MHF4U – Advanced Functions Grade 12

Course Type: University Preparation

Credit: 1.0

Ontario Curriculum: MHF4U – Advanced Functions Grade 12

Includes:

• Gizmos Simulation Labs
• Video submissions
• Whiteboard activities
• Interactive content

MHF4U Prerequisites: MCR3U – Functions, Grade 11 or MCT4C – Mathematics for College Technology, Grade 12

NCAA Approved Course

MHF4U Advanced Functions Grade 12 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 in MHF4U will also refine their use of the mathematical processes necessary for success in senior mathematics. This MHF4U 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.

MHF4U Course Outline

Unit 1 – Polynomial Functions
Students will investigate polynomial functions. They will extend their knowledge about linear and quadratic
functions to include cubic, quartic and quintic functions. Students will explore their graphs and
characteristics, also distinguish polynomial functions from sinusoidal and exponential functions, and
compare and contrast the graphs of various polynomial functions with the graphs of other types of
functions. They will develop skills in how to factorize polynomial functions to the 5
th degree and graph these
functions with transformation applied. Students will determine, through investigation with and without
technology, key features (i.e. domain and range, intercepts, positive/negative intervals,
increasing/decreasing intervals) of the graphs of polynomial functions. Students will solve problems
involving applications of polynomial functions and equations and explain the difference between the
solution to an equation in one variable and the solution to an inequality in one variable, also demonstrate
that given solutions satisfy an inequality and determine solutions to polynomial inequalities in one variable
by graphing the corresponding functions, using graphing technology, and identifying intervals for which x
satisfies the inequalities.
Unit 2 – Rational functions
Students will investigate rational functions. Students will determine, through investigation with and without
technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts,
positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the
reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical
representations of these rational functions. Students will solve problems involving applications of simple
rational functions and equations and explain the difference between the solution to an equation in one
variable and the solution to an inequality in one variable, also demonstrate that given solutions satisfy an
inequality and determine solutions to simple rational inequalities in one variable by graphing the
corresponding functions, using graphing technology, and identifying intervals for which x satisfies the
inequalities.
d of 2 π , amplitude of 1) in terms of radians. Students represent a
sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians.
Students recognize that trigonometric identities are equations that are true for every value in the domain,
prove trigonometric identities through the application of reasoning skills, using a variety of relationships,
and verify identities using technology.
Unit 5 – Algebra of Functions and Rates of Change
Students extend their knowledge about characteristics of different functions to key features as domain,
range, maximum/minimum points, number of zeros of the graphs of functions created by adding,
subtracting, multiplying, or dividing functions, and describe factors that affect these properties. Students
will also investigate the composition of two functions [i.e., f(g(x))] numerically (i.e., by using a table of
values) and graphically, with technology, for functions represented in a variety of ways (e.g., function
machines, graphs, equations), and interpret the composition of two functions in real-world applications.
Students make connections, through investigation, between the slope of a secant on the graph of a function
(e.g., quadratic, exponential, sinusoidal) and the average rate of change of the function over an interval, and
between the slope of the tangent to a point on the graph of a function and the instantaneous rate of
change of the function at that point.

Unit 1:
By the end of this course, students will:
● identify and describe some key features of polynomial functions, and make connections between
the numeric, graphical, and algebraic representations of polynomial functions;
● solve problems involving polynomial graphically and algebraically;
● demonstrate an understanding of solving polynomial inequalities.

Unit 2:
By the end of this course, students will:
● identify and describe some key features of the graphs of rational functions, and represent rational
functions graphically;
● solve problems involving simple rational equations graphically and algebraically;
● demonstrate an understanding of solving simple rational inequalities.

Unit 3:
By the end of this course, students will:
● demonstrate an understanding of the relationship between exponential expressions and logarithmic
expressions, evaluate logarithms, and apply the laws of logarithms to simplify 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.

Unit 4:
By the end of this course, students will:
● 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.

Unit 5:
By the end of this course, students will:
● 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 modeling and reasoning with
functions, including problems with solutions that are not accessible by standard algebraic
techniques.

In this course, students will experience the following activities.

Presentations with embedded videos are utilized to outline concepts, explain theory with the use of
examples and practice questions, and incorporate multi-media opportunities for students to learn more
(e.g. online simulations, quizzes, etc.).

End of unit conversations and Poodlls are opportunities for students to express their ideas, problem
solving, and thought processes with a teacher who provides timely feedback.
Reflection is an opportunity for students to look back at concepts and theories with new eyes, to relate
theory to practice, and to align learning with their own values and beliefs.

Discussions with the instructor are facilitated through video conferencing, discussing the concepts and
skills being studied. This enables two-way communication between the student and the instructor, to share
ideas and ask questions in dialogue. This also helps to build a relationship between the student and
instructor.

Instructor demonstrations (research skills, etc.) are opportunities for the instructor to lead a student
through a concept or skill through video conferencing, videos, or emailing with the student.

Discussion forums are an opportunity for students to summarize and share their ideas and perspectives
with their peers, which deepens understanding through expression. It also provides an opportunity for
peer-to-peer feedback.

Practical extension and application of knowledge are integrated throughout the course. The goal is to help
students make connections between what they learn in the classroom and how they understand and relate
to the world around them and their own lives. Learning becomes a dynamic opportunity for students to be
more aware that their learning is all around them and enable them to create more meaning in their lives.

Individual activities/assignments assessments are completed individually at a student’s own pace and are
intended to expand and consolidate the learning in each lesson. Individual activities allow the teacher to
accommodate interests and needs and to assess the progress of individual students. For this reason,
students are encouraged to discuss IEPs (Individual Education Plans) with their teacher and to ask to modify
assessments if they have a unique interest that they feel could be pursued in the assessment. The teacher
plays an important role in supporting these activities by providing ongoing feedback to students, both orally
and in writing.

Research is an opportunity to apply inquiry skills to a practical problem or question. Students perform
research to gather information, evaluate quality sources, analyze findings, evaluate their analysis, and
synthesize their findings into conclusions. Throughout, students apply both creative thinking and critical
thinking. New questions are also developed to further learning.

Writing as a learning tool helps students to think critically about course material while grasping, organizing,
and integrating prior knowledge with new concepts. Good communication skills are important both in and
out of the classroom.

Virtual simulations are interactive websites that provide students with an opportunity to ask questions,
relate variables, and examine relationships.

Diagrams are visual representations of mathematical ideas and concepts. They provide another perspective
to organize ideas. Visuals are thought to promote cognitive plasticity – meaning, they can help us change
our minds or help us to remember an idea.

Graphs and charts are visual representations of math concepts and analysis. This helps us to see the
relationships within and between sets of data.

Tables involve organizing information in terms of categories (rows and columns). This helps us to
understand the relationships between ideas and data, as well as highlight trends.

Practice problems provide students with a scenario/problem to solve by applying concepts and skills
learned in a context. This helps students to understand the relevance of their learning.

As summarized in Growing Success 2010, the primary purpose of assessment and evaluation is to improve student learning. Information gathered through assessment helps teachers to determine students’ strengths and weaknesses in their achievement of the curriculum expectations in each course. This information also serves to guide teachers in adapting curriculum and instructional approaches to students’ needs and in assessing the overall effectiveness of programs and classroom practices. As part of assessment, teachers provide students with descriptive feedback that guides their efforts towards improvement.

Evaluation refers to the process of judging the quality of student work on the basis of established criteria, and assigning a value to represent that quality. All curriculum expectations must be accounted for in instruction, but evaluation focuses on students’ achievement of the overall expectations. A students’ achievement of the overall expectations is evaluated on the basis of his or her achievement of related specific expectations. Teachers will use their professional judgement to determine which specific expectations should be used to evaluate achievement of overall expectations, and which ones will be covered in instruction and assessment but not necessarily evaluated.

In order to ensure that assessment and evaluation are valid and reliable, and that they lead to the improvement of student learning, teachers must use assessment and evaluation strategies that:

● Address both what students learn and how well they learn;
● Are based both on the categories of knowledge and skills and on the achievement level descriptions given in the achievement chart
● Are varied in nature, administered over a period of time, and designed to provide opportunities for students to demonstrate the full range of their learning;
● Are appropriate for the learning activities used, the purposes of instruction, and the needs and experiences of the students;
● Are fair to all students;
● Accommodate students with special education needs, consistent with the strategies outlined in their Individual Education Plan;
● Accommodate the needs of students who are learning the language of instruction;
● Ensure that each student is given clear directions for improvement;
● Promote students’ ability to assess their own learning and to set specific goals
● Include the use of samples of students’ work that provide evidence of their achievement;
● Are communicated clearly to students and parents at the beginning of the school year and at other appropriate points throughout the school year.

The final grade will be determined as follows:

❑ 70% of the grade will be based on evaluation conducted throughout the course. This
portion of the grade should reflect the student’s most consistent level of achievement
throughout the course, although special consideration will be given to more recent
evidence of achievement.

❑ 30% of the grade will be based on a final evaluation administered at or towards the end of
the course. This evaluation will be based on evidence from one or a combination of the
following: an examination, a performance, and/or another method of evaluation suitable to
the course content. The final evaluation allows the student an opportunity to demonstrate
comprehensive achievement of the overall expectations for the course.

(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.

To learn more about this course including tests and exams please visit our FAQ page