The Rectangular Coordinate System


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1 6.2 The Rectangular Coordinate Sstem 6.2 OBJECTIVES 1. Give the coordinates of a set of points on the plane 2. 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 equations in two variables. The net step is to graph those ordered pairs as points in a plane. Because there are two numbers (one for and one for ), we will need two number lines. One line is drawn horizontall, and the other is drawn verticall; their point of intersection (at their respective zero points) is called the origin. The horizontal line is called the ais, and the vertical line is called the ais. Together the lines form the rectangular coordinate sstem. The aes divide the plane into four regions called quadrants, which are numbered (usuall b Roman numerals) counterclockwise from the upper right. ais Quadrant II Quadrant I NOTE This sstem is also called the cartesian coordinate sstem, named in honor of its inventor, René Descartes ( ), a French mathematician and philosopher. Origin ais The origin is the point with coordinates (0, 0). Quadrant III Quadrant IV We now want to establish correspondences between ordered pairs of numbers (, ) and points in the plane. For an ordered pair (, ) coordinate coordinate the following are true: 1. If the coordinate is Positive, the point corresponding to that pair is located units to the right of the ais. Negative, the point is units to the left of the ais. Zero, the point is on the ais. is negative is positive 481
2 482 CHAPTER 6 AN INTRODUCTION TO GRAPHING 2. If the coordinate is Positive, the point is units above the ais. Negative, the point is units below the ais. Zero, the point is on the ais. is positive is negative Eample 1 illustrates how to use these guidelines to give coordinates to points in the plane. Eample 1 REMEMBER: The coordinate gives the horizontal distance from the ais. The coordinate gives the vertical distance from the ais. Identifing the Coordinates for a Given Point Give the coordinates for the given point (a) A 2 units 3 units Point A is 3 units to the right of the ais and 2 units above the ais. Point A has coordinates (3, 2). (b) 2 units B 4 units Point B is 2 units to the right of the ais and 4 units below the ais. Point B has coordinates (2, 4).
3 THE RECTANGULAR COORDINATE SYSTEM SECTION (c) 3 units 2 units C Point C is 3 units to the left of the ais and 2 units below the ais. C has coordinates ( 3, 2). (d) 2 units D Point D is 2 units to the left of the ais and on the ais. Point D has coordinates ( 2, 0). CHECK YOURSELF 1 Give the coordinates of points P, Q, R, and S. Q P P Q R S R S Reversing the process above will allow us to graph (or plot) a point in the plane given the coordinates of the point. You can use the following steps.
4 484 CHAPTER 6 AN INTRODUCTION TO GRAPHING NOTE The graphing of individual points is sometimes called point plotting. Step b Step: To Graph a Point in the Plane Step 1 Start at the origin. Step 2 Move right or left according to the value of the coordinate. Step 3 Move up or down according to the value of the coordinate. Eample 2 Graphing Points (a) Graph the point corresponding to the ordered pair (4, 3). Move 4 units to the right on the ais. Then move 3 units up from the point ou stopped at on the ais. This locates the point corresponding to (4, 3). (4, 3) Move 4 units right. Move 3 units up. (b) Graph the point corresponding to the ordered pair ( 5, 2). In this case move 5 units left (because the coordinate is negative) and then 2 units up. ( 5, 2) Move 2 units up. Move 5 units left. (c) Graph the point corresponding to ( 4, 2). Here move 4 units left and then 2 units down (the coordinate is negative). Move 2 units down. Move 4 units left. ( 4, 2)
5 THE RECTANGULAR COORDINATE SYSTEM SECTION NOTE An point on an ais will have 0 for one of its coordinates. (d) Graph the point corresponding to (0, 3). There is no horizontal movement because the coordinate is 0. Move 3 units down. 3 units down (0, 3) (e) Graph the point corresponding to (5, 0). Move 5 units right. The desired point is on the ais because the coordinate is 0. 5 units right (5, 0) CHECK YOURSELF 2 Graph the points corresponding to M(4, 3), N( 2, 4), P( 5, 3), and Q(0, 3).
6 486 CHAPTER 6 AN INTRODUCTION TO GRAPHING CHECK YOURSELF ANSWERS 1. P(4, 5), Q(0, 6), R( 4, 4), and S(2, 5) 2. N M P Q
7 Name 6.2 Eercises Section Date Give the coordinates of the points graphed below. 1. A ANSWERS D C A 2. B 3. C 4. D 5. E E B Give the coordinates of the points graphed below. T U R 6. R 7. S 8. T 9. U 10. V S 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)
8 ANSWERS 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) Graph points with coordinates (2, 3), (3, 4), and (4, 5) below. What do ou observe? Can ou give the coordinates of another point with the same propert? 24. Graph points with coordinates ( 1, 4), (0, 3), and (1, 2) below. What do ou observe? Can ou give the coordinates of another point with the same propert? 488
9 ANSWERS 25. Graph points with coordinates ( 1, 3), (0, 0), and (1, 3) below. What do ou observe? Can ou give the coordinates of another point with the same propert? Graph points with coordinates (1, 5), ( 1, 3), and ( 3, 1) below. What do ou observe? Can ou give the coordinates of another point with the same propert? 27. Environment. A local plastics compan is sponsoring a plastics reccling contest for the local communit. The focus of the contest is collecting plastic milk, juice, and water jugs. The compan will award $200 plus the current market price of the jugs collected to the group that collects the most jugs in a single month. The number of jugs collected and the amount of mone won can be represented as an ordered pair. (a) In April, group A collected 1500 pounds (lb) of jugs to win first place. The prize for the month was $350. On the graph on the net page, represents the pounds of jugs and represents the amount of mone that the group won. Graph the point that represents the winner for April. 489
10 ANSWERS (b) In Ma, group B collected 2300 lb of jugs to win first place. The prize for the month was $430. Graph the point that represents the Ma winner on the same ais ou used in part (a). (c) In June, group C collected 1200 lb of jugs to win the contest. The prize for the month was $320. Graph the point that represents the June winner on the same ais as used before. $600 $400 $ (Pounds) 28. Education. The table gives the hours,, that Damien studied for five different math eams and the resulting grades,. Plot the data given in the table (Grades) (Hours) 29. Science. The table gives the average temperature (in degrees Fahrenheit) for the first 6 months of the ear,. The months are numbered 1 through 6, with 1 corresponding to Januar. Plot the data given in the table (Degrees F) (Months) 490
11 ANSWERS 30. Business. The table gives the total salar of a salesperson,, for each of the four quarters of the ear,. Plot the data given in the table $6000 $5000 $8000 $9000 $10000 $6000 $ (Quarter) 31. Sports. The table shows the number of runs scored b the New York Yankees in the 1999 World Series. Game Runs Plot the data given in the table. Runs Game 491
12 ANSWERS Sports. The following table shows the number of wins and total points for the five teams in the Atlantic Division of the National Hocke League in the earl part of the season (a) (b) Team Wins Points New Jerse Devils 5 12 Philadelphia Flers 4 10 New York Rangers 4 9 Pittsburgh Penguins 2 6 New York Islanders 2 5 Plot the data given in the table Points Wins 33. How would ou describe a rectangular coordinate sstem? Eplain what information is needed to locate a point in a coordinate sstem. 34. Some newspapers have a special da that the devote to automobile ads. Use this special section or the Sunda classified ads from our local newspaper to find all the want ads for a particular automobile model. Make a list of the model ear and asking price for 10 ads, being sure to get a variet of ages for this model. After collecting the information, make a scatter plot of the age and the asking price for the car. Describe our graph, including an eplanation of how ou decided which variable to put on the vertical ais and which on the horizontal ais. What trends or other information are given b the graph? 35. The map shown on the net page uses letters and numbers to label a grid that helps to locate a cit. For instance, Salem is located at E4. (a) Find the coordinates for the following: White Swan, Newport, and Wheeler. (b) What cities correspond to the following coordinates: A2, F4, and A5? 492
13 ANSWERS A B C a. b. c. d. e. D f. E F Getting Read for Section 6.3 [Section 2.3] Solve each of the following equations. (a) (b) (c) (d) (e) (f) Answers 1. (5, 6) 3. (2, 0) 5. ( 4, 5) 7. ( 5, 3) 9. ( 3, 5) The points lie on a line; (1, 2) P( 2,6) J( 5,3) M(5,3) F( 3, 1) R( 4, 6) H(5, 2) 493
14 25. The points lie on a line; (2, 6) 27. (a) (1500, 350); (b) (2300, 430); (c) (1200, 320) (Degrees F) (a) B7, F2, C2; (b) Osterville, Sweet Home, Mineral 4 a. 4 b. 2 c. 3 d. e. 2 f Runs (Months) Game 494
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