e.g., function machines, graphs, equations), and interpret the composition of two functions in real-world applications
    • Composition of functions (12-A.11)
  • D.2.5 determine algebraically the composition of two functions [i.e., f(g(x))], verify that f(g(x)) is not always equal to g(f(x)) [e.g., by determining f(g(x)) and g(f(x)), given f(x) = x + 1 and g(x) = 2x], and state the domain [i.e., by defining f(g(x)) for those x-values for which g(x) is defined and for which it is included in the domain of f(x)] and the range of the composition of two functions
    • Composition of functions (12-A.11)
  • D.2.6 solve problems involving the composition of two functions, including problems arising from real-world applications
    • Composition of functions (12-A.11)
  • D.2.7 demonstrate, by giving examples for functions represented in a variety of ways (e.g., function machines, graphs, equations), the property that the composition of a function and its inverse function maps a number onto itself [i.e., f to the -1 power (f(x)) = x and f(f to the -1 power (x)) = x demonstrate that the inverse function is the reverse process of the original function and that it undoes what the function does]
    • Identify inverse functions (12-A.12)
    • Find values of inverse functions from tables (12-A.13)
    • Find values of inverse functions from graphs (12-A.14)
    • Find inverse functions and relations (12-A.15)
  • D.2.8 make connections, through investigation using technology, between transformations (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) of simple functions f(x) [e.g., f(x) = x to the 3rd power + 20, f(x) = sin x, f(x) = log x] and the composition of these functions with a linear function of the form g(x) = A(x + B)
    • Translations of functions (12-B.1)
    • Reflections of functions (12-B.2)
    • Dilations of functions (12-B.3)
    • Transformations of functions (12-B.4)
    • Function transformation rules (12-B.5)
    • Describe function transformations (12-B.6)
  • D.3 compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
    • D.3.1 compare, through investigation using a variety of tools and strategies (e.g., graphing with technology; comparing algebraic representations; comparing finite differences in tables of values) the characteristics (e.g., key features of the graphs, forms of the equations) of various functions (i.e., polynomial, rational, trigonometric, exponential, logarithmic)
      • Linear functions over unit intervals (12-A.6)
      • Exponential functions over unit intervals (12-G.9)
      • Describe linear and exponential growth and decay (12-G.10)
    • D.3.2 solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques
    • D.3.3 solve problems, using a variety of tools and strategies, including problems arising from real-world applications, by reasoning with functions and by applying concepts and procedures involving functions (e.g., by constructing a function model from data, using the model to determine mathematical results, and interpreting and communicating the results within the context of the problem)
      • Exponential growth and decay: word problems (12-G.11)
      • Compound interest: word problems (12-G.12)
      • Continuously compounded interest: word problems (12-G.13)

Ontario Curriculum: Grade 12 Calculus and Vectors, University Preparation

12.A Rate of Change

• A.1 demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit;
  • A.1.1 describe examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically)
    • Velocity as a rate of change (C-J.4)
  • A.1.2 describe connections between the average rate of change of a function that is smooth (i.e., continuous with no corners) over an interval and the slope of the corresponding secant, and between the instantaneous rate of change of a smooth function at a point and the slope of the tangent at that point
    • Average rate of change I (C-J.1)
    • Average rate of change II (C-J.2)
    • Find instantaneous rates of change (C-J.3)
    • Find the slope of a tangent line using limits (C-J.6)
  • A.1.3 make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approximate the slope of the tangent)
  • A.1.4 recognize, through investigation with or without technology, graphical and numerical examples of limits, and explain the reasoning involved (e.g., the value of a function approaching an asymptote, the value of the ratio of successive terms in the Fibonacci sequence)
    • Find limits using graphs (C-E.1)
    • Find one-sided limits using graphs (C-E.2)
    • Determine if a limit exists (C-E.3)
    • Find limits at vertical asymptotes using graphs (C-G.1)
    • Determine end behaviour using graphs (C-G.2)
    • Determine end behaviour of polynomial and rational functions (C-G.3)
  • A.1.5 make connections, for a function that is smooth over the interval a less than or equal to x less than or equal to a + h, between the average rate of change of the function over this interval and the value of the expression (f(a + h) – f(a))/h, and between the instantaneous rate of change of the function at x = a and the value of the limit of (f(a + h) – f(a))/h as h approaches 0
  • A.1.6 compare, through investigation, the calculation of instantaneous rates of change at a point (a, f(a)) for polynomial functions [e.g., f(x) = x², f(x) = x³], with and without simplifying the expression (f(a + h) – f(a))/h before substituting values of h that approach zero [e.g., for f(x) = x² at x = 3, by determining (f(3 + 1) – f(3))/1 = 7, (f(3 + 0.1) – f(3))/0.1 = 6.1, (f(3 + 0.01) – f(3))/0.001 = 6.01, and (f(3 + 0.001 – f(3))/0.001 = 6.001, and by first simplifying (f(3 + h) – f(3))/h as ((3 + h)² – 3²)/h = 6 + h and then substituting the same values of h to give the same results]
• A.2 graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative;
  • A.2.1 determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals (e.g., by using graphing technology to examine the table of values and the slopes of tangents for a function whose equation is given; by examining a given graph), and describe the behaviour of the instantaneous rate of change at and between local maxima and minima
  • A.2.2 generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f'(x) or (dy)/(dx), and make connections between the graphs of f(x) and f'(x) or y and (dy)/(dx) [e.g., when f(x) is linear, f'(x) is constant; when f(x) is quadratic, f'(x) is linear; when f(x) is cubic, f'(x) is quadratic]
  • A.2.3 determine the derivatives of polynomial functions by simplifying the algebraic expression (f(x + h)- f(x))/h and then taking the limit of the simplified expression as h approaches zero [i.e., determining the limit of (f(x + h) – f(x))/h as h approaches 0]
    • Find instantaneous rates of change (C-J.3)
    • Find values of derivatives using limits (C-J.5)
    • Find the slope of a tangent line using limits (C-J.6)
  • A.2.4 determine, through investigation using technology, the graph of the derivative f'(x) or (dy)/(dx) of a given sinusoidal function [i.e., f(x) = sin x, f(x) = cos x] (e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify graphically that when f(x) = sin x, f'(x) = cos x, and when f(x) = cos x, f'(x) = – sin x; by using a motion sensor to compare the displacement and velocity of a pendulum)
  • A.2.5 determine, through investigation using technology, the graph of the derivative f,(x) or (dy)/(dx) of a given exponential function [i.e., f(x) = a to the x power (a greater than 0, a does not equal 1)] [e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify that when f(x) = a to the x power, f'(x) = kf(x)], and make connections between the graphs of f(x) and f'(x) or y and (dy)/(dx) [e.g., f(x) and f'(x) are both exponential; the ratio (f'(x))/(f(x)) is constant, or f'(x) = kf(x); f'(x) is a vertical stretch from the x-axis of f(x)]
  • A.2.6 determine, through investigation using technology, the exponential function f(x) = a to the x power (a is greater than 0, a does not equal 1) for which f'(x) = f(x) (e.g., by using graphing technology to create a slider that varies the value of a in order to determine the exponential function whose graph is the same as the graph of its derivative), identify the number e to be the value of a for which f'(x) = f(x) [i.e., given f(x) = e to the x power, f'(x) = e to the x power], and recognize that for the exponential function f(x) = e to the x power the slope of the tangent at any point on the function is equal to the value of the function at that point
  • A.2.7 recognize that the natural logarithmic function f(x) = log base e of x, also written as f(x) = ln x, is the inverse of the exponential function f(x) = e to the x power, and make connections between f(x) = ln x and f(x) = e to the x power [e.g., f(x) = ln x reverses what f(x) = e to the x power does; their graphs are reflections of each other in the line y = x; the composition of the two functions, e to the lnx power or ln e to the x power, maps x onto itself, that is, e to the lnx power = x and ln e to the x power = x]
  • A.2.8 verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function f(x) = a to the x power is f'(x) = a to the x power ln a for various values of a [e.g., verifying numerically for f(x) = 2 to the x power that f'(x) = 2 to the x power ln 2 by using a calculator to show that the limit of (2 to the h power – 1)/h as h approaches 0 is ln 2 or by graphing f(x) = 2 to the x power, determining the value of the slope and the value of the function for specific x-values, and comparing the ratio (f'(x))/(f(x)) with ln 2]
• A.3 verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.
  • A.3.1 verify the power rule for functions of the form f(x) = x to the n power, where n is a natural number [e.g., by determining the equations of the derivatives of the functions f(x) = x, f(x) = x², f(x) = x³, and f(x) = x to the 4th power algebraically using the limit of (f(x + h) – f(x))/h, as h approaches 0, and graphically using slopes of tangents]
  • A.3.2 verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function g(x) = kf(x) and comparing the graphs of g'(x) and kf'(x); by using a table of values to verify that f'(x) + g'(x) = (f + g)'(x), given f(x) = x and g(x) = 3x], and read and interpret proofs involving the limit of (f(x + h) – f(x))/h, as h approaches 0, of the constant, constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required)
    • Sum and difference rules (C-K.1)
  • A.3.3 determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs
    • Find derivatives of polynomials (C-L.1)
  • A.3.4 verify that the power rule applies to functions of the form f(x) = x to the n power, where n is a rational number [e.g., by comparing values of the slopes of tangents to the function f(x) = x to the ½ power with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for f(x) = (5x³) to the 1/3 power by using the chain rule and by differentiating the simplified form, f(x) = (5 to the 1/3 power) times x] and the product rule using polynomial functions [e.g., by determining the same derivative for f(x) = (3x + 2)(2x² – 1) by using the product rule and by differentiating the expanded form f(x) = 6x³ + 4x² – 3x – 2]
    • Product rule (C-K.2)
    • Power rule I (C-K.4)
    • Power rule II (C-K.5)
    • Chain rule (C-K.6)
  • A.3.5 solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing f(x) = x² + 1/x – 1 as the product f(x) = (x² + 1)(x – 1) to the -1 power], radical functions [e.g., by expressing f(x) = the square root of x² + 5 as the power f(x) = (x² + 5) to the ½ power], and other simple combinations of functions [e.g., f(x) = x sin x, f(x) = sin x/cos x]
    • Find derivatives using the product rule I (C-L.9)
    • Find derivatives using the product rule II (C-L.10)
    • Find derivatives using the chain rule I (C-L.13)
    • Find derivatives using the chain rule II (C-L.14)

12.B Derivatives and Their Applications

• B.1 make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching;
  • B.1.1 sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes)
  • B.1.2 recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function
  • B.1.3 determine algebraically the equation of the second derivative f”(x) of a polynomial or simple rational function f(x), and make connections, through investigation using technology, between the key features of the graph of the function (e.g., increasing/ decreasing intervals, local maxima and minima, points of inflection, intervals of concavity) and corresponding features of the graphs of its first and second derivatives (e.g., for an increasing interval of the function, the first derivative is positive; for a point of inflection of the function, the slopes of tangents change their behaviour from increasing to decreasing or from decreasing to increasing, the first derivative has a maximum or minimum, and the second derivative is zero)
    • Find higher derivatives of polynomials (C-N.1)
    • Find higher derivatives of rational and radical functions (C-N.2)
  • B.1.4 describe key features of a polynomial function, given information about its first and/or second derivatives (e.g., the graph of a derivative, the sign of a derivative over specific intervals, the x-intercepts of a derivative), sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible
  • B.1.5 sketch the graph of a polynomial function, given its equation, by using a variety of strategies (e.g., using the sign of the first derivative; using the sign of the second derivative; identifying even or odd functions) to determine its key features (e.g., increasing/decreasing intervals, intercepts, local maxima and minima, points of inflection, intervals of concavity), and verify using technology
• B.2 solve problems, including optimization problems, that require the use of the concepts and procedures associated with the derivative, including problems arising from real-world applications and involving the development of mathematical models.
  • B.2.1 make connections between the concept of motion (i.e., displacement, velocity, acceleration) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically)
    • Velocity as a rate of change (C-J.4)
  • B.2.2 make connections between the graphical or algebraic representations of derivatives and real-world applications (e.g., population and rates of population change, prices and inflation rates, volume and rates of flow, height and growth rates)
  • B.2.3 solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications (e.g., population growth, radioactive decay, temperature changes, hours of daylight, heights of tides), given the equation of a function.
    • Find instantaneous rates of change (C-J.3)
  • B.2.4 solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations
  • B.2.5 solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results

12.C Geometry and Algebra of Vectors

• C.1 demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;
  • C.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)
    • Find the magnitude of a vector (12-U.1)
    • Find the direction angle of a vector (12-U.2)
  • C.1.2 represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways (e.g., 320º; N 40º W), and algebraically (e.g., using Cartesian coordinates; using polar coordinates), and recognize vectors with the same magnitude and direction but different positions as equal vectors
    • Find the component form of a vector (12-U.3)
    • Find the component form of a vector from its magnitude and direction angle (12-U.4)
  • C.1.3 determine, using trigonometric relationships [e.g., x = rcos Theta, y = rsin Theta, Theta = tan to the -1 power (y / x) or tan to the -1 power (y / x) + 180º, r = square root of (x² + y²)], the Cartesian representation of a vector in two-space given as a directed line segment, or the representation as a directed line segment of a vector in two-space given in Cartesian form [e.g., representing the vector (8, 6) as a directed line segment]
    • Find the magnitude of a vector (12-U.1)
    • Find the direction angle of a vector (12-U.2)
    • Find the component form of a vector from its magnitude and direction angle (12-U.4)
    • Find the magnitude and direction of a vector sum (12-U.9)
  • C.1.4 recognize that points and vectors in three-space can both be represented using Cartesian coordinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations
    • Find the magnitude of a three-dimensional vector (12-V.1)
    • Find the component form of a three-dimensional vector (12-V.2)
  • C.2 perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;
    • C.2.1 perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in two-space, and on vectors represented in Cartesian form in two-space and three-space
      • Add and subtract vectors (12-U.6)
      • Multiply a vector by a scalar (12-U.7)
      • Find the magnitude or direction of a vector scalar multiple (12-U.8)
      • Find the magnitude and direction of a vector sum (12-U.9)
      • Linear combinations of vectors (12-U.10)
      • Graph a resultant vector using the triangle method (12-U.11)
      • Graph a resultant vector using the parallelogram method (12-U.12)
      • Add and subtract three-dimensional vectors (12-V.4)
      • Scalar multiples of three-dimensional vectors (12-V.5)
      • Linear combinations of three-dimensional vectors (12-V.6)
    • C.2.2 determine, through investigation with and without technology, some properties (e.g., commutative, associative, and distributive properties) of the operations of addition, subtraction, and scalar multiplication of vectors
      • Add and subtract vectors (12-U.6)
      • Multiply a vector by a scalar (12-U.7)
    • C.2.3 solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications
    • C.2.4 perform the operation of dot product on two vectors represented as directed line segments (i.e., using vector a times vector b = (absolute value of vector a)(absolute value of vector b)(cos Theta)) and in Cartesian form (i.e., using vector a times vector b = (a base 1 of b base 1) + (a base 2 of b base 2) or (vector a times vector b) = (a base 1 of b base 1) + (a base 2 of b base 2) + (a base 3 of b base 3)) in two-space and three-space, and describe applications of the dot product (e.g., determining the angle between two vectors; determining the projection of one vector onto another)
    • C.2.5 determine, through investigation, properties of the dot product (e.g., investigate whether it is commutative, distributive, or associative; investigate the dot product of a vector with itself and the dot product of orthogonal vectors)
    • C.2.6 perform the operation of cross product on two vectors represented in Cartesian form in three-space [i.e., using vector a times vector b = ((a base 2 of b base 3) – (a base 3 of b base 2), (a base 3 of b base 1) – (a base 1 of b base 3), (a base 1 of b base 2) – (a base 2 of b base 1))], determine the magnitude of the cross product (i.e., using absolute value of (vector a times vector b) = (absolute value of vector a) (absolute value of vector b)(sin Theta)), and describe applications of the cross product (e.g., determining a vector orthogonal to two given vectors; determining the turning effect [or torque] when a force is applied to a wrench at different angles)
    • C.2.7 determine, through investigation, properties of the cross product (e.g., investigate whether it is commutative, distributive, or associative; investigate the cross product of collinear vectors)
    • C.2.8 solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determining work, torque, ground speed, velocity, force)
  • C.3 distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space;
    • C.3.1 recognize that the solution points (x, y) in two-space of a single linear equation in two variables form a line and that the solution points (x, y) in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel
      • Domain and range (12-A.1)
      • Find the slope of a linear function (12-A.3)
      • Solve a system of equations by graphing (12-J.1)
      • Solve a system of equations by graphing: word problems (12-J.2)
      • Classify a system of equations (12-J.3)
      • Solve a system of equations using substitution (12-J.4)
      • Solve a system of equations using substitution: word problems (12-J.5)
      • Solve a system of equations using elimination (12-J.6)
      • Solve a system of equations using elimination: word problems (12-J.7)
    • C.3.2 determine, through investigation with technology (i.e., 3-D graphing software) and without technology, that the solution points (x, y, z) in three-space of a single linear equation in three variables form a plane and that the solution points (x, y, z) in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel
      • Solve a system of equations in three variables using substitution (12-J.8)
      • Solve a system of equations in three variables using elimination (12-J.9)
      • Determine the number of solutions to a system of equations in three variables (12-J.10)
    • C.3.3 determine, through investigation using a variety of tools and strategies (e.g., modelling with cardboard sheets and drinking straws; sketching on isometric graph paper), different geometric configurations of combinations of up to three lines and/or planes in three-space (e.g., two skew lines, three parallel planes, two intersecting planes, an intersecting line and plane); organize the configurations based on whether they intersect and, if so, how they intersect (i.e., in a point, in a line, in a plane)
      • Determine the number of solutions to a system of equations in three variables (12-J.10)
    • C.4 represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.
      • C.4.1 recognize a scalar equation for a line in two-space to be an equation of the form Ax + By + C = 0, represent a line in two-space using a vector equation (i.e., vector r = (vector r base 0) + (t times vector m)) and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space
      • C.4.2 recognize that a line in three-space cannot be represented by a scalar equation, and represent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line)
      • C.4.3 recognize a normal to a plane geometrically (i.e., as a vector perpendicular to the plane) and algebraically [e.g., one normal to the plane 3x + 5y – 2z = 6 is (3, 5, –2)], and determine, through investigation, some geometric properties of the plane (e.g., the direction of any normal to a plane is constant; all scalar multiples of a normal to a plane are also normals to that plane; three non-collinear points determine a plane; the resultant, or sum, of any two vectors in a plane also lies in the plane)
      • C.4.4 recognize a scalar equation for a plane in three-space to be an equation of the form Ax + By + Cz + D = 0 whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes
      • C.4.5 determine, using properties of a plane, the scalar, vector, and parametric equations of a plane
      • C.4.6 determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms
      • C.4.7 solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equations) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically

Ontario Curriculum: Grade 12 Foundations for College Mathematics, College Preparation

12.A Mathematical Models

• A.1 evaluate powers with rational exponents, simplify algebraic expressions involving exponents, and solve problems involving exponential equations graphically and using common bases;
  • A.1.1 determine, through investigation (e.g., by expanding terms and patterning), the exponent laws for multiplying and dividing algebraic expressions involving exponents [e.g., (x³)(x²), x³ ÷ (x to the 5th power)] and the exponent law for simplifying algebraic expressions involving a power of a power [e.g. (x to the 6th power times y³)²]
    • Simplify expressions involving rational exponents (12-I.8)
  • A.1.2 simplify algebraic expressions containing integer exponents using the laws of exponents
  • A.1.3 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 greater than 0 and m and n are integers)
    • Evaluate rational exponents (12-I.4)
  • A.1.4 evaluate, with or without technology, numerical expressions involving rational exponents and rational bases [e.g., 2 to the -3 power, (-6)³, 4 to the ½ power, 1.01 to the 120 power]
    • Evaluate rational exponents (12-I.4)
    • Operations with rational exponents (12-I.5)
  • A.1.5 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
    • Solve exponential equations by factoring (12-G.4)
    • Solve exponential equations using common logarithms (12-G.5)
  • A.1.6 solve problems involving exponential equations arising from real-world applications by using a graph or table of values generated with technology from a given equation [e.g., h = 2((0.6) to the n power), where h represents the height of a bouncing ball and n represents the number of bounces]
    • Exponential growth and decay: word problems (12-G.11)
    • Compound interest: word problems (12-G.12)
    • Continuously compounded interest: word problems (12-G.13)
  • A.1.7 solve exponential equations in one variable by determining a common base (e.g., 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)
    • Solve exponential equations by factoring (12-G.4)
  • A.2 describe trends based on the interpretation of graphs, compare graphs using initial conditions and rates of change, and solve problems by modelling relationships graphically and algebraically;
    • A.2.1 interpret graphs to describe a relationship (e.g., distance travelled depends on driving time, pollution increases with traffic volume, maximum profit occurs at a certain sales volume), using language and units appropriate to the context
    • A.2.2 describe trends based on given graphs, and use the trends to make predictions or justify decisions (e.g., given a graph of the men’s 100-m world record versus the year, predict the world record in the year 2050 and state your assumptions; given a graph showing the rising trend in graduation rates among Aboriginal youth, make predictions about future rates)
    • A.2.3 recognize that graphs and tables of values communicate information about rate of change, and use a given graph or table of values for a relation to identify the units used to measure rate of change (e.g., for a distance–time graph, the units of rate of change are kilometres per hour; for a table showing earnings over time, the units of rate of change are dollars per hour)
    • A.2.4 identify when the rate of change is zero, constant, or changing, given a table of values or a graph of a relation, and compare two graphs by describing rate of change (e.g., compare distance–time graphs for a car that is moving at constant speed and a car that is accelerating)
    • A.2.5 compare, through investigation with technology, the graphs of pairs of relations (i.e., linear, quadratic, exponential) by describing the initial conditions and the behaviour of the rates of change (e.g., compare the graphs of amount versus time for equal initial deposits in simple interest and compound interest accounts)
    • A.2.6 recognize that a linear model corresponds to a constant increase or decrease over equal intervals and that an exponential model corresponds to a constant percentage increase or decrease over equal intervals, select a model (i.e., linear, quadratic, exponential) to represent the relationship between numerical data graphically and algebraically, using a variety of tools (e.g., graphing technology) and strategies (e.g., finite differences, regression), and solve related problems
      • Linear functions over unit intervals (12-A.6)
      • Identify linear and exponential functions (12-G.8)
      • Exponential functions over unit intervals (12-G.9)
      • Describe linear and exponential growth and decay (12-G.10)
    • A.3 make connections between formulas and linear, quadratic, and exponential relations, solve problems using formulas arising from real-world applications, and describe applications of mathematical modelling in various occupations.
      • A.3.1 solve equations of the form x to the n power = a using rational exponents (e.g., solve x³ = 7 by raising both sides to the exponent 1/3)
        • Solve a quadratic equation using square roots (12-C.5)
      • A.3.2 determine the value of a variable of degree no higher than three, using a formula drawn from an application, by first substituting known values and then solving for the variable, and by first isolating the variable and then substituting known values
      • A.3.3 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)
        • Compound interest: word problems (12-G.12)
        • Continuously compounded interest: word problems (12-G.13)
      • A.3.4 solve multi-step problems requiring formulas arising from real-world applications (e.g., determining the cost of two coats of paint for a large cylindrical tank)
      • A.3.5 gather, interpret, and describe information about applications of mathematical modelling in occupations, and about college programs that explore these applications

12.B Personal Finance

• B.1 demonstrate an understanding of annuities, including mortgages, and solve related problems using technology;
  • B.1.1 gather and interpret information about annuities, describe the key features of an annuity, and identify real-world applications (e.g., RRSP, mortgage, RRIF, RESP)
    • Compound interest: word problems (12-G.12)
  • B.1.2 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 an ordinary simple annuity (i.e., an annuity in which payments are made at the end of each period, and compounding and payment periods are the same) (e.g., long-term savings plans, loans)
    • Compound interest: word problems (12-G.12)
  • B.1.3 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
    • Compound interest: word problems (12-G.12)
  • B.1.4 demonstrate, through investigation using technology (e.g., a TVM Solver), the advantages of starting deposits earlier when investing in annuities used as long-term savings plans
    • Compound interest: word problems (12-G.12)
  • B.1.5 gather and interpret information about mortgages, describe features associated with mortgages (e.g., mortgages are annuities for which the present value is the amount borrowed to purchase a home; the interest on a mortgage is compounded semi-annually but often paid monthly), and compare different types of mortgages (e.g., open mortgage, closed mortgage, variable-rate mortgage)
  • B.1.6 read and interpret an amortization table for a mortgage
  • B.1.7 generate an amortization table for a mortgage, using a variety of tools and strategies (e.g., input data into an online mortgage calculator; determine the payments using the TVM Solver on a graphing calculator and generate the amortization table using a spreadsheet), calculate the total interest paid over the life of a mortgage, and compare the total interest with the original principal of the mortgage
  • B.1.8 determine, through investigation using technology (e.g., TVM Solver, online tools, financial software), the effects of varying payment periods, regular payments, and interest rates on the length of time needed to pay off a mortgage and on the total interest paid
• B.2 gather, interpret, and compare information about owning or renting accommodation, and solve problems involving the associated costs;
  • B.2.1 gather and interpret information about the procedures and costs involved in owning and in renting accommodation (e.g., apartment, condominium, townhouse, detached home) in the local community
  • B.2.2 compare renting accommodation with owning accommodation by describing the advantages and disadvantages of each
  • B.2.3 solve problems, using technology (e.g., calculator, spreadsheet), that involve the fixed costs (e.g., mortgage, insurance, property tax) and variable costs (e.g., maintenance, utilities) of owning or renting accommodation
• B.3 design, justify, and adjust budgets for individuals and families described in case studies, and describe applications of the mathematics of personal finance.
  • B.3.1 gather, interpret, and describe information about living costs, and estimate the living costs of different households (e.g., a family of four, including two young children; a single young person; a single parent with one child) in the local community
  • B.3.2 design and present a savings plan to facilitate the achievement of a long-term goal (e.g., attending college, purchasing a car, renting or purchasing a house)
  • B.3.3 design, explain, and justify a monthly budget suitable for an individual or family described in a given case study that provides the specifics of the situation (e.g., income; personal responsibilities; costs such as utilities, food, rent/mortgage, entertainment, transportation, charitable contributions; long-term savings goals), with technology (e.g., using spreadsheets, budgeting software, online tools) and without technology (e.g., using budget templates)
  • B.3.4 identify and describe the factors to be considered in determining the affordability of accommodation in the local community (e.g., income, long-term savings, number of dependants, non-discretionary expenses), and consider the affordability of accommodation under given circumstances
  • B.3.5 make adjustments to a budget to accommodate changes in circumstances (e.g., loss of hours at work, change of job, change in personal responsibilities, move to new accommodation, achievement of a long-term goal, major purchase), with technology (e.g., spreadsheet template, budgeting software)
  • B.3.6 gather, interpret, and describe information about applications of the mathematics of personal finance in occupations (e.g., selling real estate, bookkeeping, managing a restaurant, financial planning, mortgage brokering), and about college programs that explore these applications

12.C Geometry and Trigonometry

• C.1 solve problems involving measurement and geometry and arising from real-world applications;
  • C.1.1 perform required conversions between the imperial system and the metric system using a variety of tools (e.g., tables, calculators, online conversion tools), as necessary within applications
  • C.1.2 solve problems involving the areas of rectangles, triangles, and circles, and of related composite shapes, in situations arising from real-world applications
    • Area of a triangle: sine formula (12-N.16)
    • Area of a triangle: Heron’s formula (12-N.17)
  • C.1.3 solve problems involving the volumes and surface areas of rectangular prisms, triangular prisms, and cylinders, and of related composite figures, in situations arising from real-world applications
• C.2 explain the significance of optimal dimensions in real-world applications, and determine optimal dimensions of two-dimensional shapes and three-dimensional figures;
  • C.2.1 recognize, through investigation using a variety of tools (e.g., calculators; dynamic geometry software; manipulatives such as tiles, geoboards, toothpicks) and strategies (e.g., modelling; making a table of values; graphing), and explain the significance of optimal perimeter, area, surface area, and volume in various applications (e.g., the minimum amount of packaging material, the relationship between surface area and heat loss)
  • C.2.2 determine, through investigation using a variety of tools (e.g., calculators, dynamic geometry software, manipulatives) and strategies (e.g., modelling; making a table of values; graphing), the optimal dimensions of a two-dimensional shape in metric or imperial units for a given constraint (e.g., the dimensions that give the minimum perimeter for a given area)
  • C.2.3 determine, through investigation using a variety of tools and strategies (e.g., modelling with manipulatives; making a table of values; graphing), the optimal dimensions of a right rectangular prism, a right triangular prism, and a right cylinder in metric or imperial units for a given constraint (e.g., the dimensions that give the maximum volume for a given surface area)
• C.3 solve problems using primary trigonometric ratios of acute and obtuse angles, the sine law, and the cosine law, including problems arising from real-world applications, and describe applications of trigonometry in various occupations.
  • C.3.1 solve problems in two dimensions using metric or imperial measurements, including problems that arise from real-world applications (e.g., surveying, navigation, building construction), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios, and of acute triangles using the sine law and the cosine law
    • Trigonometric ratios: find a side length (12-N.10)
    • Trigonometric ratios: find an angle measure (12-N.11)
    • Solve a right triangle (12-N.12)
    • Law of Sines (12-N.13)
    • Law of Cosines (12-N.14)
    • Solve a triangle (12-N.15)
  • C.3.2 make connections between primary trigonometric ratios (i.e., sine, cosine, tangent) of obtuse angles and of acute angles, through investigation using a variety of tools and strategies (e.g., using dynamic geometry software to identify an obtuse angle with the same sine as a given acute angle; using a circular geoboard to compare congruent triangles; using a scientific calculator to compare trigonometric ratios for supplementary angles)
    • Find trigonometric ratios using the unit circle (12-N.6)
    • Find trigonometric ratios using reference angles (12-N.7)
    • Symmetry and periodicity of trigonometric functions (12-P.2)
  • C.3.3 determine the values of the sine, cosine, and tangent of obtuse angles
    • Coterminal and reference angles (12-N.4)
    • Find trigonometric ratios using the unit circle (12-N.6)
    • Find trigonometric ratios using reference angles (12-N.7)
  • C.3.4 solve problems involving oblique triangles, including those that arise from real-world applications, using the sine law (in non-ambiguous cases only) and the cosine law, and using metric or imperial units
    • Law of Sines (12-N.13)
    • Law of Cosines (12-N.14)
    • Area of a triangle: sine formula (12-N.16)
  • C.3.5 gather, interpret, and describe information about applications of trigonometry in occupations, and about college programs that explore these applications
    • Law of Sines (12-N.13)
    • Law of Cosines (12-N.14)
    • Solve a triangle (12-N.15)

12.D Data Management

• D.1 collect, analyse, and summarize two-variable data using a variety of tools and strategies, and interpret and draw conclusions from the data;
  • D.1.1 distinguish situations requiring one-variable and two-variable data analysis, describe the associated numerical summaries (e.g., tally charts, summary tables) and graphical summaries (e.g., bar graphs, scatter plots), and recognize questions that each type of analysis addresses (e.g., What is the frequency of a particular trait in a population? What is the mathematical relationship between two variables?)
  • D.1.2 describe characteristics of an effective survey (e.g., by giving consideration to ethics, privacy, the need for honest responses, and possible sources of bias, including cultural bias), and design questionnaires (e.g., for determining if there is a relationship between age and hours per week of Internet use, between marks and hours of study, or between income and years of education) or experiments (e.g., growth of plants under different conditions) for gathering two-variable data
    • Identify biased samples (12-AA.1)
  • D.1.3 collect two-variable data from primary sources, through experimentation involving observation or measurement, or from secondary sources (e.g., Internet databases, newspapers, magazines), and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software)
  • D.1.4 create a graphical summary of two-variable data using a scatter plot (e.g., by identifying and justifying the dependent and independent variables; by drawing the line of best fit, when appropriate), with and without technology
    • Find the equation of a regression line (12-AA.8)
  • D.1.5 determine an algebraic summary of the relationship between two variables that appear to be linearly related (i.e., the equation of the line of best fit of the scatter plot), using a variety of tools (e.g., graphing calculators, graphing software) and strategies (e.g., using systematic trials to determine the slope and y-intercept of the line of best fit; using the regression capabilities of a graphing calculator), and solve related problems (e.g., use the equation of the line of best fit to interpolate or extrapolate from the given data set)
    • Find the equation of a regression line (12-AA.8)
    • Interpret regression lines (12-AA.9)
    • Analyze a regression line of a data set (12-AA.10)
    • Analyze a regression line using statistics of a data set (12-AA.11)
  • D.1.6 describe possible interpretations of the line of best fit of a scatter plot (e.g., the variables are linearly related) and reasons for misinterpretations (e.g., using too small a sample; failing to consider the effect of outliers; interpolating from a weak correlation; extrapolating nonlinearly related data)
    • Interpret regression lines (12-AA.9)
    • Analyze a regression line of a data set (12-AA.10)
    • Analyze a regression line using statistics of a data set (12-AA.11)
  • D.1.7 determine whether a linear model (i.e., a line of best fit) is appropriate given a set of two-variable data, by assessing the correlation between the two variables (i.e., by describing the type of correlation as positive, negative, or none; by describing the strength as strong or weak; by examining the context to determine whether a linear relationship is reasonable)
    • Match correlation coefficients to scatter plots (12-AA.6)
  • D.1.8 make conclusions from the analysis of two-variable data (e.g., by using a correlation to suggest a possible cause-and-effect relationship), and judge the reasonableness of the conclusions (e.g., by assessing the strength of the correlation; by considering if there are enough data)
• D.2 demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations.
  • D.2.1 recognize and interpret common statistical terms (e.g., percentile, quartile) and expressions (e.g., accurate 19 times out of 20) used in the media (e.g., television, Internet, radio, newspapers)
    • Introduction to probability (12-Y.1)
  • D.2.2 describe examples of indices used by the media (e.g., consumer price index, S&P/TSX composite index, new housing price index) and solve problems by interpreting and using indices (e.g., by using the consumer price index to calculate the annual inflation rate)
  • D.2.3 interpret statistics presented in the media (e.g., the UN’s finding that 2% of the world’s population has more than half the world’s wealth, whereas half the world’s population has only 1% of the world’s wealth), and explain how the media, the advertising industry, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view (e.g., by making a general statement based on a weak correlation or an assumed cause-and- effect relationship; by starting the vertical scale on a graph at a value other than zero; by making statements using general population statistics without reference to data specific to minority groups)
  • D.2.4 assess the validity of conclusions presented in the media by examining sources of data, including Internet sources (i.e., to determine whether they are authoritative, reliable, unbiased, and current), methods of data collection, and possible sources of bias (e.g., sampling bias, non-response bias, a bias in a survey question), and by questioning the analysis of the data (e.g., whether there is any indication of the sample size in the analysis) and conclusions drawn from the data (e.g., whether any assumptions are made about cause and effect)
    • Identify biased samples (12-AA.1)
  • D.2.5 gather, interpret, and describe information about applications of data management in occupations, and about college programs that explore these applications