Ontario · Grade 12
Advanced Functions, Grade 12 (University)
- MHF4U
Ontario’s Grade 12 math course, from Exponential and Logarithmic Functions to Characteristics of Functions.
- Units
- 4
- Lessons
- 14
- Skills
- 72
- Provincial exam
- None
- Official curriculum
- Ontario Ministry of Education
Unit by unit
Every skill in the course, numbered by unit.
Unit 1 · 14 skills
Exponential and Logarithmic Functions
Evaluating Logarithmic Expressions
The logarithm as the inverse of exponentiation
I can define the logarithm as the exponent needed to raise a base to get a number, and evaluate simple logarithms.
Skill 1.1
Approximating logarithms with technology
I can use technology to approximate the logarithm of a number to any base.
Skill 1.2
Connecting logarithmic and exponential equations
I can rewrite a logarithmic equation as its equivalent exponential equation, and back again.
Skill 1.3
The Laws of Logarithms
Deriving and verifying the laws of logarithms
I can connect the laws of exponents to the laws of logarithms and verify the laws.
Skill 1.4
Applying the laws of logarithms to simplify and evaluate
I can use the laws of logarithms to simplify and evaluate numerical expressions.
Skill 1.5
Connecting Graphs and Equations of Logarithmic Functions
Key features of the graph of f(x) = log x
I can determine and describe the key features of the graph of f(x) = log x.
Skill 1.6
Logarithmic and exponential functions as inverses
I can show that exponential and logarithmic functions are inverses whose graphs reflect in y = x.
Skill 1.7
The roles of parameters in transformed logarithmic functions
I can describe how the parameters d, c, a, and k transform the graph of a logarithmic function.
Skill 1.8
Solving real-world exponential and logarithmic problems from a graph
I can solve real-world exponential and logarithmic problems using a graph.
Skill 1.9
Simplifying equivalent logarithmic and exponential expressions
I can recognize and simplify equivalent expressions involving logarithms and exponents.
Skill 1.10
Solving Exponential and Logarithmic Equations
Solving exponential equations by finding a common base
I can solve exponential equations by expressing both sides with a common base.
Skill 1.11
Solving exponential equations using logarithms
I can solve exponential equations by taking the logarithm of both sides.
Skill 1.12
Solving simple logarithmic equations algebraically
I can solve simple logarithmic equations algebraically.
Skill 1.13
Solving real-world problems with exponential and logarithmic equations
I can solve real-world problems using exponential and logarithmic equations.
Skill 1.14
The official wording · 14 outcomes
- 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
- 1.2
- determine, with technology, the approximate logarithm of a number to any base, including base 10 (e.g., by reasoning that log 29 is between 3 and 4 and using systematic trial to determine that log 29 is approximately 3.07)
- 1.3
- make connections between related logarithmic and exponential equations (e.g., log 125 = 3
- 1.4
- make connections between the laws of expo- nents and the laws of logarithms [e.g., use the statement 10 = 10 10 to deduce that log x + log y = log (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 1000 – log 100 and then rewriting the answer as a logarithmic term to the same base)
- 1.5
- use the laws of logarithms to simplify and evaluate numerical expressions
- 1.6
- determine, through investigation with tech- nology (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 func- tions of the form f(x) = log x, and make con- nections between the algebraic and graphical representations of these logarithmic functions
- 1.7
- recognize the relationship between an expo- nential function and the corresponding loga- rithmic function to be that of a function and its inverse, deduce that the graph of a loga- rithmic function is the reflection of the graph of the corresponding exponential function in the line y = x, and verify the deduction using technology
- 1.8
- determine, through investigation using technol- ogy, 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 func- tions of the form y = alog (kx), and describe these roles in terms of transformations on the horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
- 1.9
- pose problems based on real-world applica- tions 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
- 1.10
- recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
- 1.11
- solve exponential equations in one variable by determining a common base (e.g., solve 4 = 8 by expressing each side as a power of 2)
- 1.12
- 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 = 7 by taking the logarithm base 10 of both sides)
- 1.13
- solve simple logarithmic equations in one variable algebraically [e.g., log (5x + 6) = 2, log (x + 1) = 1]
- 1.14
- solve problems involving exponential and logarithmic equations algebraically, includ- ing problems arising from real-world applications
Key ideas
log_b(x) = y means b^y = x — the logarithm IS the exponent.
Because 10^a · 10^b = 10^(a+b), the matching log law is log x + log y = log(xy).
f(x) = log x has a vertical asymptote at x = 0, domain x > 0, and range all real numbers.
If two powers with the same base are equal, their exponents must be equal.
Taking a logarithm undoes exponentiation, the same way subtraction undoes addition.
Watch out for
- Thinking log_b(x) means 'multiply b by x' instead of 'the exponent on b that gives x.'.
- Assuming log(x + y) = log x + log y (confusing the product law with addition inside the log).
- Thinking a logarithmic graph has a horizontal asymptote instead of a vertical one.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 16 skills
Trigonometric Functions
Understanding and Applying Radian Measure
The radian as an alternative angle-measure unit
I can define the radian and relate radian measure to degree measure.
Skill 2.1
Representing radian measure in terms of π and as a rational number
I can represent radian measure both in terms of π and as a rational number.
Skill 2.2
Trigonometric ratios of angles in radians, with technology
I can use technology to determine trigonometric ratios for angles given in radians.
Skill 2.3
Exact trigonometric values for special angles in radians
I can determine exact trigonometric values for special angles in radians, without technology.
Skill 2.4
Connecting Graphs and Equations of Trigonometric Functions
Sketching sine and cosine graphs in radians
I can sketch f(x) = sin x and f(x) = cos x with angles in radians and describe key properties.
Skill 2.5
Connecting the tangent ratio to the tangent function
I can connect the tangent ratio to the tangent function and describe its key properties.
Skill 2.6
Graphing and describing the reciprocal trigonometric functions
I can graph the reciprocal trigonometric functions and describe their key properties.
Skill 2.7
Amplitude, period, and phase shift of sinusoidal functions
I can determine the amplitude, period, and phase shift of a sinusoidal function from its equation.
Skill 2.8
Sketching transformed sine and cosine graphs
I can sketch transformed sine and cosine graphs and state their key properties.
Skill 2.9
Writing the equation of a sinusoidal function from its graph or properties
I can write the equation of a sinusoidal function given its graph or its properties.
Skill 2.10
Solving real-world problems with trigonometric functions in radians
I can solve real-world problems involving a trigonometric function with a radian domain.
Skill 2.11
Solving Trigonometric Equations
Recognizing equivalent trigonometric expressions
I can recognize equivalent trigonometric expressions and verify equivalence using technology.
Skill 2.12
Compound angle formulas and exact trig values
I can use the compound angle formulas to determine exact trigonometric values.
Skill 2.13
Understanding trigonometric identities and disproving non-identities
I can recognize what makes an equation a trigonometric identity and use a counter-example to disprove a non-identity.
Skill 2.14
Proving trigonometric identities
I can prove trigonometric identities using reasoning and a variety of known relationships.
Skill 2.15
Solving linear and quadratic trigonometric equations
I can solve linear and quadratic trigonometric equations for 0 to 2π and verify with technology.
Skill 2.16
The official wording · 16 outcomes
- 2.1
- 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
- 2.2
- represent radian measure in terms of π (e.g., radians, 2π radians) and as a rational number (e.g., 1.05 radians, 6.28 radians)
- 2.3
- 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
- 2.4
- determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0, , , , , and their multiples less than or equal to 2π
- 2.5
- 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 2π, amplitude of 1) in terms of radians
- 2.6
- 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
- 2.7
- 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 proper- ties 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
- 2.8
- 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) = acos(k(x – d)) + c, with angles expressed in radians
- 2.9
- sketch graphs of y = a sin (k(x – d)) + c and y = acos(k(x – d)) + c by applying trans- formations 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
- 2.10
- represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
- 2.11
- pose problems based on applications involv- ing a trigonometric function with domain expressed in radians (e.g., seasonal changes in temperature, heights of tides, hours of day- light, 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
- 2.12
- recognize equivalent trigonometric expressions [e.g., by using the angles in a right triangle to recognize that sin x and cos ( – x)are equivalent; by using transformations to recognize that cos (x + )and –sin x are equivalent], and verify equivalence using graphing technology
- 2.13
- explore the algebraic development of the compound angle formulas (e.g., verify the formulas in numerical examples, using tech- nology; follow a demonstration of the alge- braic development [student reproduction of the development of the general case is not required]), and use the formulas to determine exact values of trigonometric ratios
- 2.14
- 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)
- 2.15
- prove trigonometric identities through the application of reasoning skills, using a variety of relationships (e.g., tan x = ; sin x + cos x = 1; the reciprocal identities; the compound angle formulas), and verify identities using technology
- 2.16
- solve linear and quadratic trigonometric equa- tions, with and without graphing technology, for the domain of real values from 0 to 2π, and solve related problems
Key ideas
One radian is the angle that subtends an arc equal in length to the radius, on a unit circle.
Both sine and cosine have a period of 2π and an amplitude of 1.
Two trig expressions that always produce the same value for every angle are equivalent — this can come from right-triangle relationships or from transformations.
π radians = 180°, giving the conversion factor between the two systems.
The graphs of sine and cosine are horizontal shifts of each other by π/2.
Watch out for
- Thinking radians and degrees measure different physical quantities instead of the same angle in different units.
- Labelling the x-axis in degrees out of habit instead of radians.
- Assuming two expressions are equivalent just because they look similar, without checking with several angle values or graphing.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 22 skills
Polynomial and Rational Functions
Connecting Graphs and Equations of Polynomial Functions
Recognizing polynomial expressions and functions
I can recognize a polynomial expression and explain why its equation defines a function.
Skill 3.1
Comparing numeric, graphical, and algebraic polynomial representations
I can compare the numeric, graphical, and algebraic representations of polynomial functions.
Skill 3.2
Key features of polynomial function graphs
I can describe the key features of polynomial function graphs.
Skill 3.3
Distinguishing polynomial functions from sinusoidal and exponential functions
I can distinguish polynomial functions from sinusoidal and exponential functions.
Skill 3.4
Connecting factored form to x-intercepts and sketching the graph
I can connect a polynomial function's factored form to its x-intercepts and sketch its graph.
Skill 3.5
Roles of a, k, d, and c as transformations of general functions
I can describe the roles of a, k, d, and c as transformations of a general function.
Skill 3.6
Determining a polynomial equation from given conditions
I can determine an equation of a polynomial function that satisfies given conditions.
Skill 3.7
The family of polynomial functions with given zeros
I can determine the equation of a family of polynomial functions with given zeros, and the member through a given point.
Skill 3.8
Properties of even and odd polynomial functions
I can determine and compare the properties of even and odd polynomial functions.
Skill 3.9
Connecting Graphs and Equations of Rational Functions
Key features of rational functions that are reciprocals of linear/quadratic functions
I can determine the key features of rational functions that are reciprocals of linear or quadratic functions.
Skill 3.10
Key features of rational functions with linear numerator and denominator
I can determine the key features of rational functions with linear numerator and denominator.
Skill 3.11
Sketching a simple rational function from its key features
I can sketch the graph of a simple rational function using its key features.
Skill 3.12
Solving Polynomial and Rational Equations
Verifying the remainder theorem and the factor theorem
I can verify the remainder theorem and the factor theorem.
Skill 3.13
Factoring polynomial expressions of degree up to four
I can factor polynomial expressions of degree up to four, selecting an appropriate strategy.
Skill 3.14
Connecting real roots of a polynomial equation to x-intercepts
I can connect the real roots of a polynomial equation to the x-intercepts of its graph.
Skill 3.15
Solving polynomial equations of degree up to four
I can solve polynomial equations of degree up to four and verify solutions with technology.
Skill 3.16
Connecting real roots of a rational equation to x-intercepts
I can connect the real roots of a rational equation to the x-intercepts of its graph.
Skill 3.17
Solving simple rational equations algebraically
I can solve simple rational equations algebraically and verify with technology.
Skill 3.18
Solving real-world problems with polynomial and rational functions
I can solve real-world problems involving polynomial and simple rational functions and equations.
Skill 3.19
Solving Inequalities
The difference between solving an equation and an inequality
I can explain the difference between solving an equation and solving an inequality.
Skill 3.20
Solving polynomial and rational inequalities by graphing
I can solve polynomial and rational inequalities by graphing.
Skill 3.21
Solving linear and factorable polynomial inequalities algebraically
I can solve linear and factorable polynomial inequalities algebraically and represent the solution.
Skill 3.22
The official wording · 22 outcomes
- 3.1
- 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 non- negative integral exponent, such as x – 5x + 2x – 1); recognize the equation of a polynomial function, give reasons why it is a function, and identify linear and quad- ratic functions as examples of polynomial functions
- 3.2
- compare, through investigation using graph- ing 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 nega- tive x-values)
- 3.3
- describe key features of the graphs of poly- nomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or nega- tive x-values)
- 3.4
- distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, g(x) = 2 ], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions
- 3.5
- 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 beha- viour; by locating positive and negative regions using test values between and on either side of the x-intercepts)
- 3.6
- determine, through investigation using tech- nology, 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 transforma- tions on the graphs of f(x) = x and f(x) = 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)
- 3.7
- determine an equation of a polynomial func- tion 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 soft- ware; using finite differences), and recognize that there may be more than one polynomial function that can satisfy a given set of condi- tions (e.g., an infinite number of polynomial functions satisfy the condition that they have three given x-intercepts)
- 3.8
- determine the equation of the family of poly- nomial 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 ≠ 0; the member of the family that passes through (–1, 24) is f(x) = –2(x – 5)(x + 3)(x + 2)]
- 3.9
- 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
- 3.10
- 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 con- nections between the algebraic and graphical representations of these rational functions
- 3.11
- 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 expres- sions in the numerator and denominator
- 3.12
- sketch the graph of a simple rational function using its key features, given the algebraic rep- resentation of the function
- 3.13
- 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 , and f(a) to verify the remainder theorem and the factor theorem
- 3.14
- factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factor- ing, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem)
- 3.15
- determine, through investigation using tech- nology (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 con- nection [e.g., the real roots of the equation x – 13x + 36 = 0 are the x-intercepts of the graph of f(x) = x – 13x + 36]
- 3.16
- solve polynomial equations in one variable, of degree no higher than four (e.g., 2x – 3x + 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
- 3.17
- determine, through investigation using tech- nology (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
- 3.18
- solve simple rational equations in one variable algebraically, and verify solutions using tech- nology (e.g., using computer algebra systems to determine the roots; using graphing tech- nology to determine the x-intercepts of the graph of the corresponding rational function)
- 3.19
- solve problems involving applications of polynomial and simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem, such as deter- mining the values of k for which the function f(x) = x + 6x + kx – 4 gives the same remain- der when divided by x – 1 and x + 2]
- 3.20
- explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solu- tion to an inequality in one variable, and demonstrate that given solutions satisfy an inequality (e.g., demonstrate numerically and graphically that the solution to < 5 is x < –1 or x > – )
- 3.21
- determine solutions to polynomial inequali- ties in one variable [e.g., solve f(x) ≥ 0, where f(x) = x – x + 3x – 9] and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing tech- nology, and identifying intervals for which x satisfies the inequalities
- 3.22
- solve linear inequalities and factorable poly- nomial inequalities in one variable (e.g., x + x > 0) in a variety of ways (e.g., by deter- mining intervals using x-intercepts and evalu- ating 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 – 5x + 4 < 0, the solution represented algebraically is – 2 < x < –1 or 1 < x < 2)
Key ideas
A polynomial expression is a sum of terms, each a constant times a non-negative integer power of x.
A rational function that is the reciprocal of a linear or quadratic function has vertical asymptotes wherever the original function equals zero.
The remainder theorem says the remainder when f(x) is divided by (x − a) equals f(a).
An equation's solution is usually a specific set of values; an inequality's solution is usually an interval or union of intervals.
Linear and quadratic functions are special cases of polynomial functions.
Watch out for
- Thinking an expression with a negative or fractional exponent on x is still a polynomial.
- Confusing a vertical asymptote with a hole in the graph (a removable discontinuity).
- Confusing the sign in (x − a): substituting a with the wrong sign when evaluating f(a).
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 4 · 20 skills
Characteristics of Functions
Understanding Rates of Change
Real-world rates of change and their representations
I can gather and describe real-world rates of change in different representations.
Skill 4.1
Zero, constant, and changing rates of change
I can distinguish situations with zero, constant, or changing rates of change.
Skill 4.2
Sketching a graph from a rate-of-change description
I can sketch a graph representing a relationship described in words involving rate of change.
Skill 4.3
Calculating and interpreting average rates of change
I can calculate and interpret average rates of change of functions from real-world applications.
Skill 4.4
Connecting instantaneous and average rates of change
I can connect instantaneous rate of change to average rate of change.
Skill 4.5
Approximating instantaneous rate of change from average rates
I can approximate an instantaneous rate of change using average rates over shrinking intervals.
Skill 4.6
Connecting secant and tangent slopes to rate of change
I can connect the slope of a secant to average rate of change, and the slope of a tangent to instantaneous rate of change.
Skill 4.7
Approximating the slope of a tangent using secants
I can approximate the slope of a tangent at a point using the slopes of secants through that point.
Skill 4.8
Solving average and instantaneous rate-of-change problems
I can solve real-world problems involving average and instantaneous rates of change.
Skill 4.9
Combining Functions
Key features of functions formed by combining functions
I can determine the key features of functions formed by adding, subtracting, multiplying, or dividing two functions.
Skill 4.10
Real-world applications of combinations of functions
I can recognize real-world applications of combinations of functions and solve related problems graphically.
Skill 4.11
Even, odd, and increasing/decreasing properties of combined functions
I can determine and explain properties (even, odd, increasing/decreasing) of combined functions.
Skill 4.12
Composing two functions numerically and graphically
I can determine the composition of two functions numerically and graphically, and interpret it in context.
Skill 4.13
Composing two functions algebraically and stating domain and range
I can determine the composition of two functions algebraically and state its domain and range.
Skill 4.14
Solving real-world problems with the composition of two functions
I can solve real-world problems involving the composition of two functions.
Skill 4.15
The composition of a function and its inverse maps a number onto itself
I can demonstrate that composing a function with its inverse maps a number onto itself.
Skill 4.16
Connecting transformations to composition with a linear function
I can connect transformations of a function to its composition with a linear function.
Skill 4.17
Using Function Models to Solve Problems
Comparing the characteristics of different function families
I can compare the characteristics of different function families.
Skill 4.18
Solving equations and inequalities not accessible algebraically
I can solve graphically and numerically equations and inequalities not accessible by standard algebra.
Skill 4.19
Solving real-world problems by reasoning with function models
I can solve real-world problems by reasoning with function models.
Skill 4.20
The official wording · 20 outcomes
- 4.1
- 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)
- 4.2
- recognize that the rate of change for a func- tion is a comparison of changes in the depen- dent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or chang- ing by examining applications, including those arising from real-world situations
- 4.3
- 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
- 4.4
- 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)
- 4.5
- 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)
- 4.6
- determine, through investigation using various representations of relationships (e.g., tables of values, graphs, equations), approximate instan- taneous 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
- 4.7
- 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
- 4.8
- determine, through investigation using a vari- ety 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, sinu- soidal) 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 con- nections to average and instantaneous rates of change
- 4.9
- solve problems involving average and instan- taneous 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)
- 4.10
- determine, through investigation using graph- ing 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 sin 4x, g(x) = x + 2 , h(x) = ], and describe factors that affect these properties
- 4.11
- recognize real-world applications of combi- nations 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 fre- quencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically
- 4.12
- 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)]
- 4.13
- 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 func- tions in real-world applications
- 4.14
- 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 deter- mining 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 com- position of two functions
- 4.15
- solve problems involving the composition of two functions, including problems arising from real-world applications
- 4.16
- demonstrate, by giving examples for func- tions 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 ( f(x)) = x and f(f (x)) = x demonstrate that the inverse function is the reverse process of the original function and that it undoes what the function does]
- 4.17
- 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 + 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)
- 4.18
- compare, through investigation using a vari- ety of tools and strategies (e.g., graphing with technology; comparing algebraic representa- tions; 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, trigono- metric, exponential, logarithmic)
- 4.19
- solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques Sample problem: Solve: 2x < 2 ; cos x = x, with x in radians.
- 4.20
- 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
Key ideas
A rate of change can be described in words, a table of numbers, a graph, or an algebraic formula — the same idea, four different views.
Combining two functions (by addition, subtraction, multiplication, or division) can change the domain, range, and number of zeros compared to either original function.
Polynomial, rational, trigonometric, exponential, and logarithmic functions each have a signature shape, domain, and end behaviour.
The rate of change compares how much the dependent variable changes for a given change in the independent variable.
Many real phenomena (a damped pendulum, telephone tone frequencies) are naturally modelled as a combination of two functions.
Watch out for
- Thinking a rate of change must always be expressed as a formula to count as a rate of change.
- Assuming the domain of a combined function is automatically the same as the domain of either original function.
- Assuming any curved graph must be a polynomial rather than considering exponential or trigonometric possibilities.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Plan the term
Pick your term and the days you study. Each unit gets its share of the weeks and comes back a week later for a review. The last two weeks go over the whole course.
Which course?
Ontario has 6 math courses for Grade 12. Here they are side by side.
- Advanced Functions, Grade 12 (University)This courseMHF4U4 units · 72 skillsNo provincial examWhat it coversExponential and Logarithmic Functions · Trigonometric Functions · Polynomial and Rational Functions · Characteristics of Functions
- Calculus and Vectors, Grade 12 (University)MCV4U3 units · 54 skillsNo provincial examWhat it coversRate of Change · Derivatives and Their Applications · Geometry and Algebra of Vectors
- Foundations for College Mathematics, Grade 12 (College)MAP4C4 units · 59 skillsNo provincial examWhat it coversMathematical Models · Personal Finance · Geometry and Trigonometry · Data Management
- Mathematics for College Technology, Grade 12 (College)MCT4C4 units · 57 skillsNo provincial examWhat it coversExponential Functions · Polynomial Functions · Trigonometric Functions · Applications of Geometry
- Mathematics for Work and Everyday Life, Grade 12 (Workplace)MEL4E3 units · 52 skillsNo provincial examWhat it coversReasoning With Data · Personal Finance · Applications of Measurement
- Mathematics of Data Management, Grade 12 (University)MDM4U5 units · 57 skillsNo provincial examWhat it coversCounting and Probability · Probability Distributions · Organization of Data for Analysis · Statistical Analysis · Culminating Data Management Investigation
What a university, college or program asks for is theirs to say: check its admission page, and your school’s course guide.
The official curriculum
This page follows Ontario’s own curriculum for Advanced Functions, Grade 12 (University).
Textbooks
The textbooks Ontario Ministry of Education lists for Advanced Functions, Grade 12 (University):
- Nelson Advanced Functions
- McGraw-Hill Ryerson Advanced Functions 12
From the Trillium List, read October 27, 2026. ontario.ca
Study it in MapleMind
MapleMind teaches Advanced Functions, Grade 12 (University) one skill at a time. Each skill gets a short lesson, practice as you go and a check worth up to three stars. It remembers where you stopped.
Course Companion
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Questions about Advanced Functions, Grade 12 (University)
What is Advanced Functions, Grade 12 (University)?
Ontario’s Grade 12 math course: 4 units and 72 skills, from Exponential and Logarithmic Functions to Characteristics of Functions.
What units are in Advanced Functions, Grade 12 (University)?
1. Exponential and Logarithmic Functions, 2. Trigonometric Functions, 3. Polynomial and Rational Functions and 4. Characteristics of Functions.
Is there a provincial exam for Advanced Functions, Grade 12 (University)?
No. Ontario gives no provincial exam for Advanced Functions, Grade 12 (University).
What textbook does Advanced Functions, Grade 12 (University) use?
The textbooks Ontario Ministry of Education lists for Advanced Functions, Grade 12 (University): Nelson Advanced Functions, Nelson Education Ltd. and McGraw-Hill Ryerson Advanced Functions 12, McGraw-Hill Ryerson.
Can MapleMind help with Advanced Functions, Grade 12 (University)?
Yes. MapleMind teaches every skill of Advanced Functions, Grade 12 (University), step by step, with practice and checks, and its tutor answers questions about any of them. It’s free with daily limits, on iPhone, Android and the web.
Get ready for Advanced Functions, Grade 12 (University)
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