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
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)

Continues below

Example 1

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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`?