3.2 The Rectangular Coordinate Sstem 3.2 OBJECTIVES 1. Graph a set of ordered pairs 2. Identif plotted points 3. Scale the aes NOTE In the eighteenth centur, René Descartes, a French philosopher and mathematician, created a wa of graphing ordered pairs. In Chapter 1, we used a number line to locate and visualize real numbers. Such a line has been used b mathematicians for so man ears that we do not know who gets credit for creating the number line. Locating and visualizing ordered pairs is a different stor. A rectangular coordinate sstem consists of two perpendicular number lines, called aes, with the positive directions defined b up and right. If the two lines have the same scale, we sometimes refer to the sstem as a Cartesian coordinate sstem in honor of René Descartes. The positive direction. NOTE The development of the coordinate sstem was part of an effort to combine the knowledge of geometr with that of algebra. (0, 0) The positive direction. (0, 0) is called the origin. Definitions: Aes The horizontal line is called the ais. The vertical line is the ais. Together the are called the and aes (pronounced aees ). Definitions: Coordinate Plane The plane (a flat surface that continues forever in ever direction) containing the and aes is called the coordinate plane. Quadrant II 8 6 4 2 8 6 4 2 2 Quadrant 4 III 6 8 Quadrant I 2 4 6 8 Quadrant IV The and aes divide the coordinate plane into four parts called quadrants. Ever point in the coordinate plane can be described using an ordered pair of real numbers. And, conversel, ever ordered pair can be plotted as a point in the plane. Given an ordered pair (2, 4), 2 is called the coordinate and 4 is called the coordinate. If either coordinate of an ordered pair is equal to 0, the associated point lies on one of the aes. Ever point that does not lie on the or ais can be plotted in one of the four quadrants. 131
132 CHAPTER 3 THE COORDINATE PLANE AND FUNCTIONS Eample 1 Graphing Ordered Pairs Graph the ordered pair (3, 5). Beginning at the origin, we move 3 units in the direction (positive is to the right) and 5 units in the direction (negative is down). (3, 5) CHECK YOURSELF 1 Graph the ordered pairs {(0, 3), ( 2, 4), (3, 1)}. In general, the following pattern shows the quadrant in which a given point is located. Quadrant Sign Pattern I (, ) II (, ) III (, ) IV (, ) If a given point lies on an ais, it fits one of the following patterns: Ais Pattern (, 0) (0, ) We will use these patterns in Eample 2. Eample 2 Locating the Graph of an Ordered Pair Identif the quadrant or ais for the graph of each point. (a) 3, 1 4 The pattern indicates quadrant IV. (b) (0, 6) The pattern indicates the ais.
THE RECTANGULAR COORDINATE SYSTEM SECTION 3.2 133 CHECK YOURSELF 2 Identif the quadrant or ais for the graph of each point. (a) ( 3, 0) (b) 7 (c) (p, 3) 2, 6 To find the ordered pair associated with a plotted point, we move verticall from the point to find the coordinate and horizontall from the point to find the coordinate. Eample 3 Identifing Plotted Points Find the ordered pair associated with each point. A C NOTE When no scale appears on the grid, we assume that each division on each ais is one unit. B D From point A, a vertical line meets the ais at 3. A horizontal line meets the ais at 4. The ordered pair is ( 3, 4). B is associated with (0, 4), C with (3, 3), and D with (3, 3). CHECK YOURSELF 3 Find the ordered pair associated with each point. C A D B
134 CHAPTER 3 THE COORDINATE PLANE AND FUNCTIONS Eample 4 Reading Ordered Pairs Given Scaled Aes NOTE The same decisions must be made when ou are using a graphing calculator. When graphing this kind of relation on a calculator, ou must decide what the appropriate viewing window should be. A surve of residents in a large apartment building was recentl taken. The following points represent ordered pairs in which the first number is the number of ears of education a person has had, and the second number is his or her income (in thousands of dollars). Estimate, and interpret, each ordered pair represented. Thousands of dollars 150 100 50 A D C B A is (9, 20), B is (16, 120), C is (15, 70), and D is (12, 30). Person A completed 9 ears of education and made $20,000 in 2001. Person B completed 16 ears of education and made $120,000 in 2001. Person C had 15 ears education and made $70,000. Person D had 12 ears and made $30,000. 5 10 15 Years It is not obvious from this graph how to predict income from ears of education, but ou might suspect that in most cases, more education results in more income. CHECK YOURSELF 4 Each ear on his son s birthda, Armand records his son s weight. The following points represent ordered pairs in which the first number represents his son s age and the second number represents his weight. For eample, point A indicates that when his son was 1 ear old, the bo weighed 14 pounds. Estimate each ordered pair represented. 30 D Weight 20 A B C 10 1 2 3 4 5 Age 6 7 8 CHECK YOURSELF ANSWERS 1. 2. (a) ais; (b) quadrant II; (c) quadrant I (0,3) 3. A( 1, 4), B(5, 2), C( 5, 3), and D(1, 0) ( 2, 4) (3, 1) C 2 1 2, 22 4. A(1, 14), B(2, 20),, and D(3, 28)
Name 3.2 Eercises Section Date In eercises 1 to 6, graph each set of ordered pairs. 1. 3, 5, 4, 6, 2, 6, 5, 6 ANSWERS 1. 2. 3. 2. 5, 1 2, (6, 0), ( 3, 4), ( 1, 4) 3. (0, 5), (2, 3), 1, 5 2, 5, 3 4 135
ANSWERS 4. 5. 4. ( 1, 2), (5, 2), ( 3, 4), 1 2, 2 6. 5. 1, 6, 3, 5, 2, 5, 1, 4 6. 3, 5, 2, 0, 0, 2, 1, 5 136
ANSWERS In eercises 7 to 18, give the quadrant in which each of the following points is located or the ais on which the point lies. 7. (4, 5) 8. ( 3, 2) 7. 8. 9. 10. 11. 9. ( 4, 3) 10. (2, 4) 12. 13. 11. (5, 0) 12. ( 5, 7) 14. 15. 16. 13. ( 4, 7) 14. ( 3, 7) 17. 18. 19. 15. (0, 7) 16. ( 3, 0) 20. 21. 53 4, 3 17. 18. 2, 45 6 22. 23. 24. In eercises 19 to 28, give the coordinates (ordered pairs) associated with the points indicated in the figure. 25. 26. 27. Q P 19. P 20. Q 21. R 28. R T S U V Y X W 22. S 23. T 24. U 25. V 26. W 27. X 28. Y 137
ANSWERS 29. 30. 29. A compan has kept a record of the number of items produced b an emploee as the number of das on the job increases. In the following figure, points correspond to an ordered-pair relationship in which the first number represents das on the job and the second number represents the number of items produced. Estimate each ordered pair produced. In our own words, state the meaning of the graph. 100 Items Produced 50 2 4 Das 6 8 30. In the following figure, points correspond to an ordered-pair relationship between height and age in which the first number represents age and the second number represents height. Estimate each ordered pair represented. 100 Height 50 5 10 Age 15 138
ANSWERS In eercises 31 and 32, plot the points whose coordinates are given in the table. Scale the aes appropriatel. 31. 32. 31. Grades. The table gives the time,, in hours invested in studing for four different algebra eams and the resulting grade,. 4 3 2 8 1 85 80 70 95 60 120 100 Grades 80 60 40 20 2 4 6 8 10 12 Hours 32. Fuel efficienc. The table gives the speed,, of a car in miles per hour and the approimate fuel efficienc,, in miles per gallon. 45 55 60 65 70 30 25 20 20 15 40 35 Miles/Gallon 30 25 20 15 10 5 10 20 30 40 50 60 70 80 Miles/Hour 139
ANSWERS 33. Graph the points with coordinates (1, 2), (2, 3), and (3, 4). What do ou observe? Give the coordinates of another point with the same propert. 33. 34. 34. Graph points with coordinates ( 1, 3), (0, 0), and (1, 3). What do ou observe? Give the coordinates of another point with the same propert. Answers 1. 3. 5. 7. Quadrant I 9. Quadrant III 11. ais 13. Quadrant II 15. ais 17. Quadrant IV 19. (3, 5) 21. ( 6, 0) 23. ( 5, 4) 25. (0, 4) 27. (6, 2) 29. Points are (1, 30), (2, 45), (3, 60), (4, 60), (5, 75), (6, 90), (7, 95) 31. 33. The coordinate is 1 more than the 160 coordinate; (4, 5) Grades 140 120 100 80 60 40 20 2 4 6 8 10 12 14 16 Hours 140