MHF4U – Advanced Functions Grade 12

Unit 1: Polynomial Functions

Students explore polynomial functions and extend their knowledge of linear and quadratic functions to higher-degree polynomials such as cubic, quartic, and quintic. They analyze graphs and key features including domain, range, intercepts, and intervals of increase/decrease. Students learn to factor polynomials, solve polynomial equations and inequalities, and use graphing technology to find zeros and analyze functions.

Unit 2: Rational Functions

Students study rational functions, written as ratios of polynomials​. They analyze graphs to identify vertical and horizontal asymptotes, intercepts, domain, and range. Students also solve rational equations and inequalities and determine restrictions. Applications help students understand how rational functions model real-world relationships.

Unit 3: Exponential and Logarithmic Functions

Students examine the relationship between exponential and logarithmic expressions, including equations. They apply the laws of logarithms to simplify expressions and solve exponential and logarithmic equations. Students also analyze graphs and apply these concepts to real-world models of growth and decay.

Unit 4: Trigonometric Functions

Students investigate trigonometric functions using radian measure and the unit circle. They analyze the graphs of functions including transformations like amplitude, period, and phase shift. Students also verify trigonometric identities and solve trigonometric equations used to model periodic phenomena.

Unit 5: Algebra of Functions and Rates of Change

Students explore how functions combine through operations​. They also study function composition and how it affects graphs and properties. The unit introduces average rate of change and instantaneous rate of change by analyzing the slopes of secant and tangent lines on graphs. These concepts prepare students for calculus and advanced mathematics.

By the end of the Grade 12 Advanced Functions course, students will develop a strong understanding of advanced mathematical functions and their applications. Students will learn to analyze polynomial, rational, exponential, logarithmic, and trigonometric functions by identifying key features such as domain, range, intercepts, asymptotes, and transformations through algebraic, graphical, and numerical representations.

Students will apply mathematical reasoning to solve equations and inequalities, including polynomial, rational, exponential, logarithmic, and trigonometric equations. They will also demonstrate an understanding of important concepts such as radian measure, trigonometric identities, and the relationships between exponential and logarithmic expressions.

In addition, students will explore how functions interact through operations. They will investigate average and instantaneous rates of change and use mathematical models to solve problems involving real-world relationships. Through these experiences, students will strengthen their analytical, problem-solving, and mathematical communication skills in preparation for future studies in calculus and other advanced mathematics courses.

The MHF4U course uses a variety of interactive strategies to support student success in this Ontario Grade 12 math course. Students in the grade 12 Advanced Functions online course develop their understanding of Grade 12 Advanced Functions through the following learning activities:

• Presentations with embedded videos that explain key concepts in the course, including examples, practice questions, and multimedia tools such as simulations and quizzes.
• End-of-unit conversations and reflections where students explain their problem-solving processes and receive feedback from their instructor.
• Instructor discussions and demonstrations through video conferencing, videos, or email to clarify concepts and support student learning.
• Discussion forums that allow students to share ideas, summarize learning, and receive peer feedback.
• Practical applications that connect Advanced Functions Ontario concepts to real-world situations.
• Independent assignments and assessments completed at a student’s own pace, allowing for individualized feedback and support.
• Research and writing activities that develop critical thinking and help students organize and communicate mathematical ideas.
• Visual learning tools such as graphs, charts, diagrams, tables, and virtual simulations to analyze relationships and mathematical patterns.

Practice problems that help students apply concepts learned in the Grade 12 Advanced Functions online course to real scenarios.

Assessment and evaluation follow the guidelines outlined in Growing Success (Ontario Ministry of Education, 2010) and are designed to support and improve student learning. Assessment helps teachers identify students’ strengths and areas for improvement in relation to the course expectations, while also guiding instructional strategies and classroom practices. Throughout the course, teachers provide descriptive feedback to help students refine their understanding and continue improving their performance.

Evaluation is the process of judging the quality of student work based on established criteria and achievement levels. While all curriculum expectations are addressed through instruction and assessment, evaluation focuses primarily on students’ achievement of the overall expectations. Teachers use professional judgement to determine which specific expectations best demonstrate overall achievement.

To ensure assessment and evaluation are effective and meaningful, teachers use strategies that:

• Address both what students learn and how well they learn
• Reflect the knowledge and skills categories outlined in the achievement chart
• Are varied and completed over time to give students multiple opportunities to demonstrate learning
• Are appropriate for the learning activities and needs of students
• Are fair and inclusive for all learners
• Provide accommodations for students with special education needs or language-learning needs
• Offer clear feedback and guidance to help students improve
• Encourage students to reflect on their learning and set goals

The final grade for the course is determined as follows:

❑ 70% based on evaluations conducted throughout the course, reflecting the student’s most consistent level of achievement, with consideration given to more recent evidence of learning.

❑ 30% based on a final evaluation at or near the end of the course. This may include an examination, performance task, or another comprehensive assessment that demonstrates overall achievement of the course expectations.

(Growing Success: Assessment, Evaluation and Reporting in Ontario Schools. Ontario

Ministry of Education Publication, 2010 p.41)

All students can succeed. Some students are able, with certain accommodations, to participate in the regular course curriculum and to demonstrate learning independently. Accommodations allow access to the course without any changes to the knowledge and skills the student is expected to demonstrate. The accommodations required to facilitate the student’s learning can be identified by the teacher, but recommendations from a School Board generated Individual Education Plan (IEP) if available can also be consulted. Instruction based on principles of universal design and differentiated instruction focuses on the provision of accommodations to meet the diverse needs of learners.

Examples of accommodations (but not limited to) include:

• Adjustment and or extension of time required to complete assignments or summative tasks
• Providing alternative assignments or summative tasks
• Use of scribes and/or other assistive technologies
• Simplifying the language of instruction

To learn more go to our Individual Education Plan (IEP) page.

What is MHF4U?

MHF4U is the Ontario curriculum code for the Grade 12 Advanced Functions course. It focuses on analyzing different types of mathematical functions and preparing students for calculus and other university-level math courses.

What prerequisite is required for MHF4U?

Students must complete either MCR3U – Functions, Grade 11 or MCT4C – Mathematics for College Technology, Grade 12 before enrolling.

Is MHF4U required for university?

Many university programs, especially in STEM, engineering, and business, require this course because it prepares students for Calculus and Vectors.

Can I take MHF4U online?

Yes. Students can take the course through an accredited Advanced Functions online course Ontario provider like Ontario eSecondary School, allowing flexible scheduling and self-paced learning.

How long does it take to complete MHF4U?

The course is designed for approximately 110 hours of learning, but online students can progress faster or slower depending on their schedule.

Is tutoring available for MHF4U?

Students at Ontario eSecondary School have access to tutoring and academic support, including on-demand assistance from subject matter experts.