Will this help with my child's Grade 12 math homework?

Yes. StudyPug covers every MHF4U strand, including logarithms, trigonometric identities, polynomial functions, and rates of change, so your child can find help for any homework question quickly.

How long are the video lessons?

Most video lessons run between 5 and 15 minutes, focusing on one concept at a time. Your child can pause, rewind, and re-watch as many times as needed to understand the material fully.

Is there a provincial test for Grade 12 math in Ontario?

Ontario's EQAO assessments cover Grades 3, 6, and 9 math. Grade 12 is not an EQAO assessment year, though MHF4U is a prerequisite for many university programs and the final course grade matters for applications.

Can my child use StudyPug on a tablet or phone?

Yes. StudyPug works on tablets, phones, and computers, so your child can review logarithmic functions or trigonometric identities at home, on the bus, or anywhere with an internet connection.

Can I try StudyPug before paying?

Yes. StudyPug offers free sample lessons so you can explore the content before subscribing, and every paid plan includes a 30-day money-back guarantee if you are not satisfied.

ON Grade 12 Math Curriculum

Support your child through Ontario's Grade 12 Advanced Functions course with lessons and practice problems covering every strand, from logarithmic functions to rates of change.

ON Grade 12 Math Curriculum

OE_IDExpectationsStudyPug Topic

ON.OE.12AF.A1.1Recognize 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?
- Converting from logarithmic form to exponential form
- Evaluating logarithms without a calculator
ON.OE.12AF.A1.2Determine, with technology, the approximate logarithm of a number to any base, including base 10
- Common logarithms
- Natural log: ln
- Evaluating logarithms using change-of-base formula
ON.OE.12AF.A1.3Make connections between related logarithmic and exponential equations and solve simple exponential equations by rewriting them in logarithmic form
- Converting from exponential form to logarithmic form
- Solving exponential equations with logarithms
ON.OE.12AF.A1.4Make 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
- Product rule of logarithms
- Quotient rule of logarithms
- Combining product rule and quotient rule in logarithms
- Power rule of logarithms
- Evaluating logarithms using logarithm rules
ON.OE.12AF.A2.1Determine, through investigation with technology, key features of the graphs of logarithmic functions
- Graphing logarithmic functions
- Graphing transformations of logarithmic functions
ON.OE.12AF.A2.2Recognize 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
- Inverse functions
- Graphing exponential functions
- Graphing transformations of exponential functions
ON.OE.12AF.A2.3Determine, 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
- Transformations of functions: Horizontal translations
- Transformations of functions: Vertical translations
- Reflection across the y-axis: y = f(-x)
- Reflection across the x-axis: y = -f(x)
- Transformations of functions: Horizontal stretches
- Transformations of functions: Vertical stretches
ON.OE.12AF.A2.4Pose 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
- Finding an exponential function given its graph
- Finding a logarithmic function given its graph
ON.OE.12AF.A3.1Recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
- Solving exponential equations using exponent rules
ON.OE.12AF.A3.3Solve simple logarithmic equations in one variable algebraically
- Solving logarithmic equations
ON.OE.12AF.A3.4Solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications
- Exponential growth and decay by a factor
- Exponential decay: Half-life
- Exponential growth and decay by percentage
- Finance: Compound interest
- Continuous growth and decay
- Logarithmic scale: Richter scale (earthquake)
- Logarithmic scale: pH scale
- Logarithmic scale: dB scale
- Finance: Future value and present value
ON.OE.12AF.B1.1Recognize 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
- Converting between degrees and radians
ON.OE.12AF.B1.2Represent radian measure in terms of ? and as a rational number
- Trigonometric ratios of angles in radians
ON.OE.12AF.B1.3Determine, with technology, the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure
- Radian measure and arc length
ON.OE.12AF.B1.4Determine, 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?
- Find the exact value of trigonometric ratios
ON.OE.12AF.B2.1Sketch 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
- Cosine graph: y = cos x
ON.OE.12AF.B2.2Make 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
ON.OE.12AF.B2.3Graph, 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
- Cotangent graph: y = cot x
- Secant graph: y = sec x
- Cosecant graph: y = csc x
ON.OE.12AF.B2.4Determine 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
- Graphing transformations of trigonometric functions
ON.OE.12AF.B2.6Represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
- Determining trigonometric functions given their graphs
ON.OE.12AF.B2.7Pose 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
- Ferris wheel trig problems
- Tides and water depth trig problems
- Spring (simple harmonic motion) trig problems
ON.OE.12AF.B3.1Recognize equivalent trigonometric expressions and verify equivalence using graphing technology
- Quotient identities and reciprocal identities
- Pythagorean identities
- Sum and difference identities
- Cofunction identities
ON.OE.12AF.B3.2Explore the algebraic development of the compound angle formulas, and use the formulas to determine exact values of trigonometric ratios
- Double-angle identities
ON.OE.12AF.B3.3Recognize 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
- Determining non-permissible values for trig expressions
ON.OE.12AF.B3.4Solve linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from 0 to 2?, and solve related problems
- Solving first degree trigonometric equations
- Solving second degree trigonometric equations
- Solving trigonometric equations involving multiple angles
- Solving trigonometric equations using pythagorean identities
- Solving trigonometric equations using sum and difference identities
- Solving trigonometric equations using double-angle identities
ON.OE.12AF.C1.1Recognize 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?
- Polynomial components
- Polynomial functions
ON.OE.12AF.C1.2Compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial functions
- Multiplying monomial by monomial
- Multiplying monomial by binomial
- Multiplying binomial by binomial
- Multiplying polynomial by polynomial
ON.OE.12AF.C1.3Describe key features of the graphs of polynomial functions
- Applications of polynomials
- Characteristics of polynomial graphs
ON.OE.12AF.C1.4Distinguish 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
- Common factors of polynomials
ON.OE.12AF.C1.5Make 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
- Find the difference of squares: (a - b)(a + b) = (a^2 - b^2)
- Evaluating polynomials
- Using algebra tiles to factor polynomials
- Factor theorem
ON.OE.12AF.C1.6Determine, 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?
- Solving polynomial equations
- Word problems of polynomials
ON.OE.12AF.C1.7Determine an equation of a polynomial function that satisfies a given set of conditions
- Factoring polynomials by grouping
- Solving polynomials with unknown coefficients
- Solving polynomials with unknown constant terms
- Determining the equation of a polynomial function
ON.OE.12AF.C1.8Determine 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
- Factoring polynomials: x^2 + bx + c
- Applications of polynomials: x^2 + bx + c
- Solving polynomials with the unknown
- Factoring polynomials: ax^2 + bx + c
- Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2
- Rational zero theorem
ON.OE.12AF.C1.9Determine, through investigation, and compare the properties of even and odd polynomial functions
- Even and odd functions
ON.OE.12AF.C2.1Determine, 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
- Graphing reciprocals of linear functions
- Graphing reciprocals of quadratic functions
ON.OE.12AF.C2.2Determine, 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
- What is a rational function?
- Graphs of rational functions
ON.OE.12AF.C2.3Sketch the graph of a simple rational function using its key features, given the algebraic representation of the function
- Simplifying rational expressions and restrictions
ON.OE.12AF.C3.1Make 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
- Remainder theorem
- Factor by taking out the greatest common factor
- Factor by grouping
- Factoring difference of squares: $x^2 - y^2$
- Factoring trinomials
- Factoring difference of cubes
- Factoring sum of cubes
ON.OE.12AF.C3.3Determine, 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
- Characteristics of quadratic functions
- Transformations of quadratic functions
- Quadratic function in general form: y = ax^2 + bx + c
- Converting from general to vertex form by completing the square
- Shortcut: Vertex formula
- Graphing quadratic functions: General form VS. Vertex form
- Finding the quadratic functions for given parabolas
ON.OE.12AF.C3.4Solve polynomial equations in one variable, of degree no higher than four, by selecting and applying strategies, and verify solutions using technology
- Solving quadratic equations by factoring
- Solving quadratic equations by completing the square
- Using quadratic formula to solve quadratic equations
- Nature of roots of quadratic equations: The discriminant
- Applications of quadratic equations
ON.OE.12AF.C3.5Determine, 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
- Point of discontinuity
- Adding and subtracting rational expressions
- Multiplying rational expressions
- Dividing rational expressions
- Vertical asymptote
ON.OE.12AF.C3.6Solve simple rational equations in one variable algebraically, and verify solutions using technology
- Solving rational equations
ON.OE.12AF.C3.7Solve problems involving applications of polynomial and simple rational functions and equations
- Applications of polynomial functions
- Applications of rational equations
- Simplifying complex fractions
- Partial fraction decomposition
ON.OE.12AF.C4.1Explain, 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
- Solving rational inequalities
- Solving quadratic inequalities
- Solving polynomial inequalities
ON.OE.12AF.C4.2Determine 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
- Graphing linear inequalities in two variables
- Graphing quadratic inequalities in two variables
- Graphing systems of quadratic inequalities
- Graphing systems of linear inequalities
ON.OE.12AF.C4.3Solve linear inequalities and factorable polynomial inequalities in one variable in a variety of ways, and represent the solutions on a number line or algebraically
- Applications of inequalities
ON.OE.12AF.D1.1Gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change
- Rate of change
ON.OE.12AF.D1.2Recognize 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
- Difference quotient: applications of functions
ON.OE.12AF.D1.3Sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology when possible
- Relationship between two variables
- Understand relations between x- and y-intercepts
- Domain and range of a function
- Identifying functions
ON.OE.12AF.D1.4Calculate and interpret average rates of change of functions arising from real-world applications, given various representations of the functions
- Function notation
ON.OE.12AF.D1.5Recognize examples of instantaneous rates of change arising from real-world situations, and make connections between instantaneous rates of change and average rates of change
- Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
- Midpoint formula: $M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)$
- Slope equation: $m = \frac{y_2-y_1}{x_2- x_1}$
- Slope intercept form: y = mx + b
- General form: Ax + By + C = 0
- Point-slope form: y - y_1 = m(x - x_1)
- Definition of derivative
ON.OE.12AF.D1.6Determine, 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
- Graphing linear functions using table of values
- Graphing linear functions using x- and y-intercepts
- Graphing from slope-intercept form y=mx+b
- Graphing linear functions using a single point and slope
- Word problems of graphing linear functions
- Power rule
ON.OE.12AF.D1.7Make 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
- Parallel and perpendicular lines in linear functions
- Applications of linear relations
- Slope and equation of tangent line
ON.OE.12AF.D1.8Determine, 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
- Determining number of solutions to linear equations
- Solving systems of linear equations by graphing
- Solving systems of linear equations by elimination
- Solving systems of linear equations by substitution
ON.OE.12AF.D1.9Solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods
- Money related questions in linear equations
- Unknown number related questions in linear equations
- Distance and time related questions in linear equations
- Rectangular shape related questions in linear equations
ON.OE.12AF.D2.1Determine, 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
- Subtracting functions
- Multiplying functions
- Dividing functions
- Operations with functions
ON.OE.12AF.D2.2Recognize real-world applications of combinations of functions, and solve related problems graphically
- Composite functions
ON.OE.12AF.D2.3Determine, through investigation, and explain some properties of functions formed by adding, subtracting, multiplying, and dividing general functions
- Inequalities of combined functions
ON.OE.12AF.D2.8Make connections, through investigation using technology, between transformations of simple functions and the composition of these functions with a linear function
- Combining transformations of functions
ON.OE.12AF.D3.1Compare, through investigation using a variety of tools and strategies, the characteristics of various functions
- Function notation (advanced)
- One to one functions
ON.OE.12AF.D3.2Solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques
- Direct variation
- Evaluating piecewise functions
- Graphing piecewise linear functions
- Graphing piecewise non-linear functions
ON.OE.12AF.D3.3Solve 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
- Absolute value functions
- Solving absolute value equations
- Solving absolute value inequalities

Grade 12 Advanced Functions (MHF4U)

About Grade 12 Ontario math

Ontario follows The Ontario Curriculum, structured around strands that build conceptual understanding and algebraic reasoning. In MHF4U, students work through exponential and logarithmic functions, trigonometric functions and identities, polynomial and rational functions, characteristics of functions including rates of change, and combinations and compositions of functions. StudyPug provides step-by-step lessons and practice problems aligned to every strand of this course.

Strands: Exponential and Logarithmic Functions, Trigonometric Functions, Polynomial and Rational Functions, Characteristics of Functions, Combining Functions
Step-by-step lessons and worked examples for every MHF4U topic
Practice problems with detailed solutions aligned to the Ontario Curriculum
Accessible on any device so your child can study at home or on the go

Curriculum topics by strand

Find lessons and practice problems for every Grade 12 Advanced Functions topic, organized by strand and aligned to The Ontario Curriculum (MHF4U).

Topic includes:

Practice

Video

Quiz

OE_ID

Expectations

StudyPug Topic

Ontario Grade 12 Math Topics

Exponential and Logarithmic Functions: Recognizing logarithms as inverse operations of exponentiation, evaluating logarithmic expressions, applying laws of logarithms, graphing logarithmic functions, and solving exponential and logarithmic equations including real-world applications.
Trigonometric Functions: Radian measure, exact values for special angles, graphs of primary and reciprocal trigonometric functions, sinusoidal functions and their transformations, compound angle formulas, trigonometric identities, and solving linear and quadratic trigonometric equations.
Polynomial and Rational Functions: Key features of polynomial function graphs, factored form and x-intercepts, the remainder and factor theorems, solving polynomial equations, rational functions derived from linear and quadratic expressions, and solving rational equations and inequalities.
Characteristics of Functions: Average and instantaneous rates of change, slope of secants and tangents, and solving problems involving rates of change from real-world applications using numerical and graphical methods.
Combining Functions: Adding, subtracting, multiplying, and dividing functions, composition of functions, inverse functions, and solving problems that require reasoning with combinations of functions.

How StudyPug Helps Ontario Grade 12 Math Students

When a student is stuck on a concept like proving trigonometric identities or applying the factor theorem, StudyPug provides step-by-step worked examples that break the problem down clearly. Each lesson focuses on one idea at a time, making it easier to spot exactly where confusion started.

Practice problems with full solutions let students check their reasoning after each lesson, building the confidence needed for class tests and the final MHF4U grade that counts toward university applications.

Aligned to Ontario Grade 12 Math Curriculum

Every lesson and practice problem in StudyPug's MHF4U content is organized to match the strands and learning expectations of The Ontario Curriculum, so what your child studies on StudyPug directly supports what is being taught in their Ontario classroom.

Ready to put your Grade 12 Math skills to the test?

Jump into practice questions for every ON Grade 12 Math topic - with instant feedback and step-by-step video solutions when you get stuck.

Start Practicing

Written by the StudyPug Math Team

Last updated June 5, 2026

3 M+

students

★★★★★

4.8 average

K-12

all grades