Section 2.2
The Graph of a Function
93
Version: Fall 2007
2.2
The Graph of a Function
Rene Descartes (1596-1650) was a French philosopher and mathematician who is well
known for the famous phrase “cogito ergo sum” (I think, therefore I am), which appears
in his
Discours de la methode pour bien conduire sa raison, et chercher la verite dans
les sciences
(Discourse on the Method of Rightly Conducting the Reason, and Seeking
Truth in the Sciences).
In that same treatise, Descartes introduces his coordinate
system, a method for representing points in the plane via pairs of real numbers. Indeed,
the Cartesian plane of modern day is so named in honor of Rene Descartes, who some
call the “Father of Modern Mathematics.”
Descartes’ work, which forever linked geometry and algebra, was continued in an
appendix to
Discourse on Method
, entitled
La Geometrie
, which some consider the
beginning of modern mathematics. Certainly both Newton and Leibniz, in developing
the Calculus, built upon the foundation provided in this work by Descartes.
A Cartesian Coordinate System consists of a pair of axes, usually drawn at right
angles to one another in the plane, one horizontal (labeled
x
) and one vertical (labeled
y
), as shown in the
Figure 1
.
The quadrants are numbered I, II, III, and IV, in
counterclockwise order, and samples of ordered pairs of the form
(
x, y
)
are shown in
each quadrant of the Cartesian coordinate system in
Figure 1
.
x
y
I
II
III
IV
x
y
(3
,
2)
(
−
3
,
3)
(
−
4
,
−
2)
(2
,
−
4)
Numbering the quadrants.
To the right and up is positive,
left and down is negative.
Figure 1.
The Cartesian coordinate system.
Now, suppose that we have a relation
R
=
{
(1
,
2)
,
(3
,
1)
,
(3
,
4)
,
(4
,
3)
}
.
Recall that
relation
is the name given to a collection of ordered pairs. In
Figure 2
(b)
we’ve plotted each of the ordered pairs in the relation
R
. This is called the
graph
of
the relation
R
.
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