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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.

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.

The procedures of long division and synthetic division used to carry out polynomial division will be studied.

The remainder theorem and factor theorem will be introduced and used to solve related mathematical problems.

This module will employ the factor theorem and rational roots theorem to factor cubic and quartic polynomial expressions in one variable.

This module will demonstrate a variety of strategies used to factor polynomial expression, such as, factoring by grouping, trinomial factoring, difference of squares, sum and difference of cubes, and techniques requiring the factor theorem.

Polynomial equations, in one variable, involving real and non-real roots, will be solved algebraically. Skills will be applied to graph polynomial functions and solve problems.

Polynomial inequalities, in one variable, will be solved using a variety of algebraic and graphical approaches.

This module demonstrates how graphing technology can be used to solve problems involving polynomial equations or inequalities.

## 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.

This module will explore the properties of exponential functions of the form \( y=c^x, c \gt 0, c \neq 1\). Transformations (reflections, stretches, and translations) of functions as they apply to exponential functions will be discussed.

Exponential equations, in one variable, will be solved using a common base.

Real-world problems involving exponential growth and decay will be modelled and solved using exponential functions.

This module will introduce logarithms and explore the properties of logarithmic functions of the form \(y=\log_c(x),c>0,c\neq1\). Transformations (reflections, stretches, and translations) of functions as they apply to logarithmic functions will be discussed.

The properties of logarithms will be discussed, and the laws of logarithms will be introduced and verified using the laws for exponents. Equivalent forms of logarithmic expressions will be identified using the properties and laws.

The properties and laws of logarithms will be used to solve exponential and logarithmic equations, in one variable.

## 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.

In this module, radian measure, an alternative way to measure angles, will be defined. The connection between radians and the circle, and radians and degrees will be established. Applications involving arc length and sector area will be solved.

The unit circle will be used to make connections between angles and trigonometric ratios. Primary and reciprocal trigonometric ratios will be defined and developed further through their connection to the unit circle.

Two special right angle triangles are used in conjunction with the unit circle to determine the exact values of the six trigonometric ratios for special angles \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\),\(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and their integer multiples. Simple trigonometric equations will also be solved.

Using the knowledge gained from the unit circle and special triangles, the graphs of the three primary trigonometric functions \(y=\sin(x)\), \(y=\cos(x)\), and \(y=\tan(x)\) will be sketched. Properties of each of these graphs will be examined and new terminology will be introduced.

In this module, the graph of each of the reciprocal functions, \(y=\csc(x)\), \(y=\sec(x)\), and \(y=\cot(x)\), will be developed from the graph of the corresponding primary trigonometric function. Properties of these graphs will be examined.

We previously developed a general equation, \(y=af[b(x-h)]+k\), for the transformations of some function, \(y=f(x)\). In this module, we apply this work to consider transformations of \(y=\sin(x)\) and \(y=\cos(x)\). New terminology specific to sinusoidal functions will be introduced.

Given specific information about a sinusoidal function, we will develop possible equations using the sine and cosine functions. Once we have a sketch and equation, we will use both to gain more information about the function.

Many real world situations can be modeled using the sine and cosine functions. We will convert the given information to a sketch and then determine a possible trigonometric equation to model the situation. The model will be useful in answering many questions arising from the specific application.

## 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.

## Operations on Functions

Graphs of functions, \( y=f(x) \pm g(x) \), formed by the addition or subtraction of two functions, will be investigated. Key properties of the functions will be discussed and factors influencing the behaviour of the graphs will be identified. Connections will be made to real-world applications to model and solve problems.

Graphs of functions, \( y=f(x)g(x) \) and \( y=\frac{f(x)}{g(x)} \), formed by the multiplication or division of two functions will be investigated. Key properties of the functions will be discussed and factors influencing the behaviour of the graphs will be identified. Connections will be made to real-world applications to model and solve problems.

The composition of two functions, \( y=f(g(x)) \), will be determined algebraically, numerically, and graphically. Problems involving the composition of functions, including real-world applications will be solved.

Equations and inequalities involving a combination of functions will be solved numerically and graphically, with and without graphing technology. In particular, a graphical and algebraic approach to solving radical equations will be discussed.

## Rates of Change

In this module, secants and tangents will be used to develop the concept of average and instantaneous rates of change. These concepts will be examined through real-world applications involving rates of change.

The topic of average and instantaneous rates of change will be examined with exponential and trigonometric functions. Real-world applications will be used to pursue these ideas.

The idea of a limit will be introduced in this module. This idea will be used to determine the slope of a tangent to a point on a quadratic function and to then determine the equation of the tangent at that point.

This module further extends the work of the previous module. The method previously introduced will be applied to other functions seen in this course.