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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.
This module introduces terminology associated with polynomial functions. The properties and graphs of the cubic and quartic power functions will be studied.
This module will review and extend knowledge of transformations (reflections, stretches, and translations) to include the cubic and quartic power functions.
This module explores the behaviour of polynomial functions (of degree \(\le6\)), using technology. The possible number of turning points and shape of the graph, the possible number of x-intercepts, and the end behaviour of the polynomial function will be studied.
Through an investigation, connections will be made between the multiplicity of the linear factors, of a polynomial function in factored form, and the behaviour of the graph of the function at its x-intercepts. This knowledge, along with the concepts covered in the previous module, will be applied to sketch the graph of a polynomial function from its equation in factored form.
This module demonstrates the procedures used to determine the general equation of a family of polynomial functions sharing a common set of zeros (real and non-real), and the specific equation of a member of the family, which passes through another given point.
This module will review and extend understanding of finite differences to include cubic and quartic polynomial functions. Connections between the constant finite differences, the degree of the polynomial, and the coefficient of the highest degree term will be identified.
This module will explore, algebraically and graphically, symmetry in polynomial functions. Properties of the equation of the function that will assist in recognizing even/odd symmetry when it occurs in polynomial functions will be identified.
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.