The Rectangular Coordinate System
When we write down a formula for some quantity, y, in terms of
another quantity, x, we are expressing a relationship between the
two quantities. For example, if we use the symbols
y = the area of a square
and
x = the length of one side of the square,
then the formula
y = x^{ 2}
tells us the relationship between the length of a side of a
square and the area of the square. It tells us that to calculate
the area of the square, we must raise the length of one side of
that square to the power 2 (which is one reason why raising a
number to the power 2 is often referred to as
“squaring” the number).
We can use this formula to calculate the value of y for any
particular value of x. This formula relating y to x is itself
informative, but often we can understand the nature of the
relationship between y and x even better if we have a visual
image of its characteristics as well. This is where graphing
formulas is helpful.
A graph of a relationship is a way of drawing
points and other geometric shapes at locations representing the
values of x and y. This is most commonly done using a socalled rectangular
coordinate system . When the formula expresses y in
terms of x, the coordinate system is usually arranged as:
 the vertical axis and the horizontal axis (often called
the yaxis and the xaxis, respectively, if the two
variables are y and x) intersect at a central point
called the origin , which corresponds to y = 0 and x = 0.
1 2 3 4 5 2 3 4 5 1 2 3 4 5 2 1 3 4 5 0 \fs34 y
x horizontal axis or x axis vertical axis or y  axis
positive values of y positive values of x negative values
of y negative values of x the “origin” (x = 0,
y = 0)
 a numerical scale is created on each axis. Values on the
horizontal axis increase from 0 at the origin though
positive values to the right, and from 0 at the origin
through negative values: 1, 2, 3, …, etc., to the
left. The scales along the axes should be uniform –
the values of x should be spaced uniformly along the
length available.
 values on the vertical scale increase from 0 at the
origin through positive values as you go upwards, and
from 0 at the origin through negative values: 1, 2, 3,
…, etc., as you go down.
These two axes complete with the explicitly labelled numerical
scales form what is called a rectangular coordinate system.
Then, the corresponding pair of values, x = a and y = b,
written as a pair of numbers in this order in brackets, (a, b),
corresponds to, or is plotted as a point at the
location where the vertical line through x = a intersects the
horizontal line through y = b:
The values x = a and y = b here are called the coordinates or
rectangular coordinates of the point. You can also think of the
coordinates, (a, b), of a point as indicating that to get to the
point from the origin, you need to move ‘a’ units
horizontally and ‘b’ units vertically. Positive
movements are to the right horizontally, and upwards vertically.
Negative movements are to the left horizontally and downwards
vertically.
Example:
Plot the points
A = (4, 2)
B = (3, 5)
C = (4, 3)
and
D = (5, 2)
on the coordinate axes shown below.
Be sure to label the axes scales and label the points you
plot.
Solution:
Recalling the meaning of this notation giving pairs of numbers
in brackets, we know that the point A is the point that occurs at
x = 4 and y = 2 – the first number in the brackets gives the
xcoordinate of the point, and the second number in brackets gives
the ycoordinate of the point.
Now, the xcoordinates for these four points range from a
minimum of 4 to a maximum of 5, so our horizontal scales must go
at least to 4 on the left to at least +5 on the right. Also, we
see that the ycoordinates must go at least to 3 on the bottom
to at least 5 on the top. The result is:
The dotted lines show how the points line up with the
appropriate scale positions on both axes. Notice that the points
are plotted as heavy dots – if you are just plotting points,
there is no need to join them by lines or add any other features
to the graph.
