Gizmos for Ontario - Mathematics: 12th Grade Advanced Functions (Ontario Curriculum adopted 2010)

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

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

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)

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)

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

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

A.3: : solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

A.3.4: : solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications

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

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)

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

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

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

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.2.4: : 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

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

B.2.6: : represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians

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.3.2: : 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

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

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)

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)

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)

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)

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]

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

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)

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]

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)

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

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)

D: : Characteristics of Functions

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

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.2.8: : 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)

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)

D.3.2: : solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques

Correlation last revised: 9/16/2020

Ontario - Mathematics: 12th Grade Advanced Functions

Ontario Curriculum | Adopted: 2010

A: : Exponential and Logarithmic Functions

B: : Trigonometric Functions

C: : Polynomial and Rational Functions

D: : Characteristics of Functions