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MHF4U Grade 12 Advanced Functions Course (University)

Course Details

  • 4. Grade 12

PREREQUISITE: Functions, Grade 11, University Preparation, or Mathematics for College Technology, Grade 12, College Preparation

GRADE: 12 (University)

AVAILABILITY: Full-time – All Campuses, Part-time – All Campuses, Private – All campuses, Summer School – Accelerated, Blyth Academy Online


Course Overview

MHF4U extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric 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. MHF4U online 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. Whether students complete this grade 12 course online or on campus, the MHF4U course extends students’ experience with advanced functions and prepares students for the next steps in their mathematics education.

Course Outline

Below is a course outline for the MHF4U online course. The Ministry of Education in Ontario sets the MHF4U curriculum, ensuring that students learn the same concepts and skills whether they complete the MHF4U course in person or online. MHF4U is a Grade 12 course at a University preparation level.

Mathematical processes are integrated into student learning throughout all areas of the course. Students will foster a variety of skills, including:

  • Solving problems through the development, selection, application, comparison, and adaptation of various problem-solving strategies.
  • Reasoning and proving, such as the development and application of reasoning skills in order to make and assess mathematical conjectures and justify conclusions.
  • Reflecting on and monitoring thinking to clarify understanding while completing investigations and solving problems, such as by assessing a chosen strategy’s effectiveness, proposing alternative approaches, etc.
  • Selecting tools and computational strategies to use various concrete, visual, and electronic learning tools and appropriate strategies.
  • Making connections between mathematical concepts, skills, and procedures.
  • Creating representations of mathematical ideas, such as numeric, algebraic, and pictorial representations.
  • Communicating mathematical thinking orally, visually, and in writing by using precise mathematical language and appropriate representations.


Essential Question: How can characteristics of polynomials be used to make connections between functions and their graphical representations?

In this unit, students will investigate the key properties and characteristics of polynomials and use their findings to form connections between algebraic and graphical representations of polynomial functions. Students will develop skills that can be used to analyze and solve polynomial equations and inequalities.

Rational Functions

Essential Question: What patterns exist in rational functions and how can they be used to make predictions about their graphs?

In this unit, students will examine the key characteristics of rational functions and use them to develop an understanding of their graphical representations. Students will investigate the different cases where horizontal asymptotes occur and use their understanding to solve rational equations and inequalities.


Essential Question: How can the properties of trigonometric functions be used to make predictions about real-world scenarios?

In this unit, students will build on their understanding of trigonometry acquired from previous studies of mathematics. Students will develop an understanding of radian measure and conceptualize special triangles and the unit circle in terms of radians. Students will explore advanced trigonometric identities and make connections to the graphs of sine and cosine. These graphs will be used to develop the graphs of tangent, cosecant, secant, and cotangent. Lastly, students will apply the understandings developed in this unit to solve trigonometric equations and real-world problems.

Exponential and Logarithmic Functions

Essential Question: How can the relationship between exponential and logarithmic functions be used to solve real-world problems?

In this unit, students will review exponential functions and make connections to the logarithmic function by applying an understanding of inverse functions. Students will then learn to write expressions in both exponential and logarithmic form. Students will learn about the laws of logarithms and how they can be applied to solve logarithmic equations. Lastly, students will combine the understandings they develop throughout this unit and apply them to solve real-world problems that can be modelled using exponential and logarithmic functions.

Combining Functions and Rates of Change

Essential Question: How can rates of change be used to make sense of motion in the world?

In this unit, students will develop techniques for combining functions and understanding of rates of change. . Students will begin by examining sums, differences, products, and quotients of functions. Students will then substitute functions into other functions in order to form composite functions. The domain, range, and key characteristics of combined functions will then be explored. Students will then turn their attention to rates of change of functions and investigate how the average rate of change can be used to approximate the instantaneous rate of change of a given function.


Course evaluation is based on a student’s achievement of curriculum expectations. The course will build on students’ experience with functions so that students may demonstrate skills required for effective learning. The student’s final grade will represent the quality of that student’s overall achievement of course expectations. The final percentage grade will be calculated as follows:

70% of the grade:  Evaluations conducted throughout the duration of the course will account for 70% of the final course grade. This portion will reflect a student’s most consistent achievement level throughout the course.

30% of the grade: A final evaluation administered following the completion of the online course will account for 30% of the final course grade. The final exam will be a proctored exam.

MHF4U will extend students’ experience with and knowledge of functions, preparing students for success in senior mathematics and university programs. Register today!

Please consult our Frequently Asked Questions Page or the Exam section within your course for more details on final exams and the exam fee. More information can also be found in our Student Handbook
Ready to get started? Register today

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