A. Rectangular Coordinates
The rectangular coordinate system is also known as
the Cartesian coordinate system after Rene
Descartes, who popularized its use in analytic
geometry. The rectangular coordinate system is
based on a grid, and every point on the plane can be
identified by unique x and y coordinates, just as any
point on the Earth can be identified by giving its
latitude and longitude.
Axes
Locations on the grid are measured relative to a fixed
point, called the origin, and are measured according
to the distance along a pair of axes. The x and y axes
are just like the number line, with positive distances
to the right and negative to the left in the case of the
x axis, and positive distances measured upwards and
negative down for the y axis. Any displacement away
from the origin can be constructed by moving a
specified distance in the x direction and then another
distance in the y direction. Think of it as if you were
giving directions to some one by saying something
like “go three blocks East and then 2 blocks North.”
Coordinates, Graphing Points
We specify the location of a point by first giving its x
coordinate (the left or right displacement from the
origin), and then the y coordinate (the up or down
displacement from the origin). Thus, every point on
the plane can be identified by a pair of numbers
(x, y), called its coordinates.
Examples:
Quadrants
Sometimes we just want to know what general part
of the graph we are talking about. The axes naturally
divide the plane up into quarters. We call these
quadrants, and number them from one to four.
Notice that the numbering begins in the upper right
quadrant and continues around in the counterclockwise
direction. Notice also that each quadrant
can be identified by the unique combination of
positive and negative signs for the coordinates of a
point in that quadrant.
B. Graphing Functions
Consider an equation such as
y = 2x – 1
We say that y is a function of x because if you choose
any value for x, this formula will give you a unique
value of y. For example, if we choose x = 3 then the
formula gives us
y = 2(3) – 1
or
y = 5
Thus we can say that the value y = 5 is generated by
the choice of x = 3. Had we chosen a different value
for x, we would have gotten a different value for y. In
fact, we can choose a whole bunch of different
values for x and get a y value for each one. This is
best shown in a table:
| x (input) |
x – Formula - y |
y (output) |
 |
This relationship between x and its corresponding y
values, produces a collection of pairs of points (x, y),
namely
(–2, –5)
(–1, –3)
(0, –1)
(1, 1)
(2, 3)
(3, 5)
Since each of these pairs of numbers can be the
coordinates of a point on the plane, it is natural to ask
what this collection of ordered pairs would look like
if we graphed them. The result is something like this:
The points seem to fall in a straight line. Now, our
choices for x were quite arbitrary. We could just as
well have picked other values, including non-integer
values. Suppose we picked many more values for x,
like 2.7, 3.14, etc. and added them to our graph.
Eventually the points would be so crowded together
that they would form a solid line:
The arrows on the ends of the line indicate that it
goes on forever, because there is no limit to what
numbers we could choose for x. We say that this line
is the graph of the function y = 2x – 1.
If you pick any point on this line and read off its x
and y coordinates, they will satisfy the equation
y = 2x – 1. For example, the point (1.5, 2) is on the
line:
and the coordinates x = 1.5, y = 2 satisfy the equation
y = 2x – 1:
2 = 2(1.5) – 1
Note: This graph turned out to be a straight line only
because of the particular function that we used as an
example. There are many other functions whose
graphs turn out to be various curves .
C. Straight Lines
Linear Equations in Two Variables
The equation y = 2x – 1 that we used as an example
for graphing functions produced a graph that was a
straight line. This was no accident. This equation is
one example of a general class of equations that we
call linear equations in two variables. The two
variables are usually (but of course don’t have to be)
x and y. The equations are called linear because their
graphs are straight lines. Linear equations are easy to
recognize because they obey the fol lowing rules :
1. The variables (usually x and y) appear only to the
first power
2. The variables may be multiplied only by real
number constants
3. Any real number term may be added (or
subtracted , of course)
4. Nothing else is permitted!
* This means that any equation containing things like
x2, y2, 1/x, xy, square roots , or any other function of x
or y is not linear.
Describing Lines
Just as there are an infinite number of equations that
satisfy the above conditions, there are also an infinite
number of straight lines that we can draw on a graph.
To describe a particular line we need to specify two
distinct pieces of information concerning that line. A
specific straight line can be determined by specifying
two distinct points that the line passes through, or it
can be determined by giving one point that it passes
through and somehow describing how “tilted” the
line is.
