# 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

### Need Graph Paper?

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

### Example 2

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

Where is the fourth vertex `D`?

### Example 3

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

### Exercises

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

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

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