# 3. Rectangular Coordinates

A good way of presenting a function is by graphical representation.

Graphs give us a visual picture of the function.

The most common way to graph a function is to use the rectangular co-ordinate system. This consists of:

The x-axis;

The y-axis;

The origin (0,0); and

The four quadrants, normally labelled I, II, III, IV.

### Where did all this come from?

Rene Descartes

The x-y coordinate system is also called the Cartesian Coordinate system, after its developer, Rene Descartes (1596 - 1650). This graphing system was incredibly important for the advancement of science and engineering.

Normally, the values of the independent variable (generally the x-values) are placed on the horizontal axis, while the values of the dependent variable (generally the y-values) are placed on the vertical axis.

The x-value, called the abscissa, is the perpendicular distance of P from the y-axis.

The y-value, called the ordinate, is the perpendicular distance of P from the x-axis.

The values of x and y together, written as (x, y) are called the co-ordinates of the point P.

It's called the "rectangular" coordinate system because the scale used along the x-axis is evenly spaced, as is the scale along the y-axis. Other systems exist where the scale is not even (see Log-log and semi-log graphs) and some are even circular (see Polar Coordinates)

### Example 1

Locate the points A(2 , 1) and B(-4 , -3) on the rectangular co-ordinate system.

Answer

To answer this properly, we need to do the following:

1. Label the axes with x and y.
2. Put a scale on the axes (the numbers) such that the points will fit on the graph.
3. Then put dots for the required points A and B.

Here is our result.

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Continues below

### Example 2

Three vertices of a rectangle are A(-3 , -2), B(4 , -2) and C(4,1).

Where is the fourth vertex D?

Answer

Here are the positions of points A, B and C:

Since the opposite sides of a rectangle are equal and parallel, we can see that:

The y co-ordinate of D must be 1

The x co-ordinate of D must be -3

We conclude the co-ordinates of D are (-3, 1).

Here's our completed rectangle:

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### Example 3

Where are all points (x , y) for which x < 0 and y < 0?

Answer

We have:

• x < 0 means that x is negative,
• and y < 0 also means that y is negative,

So the only region where both co-ordinates for all points are negative is the "third quadrant (III)".

The shaded area represents the region in question.

x
y

The negative x- and negative y-axes are dashed to indicate they are not included in the region.

Easy to understand math videos:
MathTutorDVD.com

### Exercises

Q1 Where are all the points whose abscissas equal their ordinates?

Answer

"Abscissas" means x-values, while "ordinates" means y-values.

So the question means "where on the rectangular system do we have x = y for all points (x, y)?"

In other words, we want a line connecting points like (-3, -3) and (0, 0) and (5, 5) and (700, 700).

The line we want cuts the first and third quadrants in half at 45^@. We can write this line as y = x.

y=x

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Q2 Where are all the points (x, y) for which x = 0 and y < 0?

Answer

They are on the negative part of the y-axis, indicated by the magenta line.