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Skills available for Ontario grade 12 math curriculum

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12.A Exponential and Logarithmic Functions

12.B Trigonometric Functions

12.C Polynomial and Rational Functions

  • 12.C.1 identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;

  • 12.C.2 identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;

    • 12.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]

    • 12.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

    • 12.C.2.3 sketch the graph of a simple rational function using its key features, given the algebraic representation of the function

  • 12.C.3 solve problems involving polynomial and simple rational equations graphically and algebraically;

    • 12.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

    • 12.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)

    • 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]

    • 12.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)

    • 12.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]

    • 12.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)

    • 12.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]

  • 12.C.4 demonstrate an understanding of solving polynomial and simple rational inequalities.

    • 12.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);

    • 12.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

    • 12.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)

12.D Characteristics of Functions

  • 12.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;

    • 12.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)

    • 12.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)

    • 12.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

    • 12.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)

    • 12.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)

    • 12.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

    • 12.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

    • 12.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

    • 12.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)

  • 12.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;

    • 12.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

    • 12.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

    • 12.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)]

    • 12.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

    • 12.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

    • 12.D.2.6 solve problems involving the composition of two functions, including problems arising from real-world applications

    • 12.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]

    • 12.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)

  • 12.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.