Curriculum Overview · Ontario Ministry of Education

📋 MHF4U Curriculum Overview

Advanced Functions, Grade 12 University Preparation

Aligned with The Ontario Curriculum, Grades 11 and 12: Mathematics (2007 Revised) · Growing Success (2010)

1. Course Identification

Course Code
MHF4U
Course Title
Advanced Functions
Grade / Type
12 / University (U)
Credit Value
1.0
Scheduled Hours
110
Prerequisite
MCR3U — Functions, Grade 11 University
Curriculum Document
2007 Revised
Policy Framework
Growing Success (2010)

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 going on to any one of a variety of university programs.

2. Strands & Expectations

Strand A — Exponential and Logarithmic Functions

Overall Expectations: students will demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions; identify and describe key features of logarithmic functions; and solve exponential and logarithmic equations.

Key Specific Expectations

  • A1.1 Recognize the logarithm of \( x \) to the base \( b \) (\( \log_b x \)) as the exponent to which \( b \) must be raised to obtain \( x \).
  • A1.2 Evaluate logarithmic expressions and approximate the logarithm of a number to any base using common or natural logarithms.
  • A1.3 Make connections between related logarithmic and exponential equations (\( y = b^x \Leftrightarrow x = \log_b y \)).
  • A2.1 Determine, through investigation, the roles of parameters \( a, k, d, c \) in \( f(x)=a\log_{10}(k(x-d))+c \).
  • A2.3 Pose and solve real-world problems modelled by logarithmic functions (decibel, pH, Richter scale).
  • A3.1 Recognize equivalent algebraic expressions involving logarithms; simplify using the laws of logarithms.
  • A3.4 Solve simple exponential equations by rewriting in the same base or by using logarithms.

Strand B — Trigonometric Functions

Overall Expectations: demonstrate an understanding of radian measure; make connections between trigonometric ratios and the graphical/algebraic representations of the corresponding trigonometric functions; and solve problems involving trigonometric equations and prove trigonometric identities.

Key Specific Expectations

  • B1.1 Recognize the radian as an alternative unit to the degree; convert between the two (\( 180° = \pi \) rad).
  • B1.3 Determine the exact values of \( \sin\theta, \cos\theta, \tan\theta \) for special angles \( 0, \tfrac{\pi}{6}, \tfrac{\pi}{4}, \tfrac{\pi}{3}, \tfrac{\pi}{2} \) and their multiples.
  • B2.1 Sketch graphs of \( y=\sin x, y=\cos x, y=\tan x \) using radian measure; describe period, amplitude, range, asymptotes.
  • B2.3 Determine, through investigation, the role of parameters in \( f(x)=a\sin(k(x-d))+c \) and \( f(x)=a\cos(k(x-d))+c \).
  • B2.7 Pose and solve sinusoidal modelling problems (tides, pendulum, daylight hours).
  • B3.1 Recognize equivalent trigonometric expressions; verify equivalence using the unit circle.
  • B3.2 Explore Pythagorean, quotient, compound-angle, and double-angle identities.
  • B3.4 Solve linear and quadratic trigonometric equations on a restricted domain.

Strand C — Polynomial and Rational Functions

Overall Expectations: identify and describe properties of polynomial functions; identify and describe properties of rational functions; and solve polynomial and rational equations and inequalities.

Key Specific Expectations

  • C1.1 Recognize a polynomial expression; describe its key features (degree, leading coefficient, constant term).
  • C1.3 Describe key features of graphs of polynomial functions (domain, range, end behaviour, turning points, zeros).
  • C1.4 Distinguish even/odd/neither using the symmetry test \( f(-x)=f(x) \) or \( f(-x)=-f(x) \).
  • C1.7 Determine an equation of a polynomial function given its graph or features.
  • C2.1 Determine, through investigation, key features of rational functions of the form \( f(x)=\tfrac{1}{x}, \tfrac{1}{x^2}, \tfrac{ax+b}{cx+d} \).
  • C3.4 Solve polynomial inequalities of degree at most 4 algebraically and graphically.
  • C4.1 Determine the average rate of change of a function over an interval; the instantaneous rate of change at a point using a sequence of secant slopes.

Strand D — Characteristics of Functions

Overall Expectations: demonstrate an understanding of average and instantaneous rate of change; determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions; and compare the characteristics of functions and solve problems by modelling and reasoning with functions.

Key Specific Expectations

  • D1.1 Gather, interpret, and describe information about average and instantaneous rates of change in real-world situations.
  • D1.5 Determine the equation of the line tangent to a graph at a given point as the limit of secant slopes.
  • D2.1 Determine sums, differences, products, quotients of two functions both algebraically and graphically.
  • D2.5 Determine the composition of two functions \( (f \circ g)(x)=f(g(x)) \) numerically and algebraically; state the domain of the composition.
  • D3.1 Compare the key features (rates of growth) of polynomial, exponential, logarithmic, sinusoidal functions over various intervals.
  • D3.3 Solve graphically and numerically equations involving combinations of polynomial, exponential, logarithmic, trigonometric functions.

3. Mathematical Processes

The seven mathematical processes are integrated into every strand. Students engage in:

Problem SolvingDevelop, select, apply, compare, and adapt strategies to investigate situations and solve problems.
Reasoning & ProvingDevelop and apply reasoning skills to make and investigate conjectures and to construct/defend arguments.
ReflectingDemonstrate that they are reflecting on and monitoring their thinking to clarify understanding.
Selecting Tools & StrategiesUse a variety of concrete, visual, electronic tools (Desmos, GeoGebra, graphing calculator) and computational strategies.
ConnectingMake connections among mathematical concepts and procedures, and relate ideas across strands.
RepresentingCreate a variety of representations (numeric, geometric, algebraic, graphical) and translate between them.
CommunicatingCommunicate mathematical thinking orally, visually, and in writing using appropriate vocabulary, notation, and conventions.

4. Achievement Chart

Per Growing Success (2010), each of the four categories carries equal weight (25%) within both term work and the final evaluation.

Knowledge & Understanding
25%

Subject-specific content acquired (knowledge) and the comprehension of its meaning and significance (understanding). Includes definitions, properties, computations, and identifying types of functions.

Thinking & Inquiry
25%

Use of critical and creative thinking skills and processes — planning, processing, making connections, justifying conclusions in novel and multi-step problems.

Communication
25%

Conveying meaning through various forms — clarity of expression, use of mathematical conventions, vocabulary, terminology, and labelled diagrams/graphs.

Application
25%

Use of knowledge and skills to make connections within and between contexts — real-world modelling, transfer of concepts to new situations.

Levels of achievement: Level 4 (80–100%, exceeds standard) · Level 3 (70–79%, provincial standard) · Level 2 (60–69%, approaching) · Level 1 (50–59%, limited) · R (below 50%, insufficient).

5. Evaluation Policy

Final mark = 70% Term Work + 30% Final Evaluation, per Growing Success (2010).

ComponentWeightDescription
Unit Tests (Ch 1–8, Assessment OF Learning)45%Eight chapter unit tests covering all four achievement categories.
Quizzes & Diagnostics (Assessment FOR/AS Learning)10%Formative checks; lowest score dropped.
Performance Tasks & Investigations10%Modelling, investigation, communication of process.
Culminating Performance Task5%Cross-strand application due before the final exam window.
Subtotal — Term Work70%Reported on the Provincial Report Card
Final Exam (Ch 1–8 comprehensive, 3 hours, 100 marks)30%Equally weighted across K/U, Thinking, Communication, Application (25 marks each).
Final Course Mark100%One credit toward the OSSD; eligible university prerequisite.

6. Chapter ↔ Strand Mapping

ChapterTitlePrimary StrandSpecific Expectations Addressed
Ch 1Polynomial FunctionsC, DC1.1–1.4, C1.7, D1.1, D1.5
Ch 2Polynomial Equations & InequalitiesCC3.1–3.4, C3.6–3.7
Ch 3Rational FunctionsCC2.1–2.4, C3.5
Ch 4Trigonometry (Radians, Identities)BB1.1–1.5, B3.1–3.3
Ch 5Trigonometric Functions & EquationsBB2.1–2.7, B3.4–3.5
Ch 6Exponential & Logarithmic FunctionsAA1.1–1.4, A2.1–2.3
Ch 7Solving Exp. & Log. EquationsAA3.1–3.4
Ch 8Combining FunctionsDD2.1–2.6, D3.1–3.3
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