By noting, for example, that stones fall faster than leaves, the Greek philosopher Aristotle (about 350 b.c.e.) reasoned that heavier objects fall faster than lighter ones.

For about 2000 years, this idea was generally accepted. Then in about 1600, the Italian scientist Galileo showed, by dropping objects from the Leaning Tower of Pisa, that the distance an object falls in a given time does not depend on its weight.

Galileo is generally credited with first using the experimental method by which controlled experiments are used to study natural phenomena. His aim was to find mathematical formulas that could be used to describe these phenomena. He realized that such formulas provided a way of showing a compact and precise relation between the variables.

In technology and science, determining how one quantity depends on others is a primary goal. A rule that shows such a relation is of great importance, and in mathematics such a rule is called a function. This chapter starts with a discussion of functions.

Examples of such relations in technology and science, as well as in everyday life, are numerous. Plant growth depends on sunlight and rainfall; traffic flow depends on roadway design; the sales tax on an item depends on the cost of an item; the time to access the Internet depends on how fast a computer processes data; distance traveled depends on time and speed of travel; electric voltage depends on the current and resistance. These are but a few of the innumerable possibilities.

A way of actually seeing how one quantity depends on another is by means of a graph. The basic method of graphing was devised by the French mathematicians Descartes and Fermat in the 1630s from their work to combine the methods of algebra and geometry. Their work was very influential in later developments in mathematics and technology. In this chapter, we discuss methods for graphing functions including the use of the graphing calculator.