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Grade 12 Math Courses - Ontario Curriculum

Discover Ontario's Grade 12 math options, from Advanced Functions to Data Management. Prepare for university-level mathematics and explore diverse career pathways in STEM fields.

Ontario Grade 12 Math Curriculum - Advanced Functions & More Topics

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OE_ID
Expectations
StudyPug Topic
A. Exponential and Logarithmic Functions : 1. Evaluating Logarithmic Expressions
OE.12AF.A1.1
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 of exponentiation, and evaluate simple logarithmic expressions

What is a logarithm?

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Converting from logarithmic form to exponential form

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OE.12AF.A1.2
1.2: Determine, with technology, the approximate logarithm of a number to any base, including base 10

Common logarithms

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Natural log: ln

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OE.12AF.A1.3
1.3: Make connections between related logarithmic and exponential equations and solve simple exponential equations by rewriting them in logarithmic form
OE.12AF.A1.4
1.4: 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
A. Exponential and Logarithmic Functions : 2. Connecting Graphs and Equations of Logarithmic Functions
OE.12AF.A2.1
2.1: Determine, through investigation with technology, key features of the graphs of logarithmic functions
OE.12AF.A2.2
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
OE.12AF.A2.3
2.3: 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
OE.12AF.A2.4
2.4: 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
A. Exponential and Logarithmic Functions : 3. Solving Exponential and Logarithmic Equations
OE.12AF.A3.1
3.1: Recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
OE.12AF.A3.3
3.3: Solve simple logarithmic equations in one variable algebraically
OE.12AF.A3.4
3.4: Solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications
B. Trigonometric Functions : 1. Understanding and Applying Radian Measure
OE.12AF.B1.1
1.1: 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
OE.12AF.B1.2
1.2: Represent radian measure in terms of ? and as a rational number
OE.12AF.B1.3
1.3: Determine, with technology, the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure
OE.12AF.B1.4
1.4: 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?
B. Trigonometric Functions : 2. Connecting Graphs and Equations of Trigonometric Functions
OE.12AF.B2.1
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 in terms of radians

Sine graph: y = sin x

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Cosine graph: y = cos x

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OE.12AF.B2.2
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

Tangent graph: y = tan x

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OE.12AF.B2.3
2.3: 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
OE.12AF.B2.4
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
OE.12AF.B2.6
2.6: Represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
OE.12AF.B2.7
2.7: 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
B. Trigonometric Functions : 3. Solving Trigonometric Equations
OE.12AF.B3.1
3.1: Recognize equivalent trigonometric expressions and verify equivalence using graphing technology
OE.12AF.B3.2
3.2: Explore the algebraic development of the compound angle formulas, and use the formulas to determine exact values of trigonometric ratios

Double-angle identities

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OE.12AF.B3.3
3.3: 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
OE.12AF.B3.4
3.4: 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
C. Polynomial and Rational Functions : 1. Connecting Graphs and Equations of Polynomial Functions
OE.12AF.C1.1
1.1: 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

What is a polynomial?

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Polynomial components

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Polynomial functions

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OE.12AF.C1.2
1.2: Compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial functions
OE.12AF.C1.3
1.3: Describe key features of the graphs of polynomial functions
OE.12AF.C1.4
1.4: 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
OE.12AF.C1.5
1.5: 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
OE.12AF.C1.6
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) = x and f(x) = x?
OE.12AF.C1.7
1.7: Determine an equation of a polynomial function that satisfies a given set of conditions
OE.12AF.C1.8
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
OE.12AF.C1.9
1.9: Determine, through investigation, and compare the properties of even and odd polynomial functions

Even and odd functions

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C. Polynomial and Rational Functions : 2. Connecting Graphs and Equations of Rational Functions
OE.12AF.C2.1
2.1: 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
OE.12AF.C2.2
2.2: 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
OE.12AF.C2.3
2.3: Sketch the graph of a simple rational function using its key features, given the algebraic representation of the function
C. Polynomial and Rational Functions : 3. Solving Polynomial and Rational Equations
OE.12AF.C3.1
3.1: 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
OE.12AF.C3.3
3.3: 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
OE.12AF.C3.4
3.4: Solve polynomial equations in one variable, of degree no higher than four, by selecting and applying strategies, and verify solutions using technology
OE.12AF.C3.5
3.5: 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
OE.12AF.C3.6
3.6: Solve simple rational equations in one variable algebraically, and verify solutions using technology
OE.12AF.C3.7
3.7: Solve problems involving applications of polynomial and simple rational functions and equations
C. Polynomial and Rational Functions : 4. Solving Inequalities
OE.12AF.C4.1
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
OE.12AF.C4.2
4.2: 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
OE.12AF.C4.3
4.3: 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
D. Characteristics of Functions : 1. Understanding Rates of Change
OE.12AF.D1.1
1.1: Gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change

Rate of change

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OE.12AF.D1.2
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
OE.12AF.D1.3
1.3: Sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology when possible
OE.12AF.D1.4
1.4: Calculate and interpret average rates of change of functions arising from real-world applications, given various representations of the functions

Function notation

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OE.12AF.D1.5
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
OE.12AF.D1.6
1.6: 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
OE.12AF.D1.7
1.7: 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
OE.12AF.D1.8
1.8: 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
OE.12AF.D1.9
1.9: Solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods
D. Characteristics of Functions : 2. Combining Functions
OE.12AF.D2.1
2.1: 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

Adding functions

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Subtracting functions

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Multiplying functions

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Dividing functions

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OE.12AF.D2.2
2.2: Recognize real-world applications of combinations of functions, and solve related problems graphically

Composite functions

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OE.12AF.D2.3
2.3: Determine, through investigation, and explain some properties of functions formed by adding, subtracting, multiplying, and dividing general functions
OE.12AF.D2.8
2.8: Make connections, through investigation using technology, between transformations of simple functions and the composition of these functions with a linear function
D. Characteristics of Functions : 3. Using Function Models to Solve Problems
OE.12AF.D3.1
3.1: Compare, through investigation using a variety of tools and strategies, the characteristics of various functions

One to one functions

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OE.12AF.D3.2
3.2: Solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques
OE.12AF.D3.3
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

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