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THE UNIVERSITY OF AKRON
Mathematics and Computer Science
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Lesson 8: Cartesian Coordinate System & Functions
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• Begin Lesson 8
r
IamD S
N ⊆ Z ⊆ Q ⊆ R ⊆ C
a 3 a 4 = a 7 ( ab ) 10 = a 10 b 10
− ( ab − (3 ab − 4))=2 ab − 4
( ab ) 3 ( a − 1 + b − 1 )=( ab ) 2 ( a + b )
( a − b ) 3 = a 3 − 3 a 2 b +3 ab 2 − b 3
2 x 2 − 3 x − 2=(2 x + 1)( x − 2)
1
2 x +13=0 = ⇒ x = − 26
G= { ( x,y ) | y = f ( x ) }
f ( x )= mx + b
y = sin x
Copyright c 1995–2000 D. P. Story
Last Revision Date: 4/4/2000
 
Lesson 8: Cartesian Coordinate System & Functions
Table of Contents
8. Cartesian Coordinate System & Functions
8.1. The Cartesian Coordinate System
• Referencing/Plotting Points • The Distance Formula • The
Midpoint of a Line Segment
8.2. Functions
• The Definition • The Domain of a Function • Points of
Intersection of Curves
8. Cartesian Coordinate System & Functions
8.1. The Cartesian Coordinate System
There are manyschemes for referencing points in the plane. Among
all these coordinate systems, the Cartesian Coordinate System is
the most popular and useful.
Begin bydrawing two real number lines , perpendicular to
each other, and intersecting at their zeros. One number line is
drawn horizontally, and the other vertically.
Figure 1
The two number lines, called axes , are labeled bysome appropriately
chosen symbols; usuallythe horizontal axis is labeled bythe letter x
and the vertical axis is labeled bythe letter y . The horizontal axis
is called the x -axis or the axis of abscissas and the vertical axis is
called the y -axis or the axis of ordinates .
The two perpendicular axes subdivide the plane into six sub-
sets in such a waythat anygiven point in the plane is either
(1) on the x -axis, (2) on the y -axis, (3) in the first quadrant ,
Figure 2
986093596.001.png 986093596.002.png
Section 8: Cartesian Coordinate System & Functions
(4) in the second quadrant , (5) in the third quadrant , or (6) in the
fourth quadrant . (See Figure 2 .)
• Referencing/Plotting Points
Points in the plane are referenced bytheir position relative to the
two perpendicular axes. We shall not spend a terriblylarge amount
of time on this topic because you no doubt have plotted many points
before. We shall be content with reviewing some of the definitions and
terminology.
The Method of Referencing a Point. Let P beapointinthe
plane. Draw a vertical line passing through the point P and
a horizontal line through P . The vertical line intersects the
x -axis at a certain position a and the horizontal line intersects the
y -axis at a certain position b . The Cartesian coordinates of the
point P is defined as
Figure 3
P ( a,b )
Conversely, given ordered pair of numbers, ( a,b ), there corresponds
one and onlyone point in the plane. This point is the intersection of
986093596.003.png
Section 8: Cartesian Coordinate System & Functions
the two lines obtained bydrawing a vertical line passing through a in
the x -axis and a horizontal line passing through the number b on the
y -axis.
Terminology : Let ( a,b )isthe Cartesian Coordinates of the point
P . The number a is called the first coordinate of P ,orthe x -
coordinate of P ,orthe abscissa of P ; similarly, b is called the second
coordinate ,orthe y -coordinate, or the ordinate of P .
• Question. Whydo you think we have such terminologyas the “axis
of abscissas,” the “axis of ordinates,” the “abscissa of P ” and the
“ordinate of P ”?
Having defined the method of referencing a point in the plane, the
four quadrants of the plane can be described more precisely.
Quiz. Answer each of the following about the quadrants .
1. What quadrant consists of all points P ( x,y ) satisfying x> 0
and y< 0? Quadrant ...
(a) I
(b) II
(c) III
(d) IV

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