Advanced Functions MHF4U


This 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 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.

SKU: MHF4U Category:
The MHF4U Advanced Functions course is designed for Grade 12 students preparing for university. This course significantly extends students’ knowledge of functions, preparing them for higher education in fields such as engineering, mathematics, and physical sciences.

Course Overview

In this course, students will explore in-depth the properties of polynomial, rational, logarithmic, and trigonometric functions. They will learn to develop techniques for combining functions, expand their understanding of rates of change, and enhance their ability to apply these concepts and skills effectively.


Successful completion of Functions, Grade 11, University Preparation, or Mathematics for College Technology, Grade 12, College Preparation is required.

Key Learning Objectives

  • Analyze and apply properties of polynomial and rational functions.
  • Investigate logarithmic expressions and solve related equations.
  • Examine and utilize trigonometric functions to solve practical problems.
  • Develop an understanding of complex rates of change and function combination.

Mathematical Process Expectations

Throughout MHF4U, students will integrate and apply mathematical processes such as problem solving, reasoning, connecting, and communicating to deepen their mathematical understanding.

Advanced Preparation for University

This course is essential for students aiming to undertake university programs that require a strong foundation in mathematics. It is particularly vital for those planning to study Calculus and Vectors or other advanced math courses at the university level.

Overall Expectations

By the end of this course, the student will:



  • Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
  • Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
  • Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.



  • Demonstrate an understanding of the meaning and application of radian measure;
  • Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
  • Solve problems involving trigonometric equations and prove trigonometric identities.



  • Identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
  • Identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
  • Solve problems involving polynomial and simple rational equations graphically and algebraically;
  • Demonstrate an understanding of solving polynomial and simple rational inequalities.



  • Demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
  • Determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
  • Compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.

Course Details

Course CodeMHF4U
Course TypeUniversity Preparation
PrerequisiteMCR3U or MCT4C
GradeGrade 12
Credit Value1.0

Outline of Course Content

1Polynomial Functions15 hours
2Polynomial Equations and Inequalities15 hours
3Rational Functions13 hours
4Trigonometry13 hours
5Trigonometric Functions13 hours
6Exponential and Logarithmic Functions13 hours
7Tools and Strategies for Solving Exponential and Logarithmic Equations13 hours
8Combining Functions13 hours
Final Examination2 hours
Total110 hours