# 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.

The cartesian plane

### 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.

Answer

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

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

Here is our result.

### 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

yco-ordinate of D must be `1`

The

xco-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*.

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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