Advanced Functions - 12

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; 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.

Prerequisite: MCR3U - Functions Grade 11 or MCT4C Mathematics for College Technology


Course Outline

Unit 1: Polynomial and Rational Functions (40 hours)

This unit will focus on the key properties and features of graphs of Polynomial & Rational Functions and their transformations.  Functions help you make sense of the world around you.  Many ordinary measuring devices like car odometers or bar code readers are based on mathematical functions.  Many natural occurrences like water droplets in a pond or the explosion of a supernova can be modeled by mathematical functions.  In this unit you will expand your knowledge of transformations while exploring radical and polynomial functions.  


1-1 Power Functions
1-2 Characteristics & Graphs of Polynomial Functions                          
1-3 Transformation of Polynomial Functions                                                     
1-4 Dividing Polynomials
1-5 Remainder Theorem & Factor Theorem
1-6 Polynomial Equations & Inequalities
1-7 Graphs of Reciprocal Functions
1-8 Graphs of Rational Functions
1-9 Rational Equations and Inequalities


Unit 2: Trigonometric Functions (30 hours)

In this unit, we will extend your knowledge of trigonometric ratios to develop trigonometric functions.  Trigonometry is used extensively in our daily lives from digitally recorded MP3 songs to locating your position using Global Positioning Systems (GPS).  We will learn how to graph and transform sinusoidal functions using radian measure.  Develop equations of sinusoidal functions from graphs and descriptions expressed in radian measure.  Solve problems graphically that be modelled using sinusoidal functions.  Prove complex trigonometric identities and solve linear and quadratic trigonometric equations.


2-1 Radian Measures and Special Triangles
2-2 Equivalent Trigonometric Expressions
2-3 Transformations of Sinusoidal Functions
2-4 Compound and Double Angle Formulas                                           
2-5 Trigonometric Identities
2-6 Trigonometric Equations


Unit 3: Exponential and Logarithmic Functions (20 hours)

In this unit, you will demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions.  Exponential and logarithmic functions can be used to describe and solve a wide range of problems that include determining the interest gained in bank deposits, car loan payments, the length of time medication stays in the bloodstream, etc.  You will learn to identify and describe some key features of logarithmic functions, and solve related problems graphically, solve exponential and simple logarithmic equations algebraically, including those in problems arising from real-world applications. 


3-1 Characteristics of the Exponential Function and its Inverse
3-2 Transformations of Logarithmic Functions
3-3 Properties and Laws of Logarithmic Functions                                      
3-4 Solving Exponential Equations
3-5 Solving Logarithmic Equations
3-6 Applications of Exponential and Logarithmic Functions


Unit 4: Characteristics of Functions (20 hours)

This is the final unit of the course.  Throughout the previous units, you have learned advanced techniques for interpreting a variety of functions.  Understanding functional relationships between variables is a cornerstone to further studies at university levels in disciplines such as engineering, physical sciences, computer science, and social sciences.  Relationships among these variables can be complex and may involve a combination of two or more functions.  In this unit, you will learn techniques to analyze various combinations of functions and solve real-world problems requiring these techniques.  You will also demonstrate an understanding of the average and instantaneous rate of change, and interpret the average rate of change of a function over a given interval, as well as the instantaneous rate of change of a function at a given point.


4-1 Sum & Differences of Functions
4-2 Products & Quotients of Functions                                                     
4-3 Compositions of Functions
4-4 Rate of Change