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Pre

Calculus

Chapter 3A
Chapter 3A

Rectangular Coordinate System
Introduction
:
Rectangular Coordinate System
Although the use of rectangular coordinates in such geometric applications as surveying and planning
has been practiced since ancient times, it was not until the 17th century that geometry and algebra were
joined to form the branch of mathematics called analytic geometry. French mathematician and
philosopher Rene Descartes (15961650) devised a simple plan whereby two number lines were
intersected at right angles with the position of a point in a plane determined by its distance from each of
the lines. This system is called the rectangular coordinate system (or Cartesian coordinate system).
y
x
xaxis
yaxis
origin
(0, 0)
Points are labeled with ordered pairs of real numbers
x
,
y
, called the coordinates of the point, which
give the horizontal and vertical distance of the point from the origin, respectively. The origin is the
intersection of the
x
and
y
axes. Locations of the points in the plane are determined in relationship to
this point
0,0
. All points in the plane are located in one of four quadrants or on the
x
or
y
axis as
illustrated below.
To plot a point, start at the origin, proceed horizontally the distance and direction indicated by the
x
coordinate, then vertically the distance and direction indicated by the
y
coordinate. The resulting
point is often labeled with its ordered pair coordinates and/or a capital letter. For example, the point 2
units to the right of the origin and 3 units up could be labeled
A
2,3
.
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
(, +)
(+, +)
(, )
(+,  )
(a, 0)
(0, b)
(0,0)
Notice that the Cartesian plane has been divided into fourths. Each of these fourths is called a quadrant
and they are numbered as indicated above.
©
:
Pre

Calculus
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Calculus

Chapter 3A
Example 1
:
Plot the following points on a rectangular coordinate system:
A
2,
−
3
B
0,
−
5
C
−
4,1
D
3,0
E
−
2,
−
4
Solution:
5
4
3
2
1
0
1
2
3
4
5

5

4
3

2
1
12345
C(4,1)
E(2,4)
B(0,5)
A(2,3)
D(3,0)
Example 2
:
Shade the region of the coordinate plane that contains the set of ordered pairs
x
,
y
∣
x
0
. [The set notation is read “the set of all ordered pairs
x
,
y
such that
x
0”.]
Solution: This set describes all ordered pairs where the
x
coordinate is greater than 0. Plot several
points that satisfy the stated condition, e.g.,
2,
−
4
,
7,3
,
4,0
. These points are all located to the
right of the
y
axis. To plot all such points we would shade all of Quadrants I and IV. We indicate that
points on the
y
axis are not included
x
0
by using a dotted line.
Example 3
:
Shade the region of the coordinate plane that contains the set of ordered pairs
x
,
y
∣
x
1,
−
2
≤
y
≤
3
.
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 Fall '07
 GENERALCHEMFORENG

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