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 origin `(0,0)`; and
The four quadrants, normally labelled I, II, III, IV.
The cartesian plane
Where did all this come from?
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)
Locate the points `A(2 , 1)` and `B(-4 , -3)` on the rectangular co-ordinate system.
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.
Three vertices of a rectangle are `A(-3 , -2)`, `B(4 , -2)` and `C(4,1)`.
Where is the fourth vertex `D`?
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:
Where are all points `(x , y)` for which `x < 0` and `y < 0`?
- `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.
The negative `x`- and negative `y`-axes are dashed to indicate they are not included in the region.
Q1 Where are all the points whose abscissas equal their ordinates?
"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`?
They are on the negative part of the y-axis, indicated by the magenta line.