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Advanced Functions and Pre-Calculus
This courseware extends students' experience with functions. Students will investigate the properties of polynomial, rational, exponential, logarithmic, trigonometric and radical functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. This courseware is considered prerequisite learning for the Calculus and Vectors courseware.
Units
Functions: Transformations and Properties
This unit introduces functions along with many terms and notations that will be encountered when working with them. A large portion of the unit deals with sketching functions using transformations. The unit concludes with a look at inverses.
Polynomial Functions
This unit examines key characteristics and properties of polynomial functions, supportive in determining the shape of their graphs. Focus will be placed on studying the behaviour of 3rd and 4th degree polynomial functions. Through investigation, connections will be made between the algebraic, numeric, and graphical representations of these functions.
Polynomial Equations and Inequalities
This unit develops the factoring skills necessary to solve factorable polynomial equations and inequalities in one variable. Connections are made between the real roots of a polynomial equation and the x-intercepts of the corresponding polynomial function. Skills are applied to solve problems involving polynomial functions and equations.
Rational Functions
This unit examines key characteristics and properties of rational functions, supportive in determining the shape of their graphs. Focus will be placed on studying rational functions with linear or quadratic polynomial expressions in their numerators and/or denominators. Through investigation, connections will be made between the algebraic and graphical representations of these functions.
Exponential and Logarithmic Functions
This unit examines key characteristics and properties of exponential and logarithmic functions. Techniques used to solve exponential and logarithmic equations will be taught and applied to solving problems.
Trigonometric Functions
In this unit, you will be introduced to functions whose values repeat over regular intervals. The most common such functions are called sinusoidal functions. These functions will be examined graphically and algebraically. The ultimate goal is to be able to solve realistic applications that can be modeled by this type of function.
Trigonometric Identities and Equations
This unit explores equivalent trigonometric expressions and examines strategies to prove trigonometric identities and solve a variety of trigonometric equations. Knowledge of fundamental trigonometric identities will be extended to include compound angle and double angle formulas.
This module will review the fundamental trigonometric identities introduced in the previous unit, and use the properties of the trigonometric ratios and transformations of the functions to recognize other equivalent trigonometric expressions. The non-permissible values of the variable in trigonometric expressions and identities will be discussed. Trigonometric expressions will be simplified algebraically and verified graphically.
Verifying trigonometric identities using specific values of the variable will be discussed. Identities will be proven algebraically using the fundamental identities (the reciprocal identities, the quotient identities and the Pythagorean identities) and verified graphically using technology.
The compound angle formulas will be developed algebraically using the unit circle and the cofunction identities. These formulas will be used to simplify trigonometric expressions and prove identities, determine exact values of trigonometric ratios, and solve certain trigonometric equations.
The double angle formulas follow directly from the compound angle formulas. These formulas will be used to simplify trigonometric expressions, prove identities, determine exact values of trigonometric ratios and solve related problems.
First and second degree trigonometric equations will be solved algebraically and graphically using technology. Identities will be used to transform equations, when necessary. Solutions will be determined over a specific domain and more generally for all real values of the variable.