REVIEW OF ANALYTIC GEOMETRY
The points in a plane can be identified with ordered pairs of real numbers. We start by
drawing two perpendicular coordinate lines that intersect at the origin
on each line.
Usually one line is horizontal with positive direction to the right and is called the
-axis; the other line is vertical with positive direction upward and is called the -axis.
in the plane can be located by a unique ordered pair of numbers as follows.
Draw lines through perpendicular to the - and -axes. These lines intersect the axes in
points with coordinates
as shown in Figure 1. Then the point
is assigned the
. The first number
is called the x-coordinate of ; the second number
is called the y-coordinate of
. We say that
is the point with coordinates
we denote the point by the symbol
. Several points are labeled with their coordi-
nates in Figure 2.
By reversing the preceding process we can start with an ordered pair
at the corresponding point . Often we identify the point with the ordered pair
refer to “the point
.” [Although the notation used for an open interval
same as the notation used for a point
, you will be able to tell from the context which
meaning is intended.]
This coordinate system is called the rectangular coordinate system or the Cartesian
coordinate system in honor of the French mathematician René Descartes (1596–1650),
even though another Frenchman, Pierre Fermat (1601–1665), invented the principles of
analytic geometry at about the same time as Descartes. The plane supplied with this coor-
dinate system is called the coordinate plane or the Cartesian plane and is denoted by
The - and -axes are called the coordinate axes and divide the Cartesian plane into
four quadrants, which are labeled I, II, III, and IV in Figure 1. Notice that the first quad-
rant consists of those points whose - and -coordinates are both positive.
EXAMPLE 1 Describe and sketch the regions given by the following sets.
(a) The po