Learning Resources
Overview
You have had a wealth of experience with functions thus far in your mathematical career. This chapter provides opportunities to extend your knowledge as you encounter more types of functions. Previously, you have modeled many real-life phenomena such as motions and tides. This unit of study will give you the tools needed to analyze more phenomena such as pressure and volume for a gas, stopping rate for a car, cooling rates and continuously compounded interest. As you can see, you will be making the connection between mathematics, science and finance.
The new types of functions you will encounter will include rational functions, square-root functions, absolute-value functions and other piece-wise defined functions. As well you will broaden your understanding of exponential and logarithmic functions to include those with base e.
This unit of study contains seven sections as outlined below.
Section 3.1: introduces rational functions, beginning with simple examples having numerator 1. You will explore asymptotic behaviour, both vertical and horizontal, as the graphs of these reciprocals of polynomial functions are examined. Connections to Science are made as you investigate the pressure and volume relationship for a gas.
Section 3.2: proceeds to more general rational functions in which the numerator is also a polynomial function. Oblique asymptotes and points of discontinuity will also be introduced.
Section 3.3: involves the simplification of rational expressions and the use of these skills to solve application problems that use rational equations. In addition, you will analyze rational functions to determine the domain and range, points of discontinuity, asymptotes, and intervals of increase and decrease.
Section 3.4: provides extensive work with square root functions. You will determine the domain and range, plot the graphs, and use these functions to model real-world phenomena such as the stopping distance for a car traveling at various speeds.
Section 3.5: reintroduces the absolute-function and examines it in detail. You will analyze these functions to determine their domain and range, and plot the graphs. As well, you will solve inequalities that involve absolute-value functions.
Section 3.6: provides support for the concept of continuity. Piece-wise functions will be re-examined.
Section 3.7: introduces the number e. You will extend your understanding of exponential and logarithmic functions to include y = ex and y = lnx. These functions, in turn, will be used to solve real-world applications such as continuously compounded interest.
Graph paper is essential for many of the activities in this chapter. This unit of study should take approximately 30 hours to complete.