The Advanced Functions Grade 12 course extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students in MHF4U will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.
Course curriculum
Inside %course-code%: where the math leads
The bridge to Calculus (MCV4U)
Secants, tangents and instantaneous rates of change are the exact ideas that become the derivative — and function composition sets up the chain rule. MHF4U is the direct preparation for Calculus and Vectors.
Modelling the real world
Exponential growth and decay, sinusoidal cycles, logarithmic scales — you solve problems arising from real-world applications, graphically and algebraically, with and without technology.
The function fluency universities assume
Graphs, asymptotes, intercepts and inequalities across polynomial, rational, logarithmic and trigonometric functions — the algebra facility first-year STEM, business and economics courses expect on day one.
Proof and mathematical reasoning
Proving trigonometric identities and reasoning about why solutions work — not just computing them. The habits of precise argument that senior mathematics is built on.
Simulation Labs — see the math move
Interactive virtual labs and simulations throughout the course: explore hypotheses, relate variables and watch functions respond as you change them — hands-on practice no other online math course offers.

MHF4U Course outline
Full unit-by-unit breakdown.
Self-reflection, skill-building and strategy development to prepare for online math learning — students assess previous experience, identify areas for improvement, and set strategies for success.
Key topics
Learning-skills self-assessment · goal setting · strategies for online math learning
Assessed by
Preparation activities — not graded
Video lessons and practice problems extend linear and quadratic knowledge to cubic, quartic and quintic functions — factoring to the 5th degree, transformations, key features of graphs, and solving polynomial equations and inequalities.
Key topics
Cubic, quartic and quintic functions · factoring · transformations · domain, range and intervals · polynomial inequalities
Assessed by
Assignments (incl. a Desmos rollercoaster design) · question-response interview · proctored unit test
Investigate reciprocals of linear and quadratic functions: vertical and horizontal asymptotes, domain and range, intercepts and intervals — connecting algebraic and graphical representations to solve rational equations and inequalities.
Key topics
Reciprocals of linear and quadratic functions · vertical and horizontal asymptotes · rational equations and inequalities
Assessed by
Quiz and assignments · culminating assignment · question-response interview · proctored unit test
Connect exponential and logarithmic forms, verify and apply the laws of logarithms, and solve exponential and logarithmic equations algebraically — including problems from real-world applications.
Key topics
Laws of logarithms · evaluating logarithms · exponential and logarithmic equations · growth, decay and logarithmic scales
Assessed by
Equations assignment · culminating assignment · question-response interview · proctored unit test
Radian measure from the unit circle, the graphs and properties of sine and cosine, sinusoidal modelling — then proving trigonometric identities through reasoning and verifying them with technology.
Key topics
Radian measure and the unit circle · sine and cosine graphs (period, amplitude) · reciprocal trig functions · trigonometric identities and equations
Assessed by
Quizzes (transformations, reciprocal functions, trig equations) · trig identities assignment · question-response interview · proctored unit test
Combine functions through addition, subtraction, multiplication, division and composition, then connect secant slopes to average rates of change and tangent slopes to instantaneous rates — the doorway into calculus.
Key topics
Operations on functions · composition f(g(x)) · secant slope and average rate of change · tangent slope and instantaneous rate of change
Assessed by
Quizzes (operations, composition of functions) · question-response interview · proctored unit test
