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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.
Through investigation, connections will be made between the graphs of a linear or quadratic polynomial function, \(y=f(x)\), and its reciprocal function, \(y=\frac{1}{f(x)}\). The relationship between key features of the polynomial function and its reciprocal rational function will be used to graph rational functions of this form.
The focus of this module is on identifying the vertical asymptotes and/or points of discontinuity of a rational function and studying the behaviour of the graph to the left and to the right of the discontinuity.
The focus of this module is on identifying any horizontal or oblique asymptote of a rational function, if it exists, and studying the end behaviour of the graph of the function about these asymptotes.
Rational functions with linear expressions in the numerator and denominator will be analyzed and graphed, applying knowledge of domain, asymptotes, and \(x\)- and \(y\)-intercepts. Connections are made between functions of this form and transformed reciprocal functions of the form \(y=\frac{a}{b(x-h)}+k\).
This module will employ graphing skills covered previously to analyze and compare the graphs of a set of rational functions identifying similarities and differences, or make predictions about the graphs of rational functions with quadratic expressions in the numerator and/or denominator.
Rational equations, in one variable, are solved algebraically and using graphing technology. Procedures are applied to solve real-life problems.
Rational inequalities, in one variable, are solved using algebraic and graphical approaches and using graphing technology.
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.