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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.

The module begins with the definition of a function and how functions are represented. Domain and range, function notation, and composite functions are introduced and developed in this module.

Often in mathematics, solutions lie on an interval. Three techniques used to describe intervals are presented in this module. The module concludes with an example that is used to present definitions of concepts that will be used throughout the course.

Is there a difference in how an inequality is solved versus the solution of an equation? In this module, this question is examined and a variety of different examples are solved.

In this module we look at the graphs of five base functions: the quadratic function, the square root function, the reciprocal function, the exponential function, and the absolute value function. For each function, we will look at efficient ways to sketch the graph, discuss domain and range, and make observations about some features of each graph.

Our examination of transformations begins with a look at translations. In this module, we develop a general rule for translations, we sketch graphs given a base function and translation, we determine equations of graphs that have been translated, and we find the translation that has been applied to one function to obtain another.

How is the graph or equation of a function affected by a reflection about the \(x\)- or \(y\)-axis? In this module, we develop a general rule for these reflections, we examine even and odd functions, we sketch images which are reflections of base functions, and we determine equations of functions which have been reflected.

When a function is stretched about the \(x\)- or \(y\)-axis, what is the effect on the graph and the equation? In this module, we develop a general rule for these types of transformations, we sketch various functions given a base graph and stretch using words or mapping notation, and we determine the transformation given the equation or graph of the pre-image and image.

In this module, we put all of the transformation pieces together. By looking at an equation, we determine which base function is being transformed and which transformations have been applied. Two methods are presented for sketching functions using transformations.

### 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.