### Introduction

Engineers often find mathematical ideas easier to understand when these are portrayed visually as opposed to algebraically. Graphs are a convenient and widely-used way of portraying functions. By inspecting a graph it is easy to describe a number of properties of a function. For example, where is the function positive, and where is it negative? Where is it increasing and where is it decreasing? Do function values repeat? Questions like these can be answered once the graph of a function has been drawn. In this Section we will describe how the graph of a function is obtained and introduce various terminology associated with graphs.

We have seen in Section 2.1 that it is possible to represent a function using the form
$y=f\left(x\right)$
. An alternative representation is to write expressions for both
$y$
and
$x$
in terms of a third variable known as a
**
parameter
**
. The variables
$t$
or
$\theta $
are normally used to denote the parameter.

For example, when a projectile such as a ball or rocket is thrown or launched, the $x$ and $y$ coordinates of its path can be described by a function in the form $y=f\left(x\right)$ . However, it is often useful to also give its $x$ coordinate as a function of the time after launch, that is $x\left(t\right)$ , and the $y$ coordinate similarly as $y\left(t\right)$ . Here time $t$ is the parameter.

#### Prerequisites

- understand what is meant by a function

#### Learning Outcomes

- draw the graphs of a variety of functions
- explain what is meant by the domain and range of a function