Math 35
2.1 "The Rectangular Coordinate System"
Goals:
* Plot ordered pairs and determine the coordinates of a point.
* Graph paired data and read graphs.
* Find midpoints of line segments.
Plot Ordered Pairs and Determine the Coordinates of a Point
The idea of associating an ordered pair of numbers with points on a grid is attributed to the 17th century French
mathematician Rene Descartes. Such a grid is called: Rectangular (Cartesian) Coordinate System.
The point where the axes intersect is called
.
The two axes form a
and divide it into
four regions called
:
Every point on a coordinate plane can be identi…ed by an
of real numbers x and y;written as:
:
Example 1: (Plotting points)
Plot each point on a rectangular coordinate system and name the quadrant or axis in which the point lies.
y
4
2
a) (2; 5)
b) (0; 3)
c) ( 4; 1)
-4
-2
2
4
x
-2
-4
Example 2: (Determining the coordinates of points)
Halley’s comet passes Earth every 76 years as it travels in an orbit about the sun. Use the graph to determine the comet’s
position for the years 1912; 1930, 1948; 1966, and 1978:
Year
Coordinates
1912
1930
1948
1966
1978
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.1
Graph Paired Data
Every day, we deal with quantities that are related. For example, the distance we travel depends on how fast we are
going, your test score depends on the amount of time you study, and the height of a toy rocket depends on the time since it
was launched.
Example 3: (Using tables to draw graphs)
The following table shows the distance a girl is from the ground (in relation to time) as she bounds into the air and back
down to the trampoline.
a) Plot the ordered pairs in the table on a rectangular coordinate system.
b) Draw a smooth curve through the points.
Time (sec)
y 20
Height (f t)
0
2
0.25
9
0.5
14
1.0
18
1.5
14
1.75
9
2.0
2
15
10
5
-1
1
2
3
x
Read Graphs
Note: Since graphs are becoming a popular way to present information, it’s important to know how to read them and
interpret them.
Example 4: (Reading graphs)
The graph below shows the water level of a reservoir before, during, and after a storm. On the x
axis, 0 represents the
day the storm began. On the y axis, 0 represents the normal water level that operators try to maintain.
a) In anticipation of the storm, operators released water to
lower the level of the reservoir. By how many feet was the
water lower prior to the storm?
b) After the storm ended, on what day did the water level
begin to fall?
c) When was the water level 2 feet above normal?
Page: 2
Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.1
Step Graphs
Example 5: (Step graphs)
The graph shown in the illustration gives the …rst-class postage rates for mailing parcels weighing up to 5 ounces.
a) Find the cost of postage to mail each of the following letters …rst class:
a 1-ounce letter, a 4-ounce letter, and a 2 12 -ounces letter.
b) Find the di¤erence in postage for a 3.75-ounce letter and a 4.75-ounce letter.
c) What is the heaviest letter that can be mailed …rst class for 63c/?
Find the Midpoint of a Line Segment
Midpoint Formula:
The midpoint of a line segment with endpoints (x1 ; y1 ) and (x2 ; y2 ) is the point with coordinates:
Example 6: (Finding midpoints of line segments)
Find the midpoint of the line segment with the endpoints
Example 7:
( 1; 8) and (5; 2)
(Finding points using midpoint formula)
If ( 1; 2) is the midpoint of segment QP and the coordinates of Q are (6; 3). Find the coordinates of P .
Page: 3
Bibiana Lopez