OntarioGrade 12 Math Courses - Ontario Curriculum
Explore Grade 12 Advanced Functions (MHF4U) designed for university preparation. This comprehensive courses covers logarithmic functions, trigonometry, and polynomial functions to prepare students for post-secondary mathematics.
Ontario Grade 12 Math Curriculum - Advanced FunctionsHelp
OE_ID | Expectations | StudyPug Topic |
|---|---|---|
ON.OE.12AF.A1.1 | 12AF.A1.1: 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 of exponentiation, and evaluate simple logarithmic expressions |
ON.OE.12AF.A1.2 | 12AF.A1.2: Determine, with technology, the approximate logarithm of a number to any base, including base 10 |
ON.OE.12AF.A1.3 | 12AF.A1.3: Make connections between related logarithmic and exponential equations and solve simple exponential equations by rewriting them in logarithmic form |
ON.OE.12AF.A1.4 | 12AF.A1.4: Make connections between the laws of exponents and the laws of logarithms, verify the laws of logarithms, and use the laws of logarithms to simplify and evaluate numerical expressions |
ON.OE.12AF.A2.1 | 12AF.A2.1: Determine, through investigation with technology, key features of the graphs of logarithmic functions |
ON.OE.12AF.A2.2 | 12AF.A2.2: 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 |
ON.OE.12AF.A2.3 | 12AF.A2.3: Determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = log (x ? d) + c and the roles of the parameters a and k in functions of the form y = alog (kx), and describe these roles in terms of transformations on the graph of f(x) = log x |
ON.OE.12AF.A2.4 | 12AF.A2.4: Pose problems based on real-world applications of exponential and logarithmic functions and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation |
ON.OE.12AF.A3.1 | 12AF.A3.1: Recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types |
ON.OE.12AF.A3.3 | 12AF.A3.3: Solve simple logarithmic equations in one variable algebraically |
ON.OE.12AF.A3.4 | 12AF.A3.4: Solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications |
ON.OE.12AF.B1.1 | 12AF.B1.1: Recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle, and develop and apply the relationship between radian and degree measure |
ON.OE.12AF.B1.2 | 12AF.B1.2: Represent radian measure in terms of ? and as a rational number |
ON.OE.12AF.B1.3 | 12AF.B1.3: Determine, with technology, the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure |
ON.OE.12AF.B1.4 | 12AF.B1.4: Determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0, ?/6, ?/4, ?/3, ?/2, and their multiples less than or equal to 2? |
ON.OE.12AF.B2.1 | 12AF.B2.1: 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 in terms of radians |
ON.OE.12AF.B2.2 | 12AF.B2.2: 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 |
ON.OE.12AF.B2.3 | 12AF.B2.3: Graph, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions for angle measures expressed in radians, determine and describe key properties of the reciprocal functions, and recognize notations used to represent the reciprocal functions |
ON.OE.12AF.B2.4 | 12AF.B2.4: 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 |
ON.OE.12AF.B2.6 | 12AF.B2.6: Represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians |
ON.OE.12AF.B2.7 | 12AF.B2.7: Pose problems based on applications involving a trigonometric function with domain expressed in radians, 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 |
ON.OE.12AF.B3.1 | 12AF.B3.1: Recognize equivalent trigonometric expressions and verify equivalence using graphing technology |
ON.OE.12AF.B3.2 | 12AF.B3.2: Explore the algebraic development of the compound angle formulas, and use the formulas to determine exact values of trigonometric ratios |
ON.OE.12AF.B3.3 | 12AF.B3.3: Recognize that trigonometric identities are equations that are true for every value in the domain, prove trigonometric identities through the application of reasoning skills, using a variety of relationships, and verify identities using technology |
ON.OE.12AF.B3.4 | 12AF.B3.4: Solve linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from 0 to 2?, and solve related problems |
ON.OE.12AF.C1.1 | 12AF.C1.1: Recognize a polynomial expression and the equation of a polynomial function, give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions |
ON.OE.12AF.C1.2 | 12AF.C1.2: Compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial functions |
ON.OE.12AF.C1.3 | 12AF.C1.3: Describe key features of the graphs of polynomial functions |
ON.OE.12AF.C1.4 | 12AF.C1.4: Distinguish polynomial functions from sinusoidal and exponential functions, and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions |
ON.OE.12AF.C1.5 | 12AF.C1.5: Make connections, through investigation using graphing technology, between a polynomial function given in factored form and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features |
ON.OE.12AF.C1.6 | 12AF.C1.6: 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) = x and f(x) = x? |
ON.OE.12AF.C1.7 | 12AF.C1.7: Determine an equation of a polynomial function that satisfies a given set of conditions |
ON.OE.12AF.C1.8 | 12AF.C1.8: 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 |
ON.OE.12AF.C1.9 | 12AF.C1.9: Determine, through investigation, and compare the properties of even and odd polynomial functions |
ON.OE.12AF.C2.1 | 12AF.C2.1: Determine, through investigation with and without technology, key features 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 |
ON.OE.12AF.C2.2 | 12AF.C2.2: Determine, through investigation with and without technology, key features of the graphs of rational functions that have linear expressions in the numerator and denominator, and make connections between the algebraic and graphical representations of these rational functions |
ON.OE.12AF.C2.3 | 12AF.C2.3: Sketch the graph of a simple rational function using its key features, given the algebraic representation of the function |
ON.OE.12AF.C3.1 | 12AF.C3.1: Make connections, through investigation using technology, 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 |
ON.OE.12AF.C3.3 | 12AF.C3.3: Determine, through investigation using technology, 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 |
ON.OE.12AF.C3.4 | 12AF.C3.4: Solve polynomial equations in one variable, of degree no higher than four, by selecting and applying strategies, and verify solutions using technology |
ON.OE.12AF.C3.5 | 12AF.C3.5: Determine, through investigation using technology, 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 |
ON.OE.12AF.C3.6 | 12AF.C3.6: Solve simple rational equations in one variable algebraically, and verify solutions using technology |
ON.OE.12AF.C3.7 | 12AF.C3.7: Solve problems involving applications of polynomial and simple rational functions and equations |
ON.OE.12AF.C4.1 | 12AF.C4.1: 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 |
ON.OE.12AF.C4.2 | 12AF.C4.2: Determine solutions to polynomial inequalities in one variable 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 |
ON.OE.12AF.C4.3 | 12AF.C4.3: Solve linear inequalities and factorable polynomial inequalities in one variable in a variety of ways, and represent the solutions on a number line or algebraically |
ON.OE.12AF.D1.1 | 12AF.D1.1: Gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change |
ON.OE.12AF.D1.2 | 12AF.D1.2: 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 |
ON.OE.12AF.D1.3 | 12AF.D1.3: Sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology when possible |
ON.OE.12AF.D1.4 | 12AF.D1.4: Calculate and interpret average rates of change of functions arising from real-world applications, given various representations of the functions |
ON.OE.12AF.D1.5 | 12AF.D1.5: 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 |
ON.OE.12AF.D1.6 | 12AF.D1.6: Determine, through investigation using various representations of relationships, approximate instantaneous rates of change arising from real-world applications by using average rates of change and reducing the interval over which the average rate of change is determined |
ON.OE.12AF.D1.7 | 12AF.D1.7: Make connections, through investigation, between the slope of a secant on the graph of a function 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 |
ON.OE.12AF.D1.8 | 12AF.D1.8: Determine, through investigation using a variety of tools and strategies, the approximate slope of the tangent to a given point on the graph of a function by using the slopes of secants through the given point |
ON.OE.12AF.D1.9 | 12AF.D1.9: Solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods |
ON.OE.12AF.D2.1 | 12AF.D2.1: Determine, through investigation using graphing technology, key features of the graphs of functions created by adding, subtracting, multiplying, or dividing functions, and describe factors that affect these properties |
ON.OE.12AF.D2.2 | 12AF.D2.2: Recognize real-world applications of combinations of functions, and solve related problems graphically |
ON.OE.12AF.D2.3 | 12AF.D2.3: Determine, through investigation, and explain some properties of functions formed by adding, subtracting, multiplying, and dividing general functions |
ON.OE.12AF.D2.8 | 12AF.D2.8: Make connections, through investigation using technology, between transformations of simple functions and the composition of these functions with a linear function |
ON.OE.12AF.D3.1 | 12AF.D3.1: Compare, through investigation using a variety of tools and strategies, the characteristics of various functions |
ON.OE.12AF.D3.2 | 12AF.D3.2: Solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques |
ON.OE.12AF.D3.3 | 12AF.D3.3: 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 |