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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)

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.

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.

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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.

1 2 3 4 -1 -2 -3 -4 1 2 -1 -2 -3 -4 x y
`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.

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