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MHF4U - Advanced Functions


Course Title: Advanced Functions
Course Code: MHF4U
Grade: 12
Course Type: University Preparation
Credit Value: 1
Prerequisite: Functions, MCR3U; or Mathematics for College Technology, MCT4C
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2007 (Revised)
Text: Nelson, Advanced Functions,
© 2009 ISBN-13:9780176374433

Department: Mathematics
Course Developer: Jenny Li
Development Date: May 2013

Course Description:

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.

Overall Expectations: MHF4U


Overall Expectations

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.


Overall Expectations

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.


Overall Expectations

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.


Overall Expectations

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.


Unit details:


Titles and Descriptions

Time and Sequence

Unit 1

Characteristics of Functions

Review and consolidate your knowledge of the properties and characteristics of basic functions and their inverses. Review and consolidate your knowledge of graphing functions using transformations. Investigate the characteristics of piecewise functions. Calculate an average rate of change of a Function. Estimate the instantaneous rate of change of a function. Interpret the average rate of change and the instantaneous rate of change of a function. Solve problems that involve rate of change.

18 hours

Unit 2

Polynomial Functions

Identify and use key features of polynomial functions. Solve problems using a variety of tools and strategies related to polynomial functions. Determine and interpret average and instantaneous rates of change for polynomial functions.

15 hours

Unit 3

Rational Functions

Identify the key characteristics of rational functions from their equations and use these characteristics to sketch their graphs. Solve rational equations and inequalities with and without graphing technology. Determine average and instantaneous rates of change in situations that are modelled by rational functions.

12 hours

Unit 4


Explore, define and use radian measure. Determine the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure. Recognize equivalent trigonometric expressions and verify equivalence using graphing technology. Explore the algebraic development of compound angle formulas and use the formulas to determine exact values of trigonometric ratios. Prove trigonometric identities through the application of reasoning skills, using a variety of relationships.

15 hours

Unit 5

Trigonometry Functions

Graph and transform primary trigonometric functions and their reciprocals in radians and identify key features of the functions. Solve linear and quadratic trigonometric equations using radian measures. Make connections between graphic and algebraic representations of trigonometric relationships. Determine and interpret average and instantaneous rates of change for trigonometric functions. Solve problems, using a variety tools and strategies, including problems arising from real-world applications, by reasoning with functions and by applying concepts and procedures involving functions.

15 hours

Unit 6

Exponential and Logarithmic Functions

Develop the understanding that the logarithmic function is the inverse of the exponential function. Simplify exponential and logarithmic expressions using exponent rules. Identify features of the exponential and logarithmic function including rates of change. Transform exponential and logarithmic functions. Evaluate exponential and logarithmic expressions and equations. Solve problems that can be modelled using exponential or logarithmic functions.

21 hours

Unit 7

Combinations of Functions

Create new functions by adding, subtracting, multiplying, or dividing functions. Create composite functions. Determine key properties of the new functions. Generalize their understanding of a function.

12 hours


Final Evaluation

The final assessment task is a proctored three hour exam worth 20% of the student's final mark.

2 hours



110 hours

Teaching / Learning Strategies:

Students will follow a similar pattern of instructions in all units. To begin students will be involved in the exploration of an investigation of a concept. Then they will apply what they have learned in several real life scenarios or applications of the concept. Students will see solutions to applications after they try to solve them for themselves. Then students will complete assignments where no solutions are provided and submit these for assessment. Finally the unit ends with a test or other suitable assessment of learning such as projects. Since the over-riding aim of this course is to help students use the language of mathematics skillfully, confidently and flexibly, a wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests and ability levels.

Seven mathematical processes will form the heart of the teaching and learning strategies used.

Communicating: To improve student success there will be several opportunities for students to share their understanding both in oral as well as written form.

Problem solving: Scaffolding of knowledge, detecting patterns, making and justifying conjectures, guiding students as they apply their chosen strategy, directing students to use multiple strategies to solve the same problem, when appropriate, recognizing, encouraging, and applauding perseverance, discussing the relative merits of different strategies for specific types of problems.

Reasoning and proving: Asking questions that get students to hypothesize, providing students with one or more numerical examples that parallel these with the generalization and describing their thinking in more detail.

Reflecting: Modeling the reflective process, asking students how they know.

Selecting Tools and Computational Strategies: Modeling the use of tools and having students use technology to help solve problems.

Connecting: Activating prior knowledge when introducing a new concept in order to make a smooth connection between previous learning and new concepts, and introducing skills in context to make connections between particular manipulations and problems that require them.

Representing: Modeling various ways to demonstrate understanding, posing questions that require students to use different representations as they are working at each level of conceptual development - concrete, visual or symbolic, allowing individual students the time they need to solidify their understanding at each conceptual stage.

Other strategies used include; Guided Exploration, Problem Solving, Graphing, Visuals, Direct Instruction, Independent Reading, Independent Study, Ideal Problem Solving, Model analysis, Logical Mathematical Intelligence, Graphing Applications, and Problem Posing.

Assessment and Evaluation Strategies of Student Performance:

Assessment is the process of gathering information that accurately reflects how well a student is achieving the curriculum expectations in a subject or course. The primary purpose of assessment is to improve student learning. Assessment for the purpose of improving student learning is seen as both “assessment for learning” and “assessment as learning”. As part of assessment for learning, teachers provide students with descriptive feedback and coaching for improvement. Teachers engage in assessment as learning by helping all students develop their capacity to be independent, autonomous learners who are able to set individual goals, monitor their own progress, determine next steps, and reflect on their thinking and learning.


Teachers will obtain assessment information through a variety of means, which may include formal and informal observations, discussions, learning conversations, questioning, conferences, homework, tasks done in groups, demonstrations, projects, portfolios, developmental continua, performances, peer and self-assessments, self-reflections, essays, and tests.


As essential steps in assessment for learning and as learning, teachers need to:

• plan assessment concurrently and integrate it seamlessly with instruction;

• share learning goals and success criteria with students at the outset of learning to ensure that students and teachers have a common and shared understanding of these goals and criteria as learning progresses;

• gather information about student learning before, during, and at or near the end of a period of instruction, using a variety of assessment strategies and tools;

• use assessment to inform instruction, guide next steps, and help students monitor their progress towards achieving their learning goals;

• analyze and interpret evidence of learning;

• give and receive specific and timely descriptive feedback about student learning;

• help students to develop skills of peer and self-assessment.


Teachers will also ensure that they assess students’ development of learning skills and work habits, using the assessment approaches described above to gather information and provide feedback to students.

The Final Grade:

The evaluation for this course is based on the student's achievement of curriculum expectations and the demonstrated skills required for effective learning. The percentage grade represents the quality of the student's overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline. A credit is granted and recorded for this course if the student's grade is 50% or higher. The final grade for this course will be determined as follows:

  • 70% of the grade will be based upon evaluations and assessments of learning conducted throughout the course. This portion of the grade will reflect the student's most consistent level of achievement throughout the course, although special consideration will be given to more recent evidence of achievement. All assessments of learning will be based on evaluations developed from the four categories of the Achievement Chart for the course.


  • 30% of the grade will be based on a final evaluation administered at the end of the course and may be comprised of one or more strategies including tests and projects.. This final evaluation will be based on an evaluation developed from all four categories of the Achievement Chart for the course and of expectations from all units of the course. The weighting of the four categories of the Achievement Chart for the entire course including the final evaluation will be as follows.


Knowledge & Understanding

Thinking, Inquiry & Problem Solving









l  70% for assessment of learning throughout the course

ü  5 Tests: 35%=5 * 7%

ü  2 Assignments: 14 %=2 * 7%

ü  3 Projects: 21% = 3* 7%


l  30% for final evaluations conducted near/at the end of the course

ü  Project= 10%

ü  Final exam= 20%

The Report Card:

The report card will focus on two distinct but related aspects of student achievement; the achievement of curriculum expectations and the development of learning skills. The report card will contain separate sections for the reporting of these two aspects.

A Summary Description of Achievement in Each Percentage Grade Range
and Corresponding Level of Achievement

Percentage Grade Range

Achievement Level

Summary Description


Level 4

A very high to outstanding level of achievement. Achievement is above the provincial standard.


Level 3

A high level of achievement. Achievement is at the provincial standard.


Level 2

A moderate level of achievement. Achievement is below, but approaching, the provincial standard.


Level 1

A passable level of achievement. Achievement is below the provincial standard.

below 50%

Level R

Insufficient achievement of curriculum expectations. A credit will not be granted.





Program Planning Considerations for Mathematics:

Teachers who are planning a program in Mathematics must take into account considerations in a number of important areas. The areas of concern to all teachers that are outlined include the following:

  • Teaching Approaches
  • Program Considerations for English Language Learners
  • Literacy and Inquiry/Research Skills
  • The Role of Information and Communication Technology in Mathematics
  • Career Education in Mathematics

Considerations relating to the areas listed above that have particular relevance for teachers planning programs in Mathematics:

Teaching Approaches. To make learning accessible to students, teachers must draw upon the prior knowledge and skills possessed by students. Students must have a solid conceptual foundation in mathematics. Students must be provided with the opportunity to learn the expectations of their mathematical curriculum in diverse ways. Teachers should make use of manipulatives in their teaching of mathematics which allow students to represent abstract ideas of math in concrete ways. Teachers will provide a rich math curriculum which will allow students to investigate and identify thus gaining experience with applications of the new math curriculum. Teachers need to promote attitudes conducive to the learning of math by showing students multiple ways of solving problems so that they gain confidence in problem solving.

Program Considerations for English Language Learners. This Mathematics course can provide a wide range of options to address the needs of ESL/ELD students. Assessment and evaluation exercises will help ESL students in mastering the English language and all of its idiosyncrasies. In addition, since all occupations require employees with a wide range of English skills and abilities, many students will learn how the operation of their own physical world can contribute to their success in their social world. Assessment and evaluation accommodations, as well as other program accommodations can and will be made to facilitate the success of the ESL or ELD students.

Literacy and Inquiry/Research Skills. Communication skills are fundamental to the development of mathematical literacy. Fostering students' communication skills is an important part of the teacher's role in the math curriculum. When reading in mathematics, students use a different set of skills than they do when reading fiction or general non-fiction. They need to understand vocabulary and terminology that are unique to mathematics, and must be able to interpret symbols, charts, diagrams, and graphs. In all math courses, students are expected to use appropriate and correct terminology, and are encouraged to use language with care and precision in order to communicate effectively. Math courses also encourage students to communicate with precision in order to communicate effectively. Students are encouraged throughout their online mathematics course to ask questions to their peers and teacher and, as well, to become proactive in the solving of their own questions through investigations.

The Role of Information and Communication Technology in Mathematics. Information and communication technology (ICT) is considered a learning tool that must be accessed by Mathematics students when the situation is appropriate. As a result, students will develop transferable skills through their experience with word processing, internet research, presentation software, and equation editors as might be expected in any environment. By using ICT tools, the students will be able to reduce the time required to perform mundane or repetitive tasks thus creating more time to be spent on higher order tasks such as thinking or concept development. The nature of the online course itself, with students enrolled from all over the world, brings the global community into the classroom.

Career Education in Mathematics. Mathematics definitely helps prepare students for employment in a huge number of diverse areas - Engineering, Science, Business, etc. The skills, knowledge and creativity that students acquire through this course are essential for a wide range of careers. Being able to express oneself in a clear concise manner without ambiguity, solve problems, make connections between this Mathematics course and the larger world, etc., would be an overall intention of this Mathematics course, as it helps students prepare for success in their working lives.

placements need to assess placements for safety and ensure that students can read and understand the importance of issues relating to health and safety in the workplace.



Nelson, Advanced Functions, © 2009 ISBN-13:9780176374433

McGraw-Hill Ryerson, Advanced Functions, © 2008 ISBN-13: 9780070266360

Access to a scanner or digital camera

Graphing Calculator,