Apex Academy Courses

Advanced Functions, Grade 12


Advanced Functions, Grade 12

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.

Prerequisite: Functions, Grade 11, University Preparation or Mathematics for College Technology, Grade 12, College Preparation

Course Outline

Module 1 Overview: Polynomials

Guiding Question:
How can the properties of polynomial functions determine their graphs and applications?

In this module, we extend our understanding of quadratic functions to higher degree polynomials. We focus on the characteristics of cubic and quartic polynomials. We consider patterns in their graphs as well as strategies to solve polynomial equations and inequalities.

Module 2 Overview: Rational Functions

Guiding Question:
How do the parameters in a rational function determine its properties and its graph?

In this module, we extend our understanding of polynomial functions to examine the characteristics of rational functions. We will practice graphing rational functions, solving rational equations and using tables for rational inequalities.

Module 3 Overview: Exponential and Logarithmic Functions

Guiding Question:
What is the inverse relationship between exponential and logarithmic functions?

In this module, we will review exponent laws, properties, and graphs. We will investigate the inverse function of an exponential function, called a logarithmic function. We can use the properties of exponents and logarithms to solve equations and to model word problems.

Module 4 Overview: Trigonometric Functions I

Guiding Question:
How can the sine and cosine graphs (in radian measure) model periodic behaviour?

In this module, we use radians to describe angles, to develop trigonometric graphs and to solve word problems about velocity. We compare the primary trigonometric graphs to the reciprocal graphs.

Module 5 Overview: Trigonometric Functions II

Guiding Question:
What is the difference between solving a trigonometric equation and a trigonometric identity?

In this module, we use radian measure, special triangles and the CAST rule to solve trigonometric equations. We also develop compound angle and double angle formulas to calculate exact trigonometric ratios. We use identities and other algebraic processes to prove new identities.

Module 6 Overview: Rates of Change and Combinations of Functions

Guiding Question:
How do rates of change describe functions? How can functions be combined through different operations?

In this module, we combine pairs of functions from the previous modules (polynomial, rational, exponential, logarithmic, sinusoidal) using operations of addition, subtraction, multiplication, division, as well as the composition of functions.

Assessment Breakdown

Course Work, Assignments, and Proctored Tests: 70%
Culminating Independent Study Project: 10%
Final Exam: 20%