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481

The Rectangular CoordinateSystem

6.2

6.2 OBJECTIVES

1. Give the coordinates of a set of points on the plane2. Graph the points corresponding to a set of ordered

pairs

In Section 6.1, we saw that ordered pairs could be used to write the solutions of equationsin two variables. The next step is to graph those ordered pairs as points in a plane.

Because there are two numbers (one for x and one for y), we will need two number lines.One line is drawn horizontally, and the other is drawn vertically; their point of intersection(at their respective zero points) is called the origin. The horizontal line is called the x axis,and the vertical line is called the y axis. Together the lines form the rectangular coordi-nate system.

The axes divide the plane into four regions called quadrants, which are numbered (usu-ally by Roman numerals) counterclockwise from the upper right.

We now want to establish correspondences between ordered pairs of numbers (x, y) andpoints in the plane.

For any ordered pair

(x, y)

x coordinate y coordinate

the following are true:

1. If the x coordinate is

Positive, the point corresponding to that pair is located x units to the right of the y axis.Negative, the point is x units to the left of the y axis.Zero, the point is on the y axis.

y axis

x axis

Quadrant IQuadrant II

Quadrant III Quadrant IV

Origin

NOTE This system is also calledthe cartesian coordinatesystem, named in honor of itsinventor, René Descartes(1596–1650), a Frenchmathematician andphilosopher.

x is x is

negative positive

y

x

The origin is the point withcoordinates (0, 0).

482 CHAPTER 6 AN INTRODUCTION TO GRAPHING

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Example 1

Identifying the Coordinates for a Given Point

Give the coordinates for the given point

(a)

Point A is 3 units to the right of the y axis and 2 units above the x axis. Point A has coordi-nates (3, 2).

(b)

Point B is 2 units to the right of the y axis and 4 units below the x axis. Point B has coordi-nates (2, �4).

2 units

4 units

B

y

x

2 units

3 units

A

y

x

REMEMBER: The xcoordinate gives the horizontaldistance from the y axis. The ycoordinate gives the verticaldistance from the x axis.

2. If the y coordinate is

Positive, the point is y units above the x axis.Negative, the point is y units below the x axis.Zero, the point is on the x axis.

Example 1 illustrates how to use these guidelines to give coordinates to points in theplane.

y is

y is negative

positive

y

x

THE RECTANGULAR COORDINATE SYSTEM SECTION 6.2 483©

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Give the coordinates of points P, Q, R, and S.

P _________

Q _________

R _________

S _________

C H E C K Y O U R S E L F 1

PQ

RS

y

x

(c)

Point C is 3 units to the left of the y axis and 2 units below the x axis. C has coordinates(�3, �2).

(d)

Point D is 2 units to the left of the y axis and on the x axis. Point D has coordinates(�2, 0).

2 units

D

y

2 units

3 units

C

y

x

Reversing the process above will allow us to graph (or plot) a point in the plane giventhe coordinates of the point. You can use the following steps.

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Example 2

Graphing Points

(a) Graph the point corresponding to the ordered pair (4, 3).

Move 4 units to the right on the x axis. Then move 3 units up from the point you stoppedat on the x axis. This locates the point corresponding to (4, 3).

Move 2 units up.

Move 5 units left.

y

(�5, 2)

x

Move 4 units left.

y

x

(�4, �2)

Move 2 units down.

Move 4 units right.

Move 3 units up.

y

x

(4, 3)

(b) Graph the point corresponding to the ordered pair (�5, 2).

In this case move 5 units left (because the x coordinate is negative) and then 2 units up.

(c) Graph the point corresponding to (�4, �2).

Here move 4 units left and then 2 units down (the y coordinate is negative).

Step 1 Start at the origin.Step 2 Move right or left according to the value of the x coordinate.Step 3 Move up or down according to the value of the y coordinate.

Step by Step: To Graph a Point in the PlaneNOTE The graphing ofindividual points is sometimescalled point plotting.

THE RECTANGULAR COORDINATE SYSTEM SECTION 6.2 485©

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(d) Graph the point corresponding to (0, �3).

There is no horizontal movement because the x coordinate is 0. Move 3 units down.

3 units down

y

x

(0, �3)

(e) Graph the point corresponding to (5, 0).

Move 5 units right. The desired point is on the x axis because the y coordinate is 0.

Graph the points corresponding to M(4, 3), N(�2, 4), P(�5, �3), and Q(0, �3).

C H E C K Y O U R S E L F 2

y

x

NOTE Any point on an axis willhave 0 for one of itscoordinates.

5 units right

(5, 0)

y

x

486 CHAPTER 6 AN INTRODUCTION TO GRAPHING

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1. P(4, 5), Q(0, 6), R(�4, �4), and S(2, �5)

2.

C H E C K Y O U R S E L F A N S W E R S

y

x

NM

QP

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6.2

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

487

Exercises

Give the coordinates of the points graphed below.

1. A

2. B

3. C

4. D

5. E

Give the coordinates of the points graphed below.

6. R

7. S

8. T

9. U

10. V

Plot points with the following coordinates on the graph below.

11. M(5, 3) 12. N(0, �3)

13. P(�2, 6) 14. Q(5, 0)

15. R(�4, �6) 16. S(�3, �4)

y

x

R

V

T

S

U

y

x

C

B

A

E

D

y

x

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ANSWERS

17.

18.

19.

20.

21.

22.

23.

24.

488

Plot points with the following coordinates on the graph below.

17. F(�3, �1) 18. G(4, 3)

19. H(5, �2) 20. I(�3, 0)

21. J(�5, 3) 22. K(0, 6)

23. Graph points with coordinates (2, 3), (3, 4), and (4, 5) below. What do you observe?Can you give the coordinates of another point with the same property?

24. Graph points with coordinates (�1, 4), (0, 3), and (1, 2) below. What do youobserve? Can you give the coordinates of another point with the same property?

y

x

y

x

y

x

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25.

26.

27.

489

25. Graph points with coordinates (�1, 3), (0, 0), and (1, �3) below. What do youobserve? Can you give the coordinates of another point with the same property?

26. Graph points with coordinates (1, 5), (�1, 3), and (�3, 1) below. What do youobserve? Can you give the coordinates of another point with the same property?

27. Environment. A local plastics company is sponsoring a plastics recycling contestfor the local community. The focus of the contest is collecting plastic milk, juice,and water jugs. The company will award $200 plus the current market price of thejugs collected to the group that collects the most jugs in a single month. The numberof jugs collected and the amount of money won can be represented as an ordered pair.

(a) In April, group A collected 1500 pounds (lb) of jugs to win first place. The prizefor the month was $350. On the graph on the next page, x represents the poundsof jugs and y represents the amount of money that the group won. Graph thepoint that represents the winner for April.

y

x

x

y

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ANSWERS

28.

29.

490

(b) In May, group B collected 2300 lb of jugs to win first place. The prize for themonth was $430. Graph the point that represents the May winner on the sameaxis you used in part (a).

(c) In June, group C collected 1200 lb of jugs to win the contest. The prize for themonth was $320. Graph the point that represents the June winner on the sameaxis as used before.

28. Education. The table gives the hours, x, that Damien studied for five different mathexams and the resulting grades, y. Plot the data given in the table.

$600

$400

$200

1000 2000 3000

(Pounds)

x 4 5 5 2 6

y 83 89 93 75 95

x 1 2 3 4 5 6

y 4 14 26 33 42 51

29. Science. The table gives the average temperature y (in degrees Fahrenheit) for thefirst 6 months of the year, x. The months are numbered 1 through 6, with 1corresponding to January. Plot the data given in the table.

80

75

85

95

90

100

2 1 3 5 4 6

(Hours)

(Gra

des)

20

40

60

2 4 6

(Months)

(Deg

rees

F)

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30.

31.

491

30. Business. The table gives the total salary of a salesperson, y, for each of the fourquarters of the year, x. Plot the data given in the table.

x 1 2 3 4

y $6000 $5000 $8000 $9000

Game 1 2 3 4

Runs 4 7 6 4

31. Sports. The table shows the number of runs scored by the New York Yankees in the1999 World Series.

Plot the data given in the table.

$10000

$6000

$2000

42

(Quarter)

2

4

6

8

1 2 3 4

Run

s

Game

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ANSWERS

32.

33.

34.

35. (a)

(b)

492

Plot the data given in the table.

33. How would you describe a rectangular coordinate system? Explain what informationis needed to locate a point in a coordinate system.

34. Some newspapers have a special day that they devote to automobile ads. Use thisspecial section or the Sunday classified ads from your local newspaper to findall the want ads for a particular automobile model. Make a list of the model yearand asking price for 10 ads, being sure to get a variety of ages for this model.After collecting the information, make a scatter plot of the age and the askingprice for the car.

Describe your graph, including an explanation of how you decided whichvariable to put on the vertical axis and which on the horizontal axis. What trendsor other information are given by the graph?

35. The map shown on the next page uses letters and numbers to label a grid that helps tolocate a city. For instance, Salem is located at E-4.

(a) Find the coordinates for the following: White Swan, Newport, and Wheeler.

(b) What cities correspond to the following coordinates: A2, F4, and A5?

1 2 3 4 5

2

4

6

8

10

12

Poin

ts

Wins

Team Wins Points

New Jersey Devils 5 12Philadelphia Flyers 4 10New York Rangers 4 9Pittsburgh Penguins 2 6New York Islanders 2 5

32. Sports. The following table shows the number of wins and total points for the fiveteams in the Atlantic Division of the National Hockey League in the early part of the1999–2000 season.

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a.

b.

c.

d.

e.

f.

493

Solve each of the following equations.

(a) 2x � 2 � 6 (b) 2 � 5x � 12

(c) 7y � 10 � �11 (d) �3 � 5x � 1

(e) 6 � 3x � 8 (f) �4y � 6 � 3

Answers1. (5, 6) 3. (2, 0) 5. (�4, �5) 7. (�5, �3) 9. (�3, 5)11–21. 23. The points lie on a line; (1, 2)

y

x

P(�2,6)

J(�5,3)

F(�3,�1)

R(�4,�6)

H(5,�2)

M(5,3)

y

x

Getting Ready for Section 6.3 [Section 2.3]

A

B

C

D

E

F

1 2 3 4 5 6 7

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494

25. The points lie on a line; (2, �6)

27. (a) (1500, 350); (b) (2300, 430); (c) (1200, 320)29. 31.

33. 35. (a) B7, F2, C2; (b) Oysterville, Sweet Home, Mineral

a. 4 b. �2 c. �3 d. e. f. 3

4�

2

3

4

5

2

4

6

8

1 2 3 4

Run

s

Game

20

40

60

2 4 6

(Months)

(Deg

rees

F)

x

y