Graphs

An introduction to Coordinate Geometry

Introduction

Mathematics can be applied to many aspects in life and play very important rolls in are day to day activity like personal budget matters.

In mathematics, there are many functions in algebra involving two variables, (usually x and y). These are known as Functions of Two Variables. Below are some examples of these types of functions:

y = 2x + 1
y = 3x² - 5x + 3
y = Sin(x)
x² + y² = 4
y² = x³ - Tan(x)

These algebraic functions are all different and have different properties. It is often useful to have a visual way to look at these functions. Graphs provide such a tool.

In this essay we will study graphs, how they are constructed and the kind of information they yield.

Cartesian Coordinates

Because we are looking at two variables we can look at them in two dimensions on a flat surface. We draw two lines at right angles as shown below.

The horizontal line (from left to right) is called the x axis. The vertical line (up and down) is called the y axis. The point where the two axes meet, is called the Origin.

Both axes are numbered. The x axis has values that increase from left to right and decrease from right to left. The y axis has values that increase upwards and decrease downwards.

This is called the Cartesian Coordinate System. The word Cartesian is a Latinised version of the name of the philosopher, Rene Descartes who invented the system.

A point can be located on this coordinate system by giving its x value followed by its y value. This is written in the form

(x, y)

The diagram below shows the location of four points, P, Q, R and S.

For example, point P has coordinates (2, 3). This means that to get to P, you must begin at the origin and move two units to the right along the x axis then three units up along the y axis. Point S has coordinates (-5, 4). This means five units to the left along the x axis followed by four units up along the y axis. Any point can be described with a pair of numbers.

Drawing Graphs

We will now go through the process of drawing a simple graph. We will use the equation

y = 2x + 1

We do this by substituting different values of x into this equation and finding the corresponding values of y. These calculations are performed in the table below.

x	2x	y = 2x + 1
-3	-6	-5
-2	-4	-3
-1	-2	-1
0	0	1
1	2	3
2	4	5

This table gives a list of coordinates for pairs. For example, when x is 1, y has the value of 3. This gives the point (1, 3), etc. The points from the table are drawn on the diagram below.

These points can be joined together to give a straight line. This straight line is the graph of the equation y = 2x + 1.

Any equation of the form y = ax + b (where a and b are numbers) will give a graph that is a straight line.

The value of a is the slope of the line. A positive value of a makes the graph slope up from left to right. A negative value of a makes the graph slope down from left to right.

The value of b indicates the point on the y axis that the line will cross.

In the graph above (y = 2x + 1), a has the value 2 so the slope is 2 and is positive. The value of b is 1 so the line crosses the y axis at the point y = 1.

The straight line is the simplest type of graph. Other types of equations will give graphs of different shapes. Some of these are shown next.

Selected Graphs

The following table shows selected graphs. The letters a, b, c, d, e represent numbers.

Graph	Name and Sample Equation
	Straight Line y = ax + b
	Straight Line y = -ax + b
	Parabola y = ax² + bx + c
	Parabola y = -ax² + bx + c
	Cubic Parabola y = ax³ + bx² + cx + d
	Circle x² + y² = a²
	Ellipse (x² / a²) + (y² / b²) = 1
	Rectangular Hyperbola y = (a / x)
	Sine Wave y = a.Sin(bx + c) OR y = a.Cos(bx + c)
	Tangent y = a.Tan(bx + c)
	Hyperbolic Sine y = a.Sinh(bx + c)
	Catenery y = a.Cosh(bx + c)
	Gauss Distribution y = Exp(-x²)