Forest Hill Collegiate Institute

Ontario Science Centre Science School Course Syllabus MHF4U1

1. Course Details Instructor: Gillian Evans (2010-2011) Course title: Advanced Functions, Grade 12 University Preparation (MHF 4U). Credit Value 1.0 Prerequisites(s): Functions, Grade 11 University Preparation (MCR 3U) or Mathematics for College Technology, Grade 12 College Preparation (MCT 4C) Textbook: Advanced Functions 12 University Guide, McGraw-Hill Ryerson, 2008

2. Overall Goals 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.

In polynomial and rational functions . identify and describe some key features of polynomial functions, and make connections between the numerical, 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.

In trigonometric functions . 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 solve trigonometric identities.

In exponential and logarithmic functions . 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.

In characteristics of functions . 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. • Specific Curriculum Expectations Please refer to Ontario Ministry of Education curriculum document for details of Overall and Specific Expectations, found at http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf

3. Assessment and Evaluation The primary purpose of assessment and evaluation is to improve student learning. Assessment is the process of gathering information from assignments, demonstrations, projects, performances, and tests that accurately reflects how well a student is achieving the curriculum expectations in a course. As part of assessment, teachers provide students with 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. In Ontario secondary schools, the value assigned will be in the form of a percentage grade. In this course, evaluation strategies including tests, quizzes, assignments, investigations, the innovation project and a final exam will be used.

i) Achievement Chart The achievement chart provides a standard, province-wide method for teachers to use in assessing and evaluating their students’ achievement. Students are evaluated according to the major categories or strands in each course. Ministry curriculum documents provide detailed description of student achievement levels.

In this course, students are evaluated in four strands, according to the weightings shown: Knowledge/ Thinking/Inquiry Communication Application Understanding 35% 15% 15% 35%

ii) Course Work (70% of final grade) • Students need to demonstrate achievement of all the overall expectations of the course. 70% of the final mark in the course will be based on work done prior to the culminating activities. Evaluations that are late, missing, and/or incomplete will affect a student’s 70% grade.

iii) Course Culminating Activities (30% of final grade) • A formal final examination worth 20% will be written in January. The Innovation Project is worth 10%.

Evaluation Format Weighting Tests, Quizzes, Assignments, Investigations 63% Portfolio Project 7% Innovation Project 10% Final Exam 20%

iv) Missed tests/quizzes policy Unit tests will be announced at least one week in advance.

It is expected that all students will write all tests and cumulative tests as a class group. If a student is unable to write the evaluation with the class because of a) previously scheduled appointments; b) school-sanctioned excursions or sporting events; c) recognized religious events; d) a death in the family; or e) a court date, then the student must inform the teacher at least two school days in advance of the test so that alternate arrangements can be made.

If these procedures are not followed, it is possible that a mark of zero will be assigned. Course Calendar: NOTE: The order of lessons described herein is subject to change.

UNIT 1: Polynomial Functions

Factoring Review

Characteristics of Functions

Introduction to Polynomial Functions

Equations and Graphs of Polynomial Functions

Transformations (Assignment)

Even/Odd Revisited

Quest - Introduction to Functions

UNIT 2: Polynomial Equations and Inequalities

The Remainder Theorem

The Factor Theorem

Factoring Sum and Difference of Cubes

Quiz – Factoring Polynomials

Polynomial Equations

Families of Polynomial Functions

Solving Inequalities Using Technology (Assignment)

Solve Factorable Polynomial Inequalities Algebraically

Review

Unit Test

UNIT 3: Rational Functions

Properties of Rational Functions

Graphing Rational Functions (Assignment)

Quiz – Rational Functions

Solve Rational Equations and Inequalities

Making Connections With Rational Functions and Equations

Review Unit Test

UNIT 4: Trigonometry and Trigonometric Functions

Trigonometry Review

Radian Measure

Graphs of Sine, Cosine, and Tangent Functions

Graphs of Reciprocal Trigonometric Functions (Assignment)

Sinusoidal Functions of the Form f (x)  asin[k(x  d)]  c f (x)  a cos[k(x  d)]  c

Review

Mid-Unit Test

Trigonometric Ratios and Special Angles

Equivalent Trigonometric Expressions

Compound Angle Formulas

Prove Trigonometric Identities

Solve Trigonometric Equations

Making Connections

Review

UNIT 5: Exponential and Logarithmic Functions

Equivalent Forms of Exponential Equations

Techniques for Solving Exponential Equations

Introduction to Exponential Functions

Quiz – Exponential Equations and Functions

Logarithms

Transformations of Logarithmic Functions

Power Law of Logarithms Product and Quotient Laws for Logarithms

Techniques for Solving Logarithmic Equations (Assignment)

Making Connections: Logarithmic Scales in the Physical Sciences

Making Connections: Mathematical Modelling With Exponential and Logarithmic Equations (Assignment)

Review

UNIT 6: Combining Functions

Sums and Differences of Functions

Products and Quotients of Functions

Composite Functions

Inequalities of Combined Functions

Making Connections: Modelling With Combined Functions

Review

UNIT 7: Rates of Change

Slopes of Secants and Average Rate of Change

Slopes of Tangents and Instantaneous Rate of Change

Finding Rate of Change for Various Function Types

Applications of Rate of Change

Review

Final Exam