P.5 The Cartesian Plane
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1 7_0P0.qp //07 8: AM Page 8 8 Chapter P Prerequisites P. The Cartesian Plane The Cartesian Plane Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points in a plane called the rectangular coordinate sstem, or the Cartesian plane, after the French mathematician René Descartes (9 0). The Cartesian plane is formed b using two real number lines intersecting at right angles, as shown in Figure P.0. The horizontal real number line is usuall called the -ais, and the vertical real number line is usuall called the -ais. The point of intersection of these two aes is the origin, and the two aes divide the plane into four parts called quadrants. Quadrant II Origin -ais Quadrant I (Vertical number line) (Horizontal number line) Quadrant III Quadrant IV -ais -ais Directed distance (, ) Directed distance Figure P.0 The Cartesian Plane Figure P. Ordered Pair (, ) -ais What ou should learn Plot points in the Cartesian plane and sketch scatter plots. Use the Distance Formula to find the distance between two points. Use the Midpoint Formula to find the midpoint of a line segment. Find the equation of a circle. Translate points in the plane. Wh ou should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, Eercise 8 on page 8 shows how to represent graphicall the numbers of recording artists inducted into the Rock and Roll Hall of Fame from 98 to 00. Ale Bartel/Stock Boston Each point in the plane corresponds to an ordered pair, of real numbers and, called coordinates of the point. The -coordinate represents the directed distance from the -ais to the point, and the -coordinate represents the directed distance from the -ais to the point, as shown in Figure P.. Directed distance from -ais, Directed distance from -ais The notation (, ) denotes both a point in the plane and an open interval on the real number line. The contet will tell ou which meaning is intended. (, ) (, ) Eample Plotting Points in the Cartesian Plane Plot the points,,,, 0, 0,, 0, and,. (0, 0) (, 0) To plot the point,, imagine a vertical line through on the -ais and a horizontal line through on the -ais. The intersection of these two lines is the point,. This point is one unit to the left of the -ais and two units up from the -ais. The other four points can be plotted in a similar wa (see Figure P.). Now tr Eercise. (, ) Figure P.
2 7_0P0.qp /7/0 9: AM Page 9 The beaut of a rectangular coordinate sstem is that it enables ou to see relationships between two variables. It would be difficult to overestimate the importance of Descartes s introduction of coordinates to the plane. Toda, his ideas are in common use in virtuall ever scientific and business-related field. In the net eample, data is represented graphicall b points plotted on a rectangular coordinate sstem. This tpe of graph is called a scatter plot. Eample Sketching a Scatter Plot The amounts A (in millions of dollars) spent on archer equipment in the United States from 999 to 00 are shown in the table, where t represents the ear. Sketch a scatter plot of the data b hand. (Source: National Sporting Goods Association) Section P. The Cartesian Plane 9 Year, t Amount, A Before ou sketch the scatter plot, it is helpful to represent each pair of values b an ordered pair t, A, as follows. 999,, 000, 9, 00, 7, 00, 79, 00, 8, 00, 8 To sketch a scatter plot of the data shown in the table, first draw a vertical ais to represent the amount (in millions of dollars) and a horizontal ais to represent the ear. Then plot the resulting points, as shown in Figure P.. Note that the break in the t-ais indicates that the numbers 0 through 998 have been omitted. STUDY TIP In Eample, ou could have let t represent the ear 999. In that case, the horizontal ais of the graph would not have been broken, and the tick marks would have been labeled through (instead of 999 through 00). Figure P. Now tr Eercise.
3 7_0P0.qp /7/0 9: AM Page 0 0 Chapter P Prerequisites TECHNOLOGY T I P You can use a graphing utilit to graph the scatter plot in Eample. First, enter the data into the graphing utilit s list editor as shown in Figure P.. Then use the statistical plotting feature to set up the scatter plot, as shown in Figure P.. Finall, displa the scatter plot (use a viewing window in which and 0 00), as shown in Figure P.. TECHNOLOGY SUPPORT For instructions on how to use the list editor, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. 00 Figure P. Figure P Figure P. Some graphing utilities have a ZoomStat feature, as shown in Figure P.7. This feature automaticall selects an appropriate viewing window that displas all the data in the list editor, as shown in Figure P Figure P.7 Figure P.8 The Distance Formula Recall from the Pthagorean Theorem that, for a right triangle with hpotenuse of length c and sides of lengths a and b, ou have a b c, as shown in Figure P.9. (The converse is also true. That is, if a b c, then the triangle is a right triangle.) Suppose ou want to determine the distance d between two points, and, in the plane. With these two points, a right triangle can be formed, as shown in Figure P.0. The length of the vertical side of the triangle is and the length of the horizontal side is B the Pthagorean Theorem, d., a Figure P.9 a + b = c c b d (, ) d. This result is called the Distance Formula. The Distance Formula The distance d between the points, and, in the plane is d. Figure P.0 d (, ) (, )
4 7_0P0.qp /7/0 9: AM Page Section P. The Cartesian Plane Eample Finding a Distance Find the distance between the points, and,. Algebraic Let,, and,,. Then appl the Distance Formula as follows. d Distance Formula Substitute for,,, and. Simplif..8 Simplif. So, the distance between the points is about.8 units. You can use the Pthagorean Theorem to check that the distance is correct. d?? Now tr Eercise. Pthagorean Theorem Substitute for d. Distance checks. Graphical Use centimeter graph paper to plot the points A, and B,. Carefull sketch the line segment from A to B. Then use a centimeter ruler to measure the length of the segment. Figure P. Cm The line segment measures about.8 centimeters, as shown in Figure P.. So, the distance between the points is about.8 units. Eample Verifing a Right Triangle Show that the points,,, 0, and, 7 are the vertices of a right triangle. The three points are plotted in Figure P.. Using the Distance Formula, ou can find the lengths of the three sides as follows. d 7 9 d 0 d Because d d 0 d, ou can conclude that the triangle must be a right triangle. Now tr Eercise 7. The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. 7 (, 7) d (, ) Figure P. = d (, 0) = 0 d = 7 An overhead projector is useful for showing how to plot points and equations. Tr projecting a grid onto the chalkboard, or tr using overhead markers and graph directl on the transparenc. A viewscreen, a device used with an overhead projector to project a graphing calculator s screen image, is also useful.
5 7_0P0.qp /7/0 9: AM Page Chapter P Prerequisites The Midpoint Formula (See the proof on page 7.) The midpoint of the line segment joining the points, and, is given b the Midpoint Formula Midpoint,. Eercises 7 and 8 on page 7 help develop a general understanding of the Midpoint Formula. Eample Finding a Line Segment s Midpoint Find the midpoint of the line segment joining the points, and 9,. Let,, and, 9,. Midpoint, 9,, 0 Midpoint Formula Substitute for,,, and. Simplif. The midpoint of the line segment is, 0, as shown in Figure P.. Now tr Eercise 9. (9, ) (, 0) 9 (, ) Midpoint Figure P. Eample Estimating Annual Sales Kraft Foods Inc. had annual sales of $9.7 billion in 00 and $.7 billion in 00. Without knowing an additional information, what would ou estimate the 00 sales to have been? (Source: Kraft Foods Inc.) One solution to the problem is to assume that sales followed a linear pattern. With this assumption, ou can estimate the 00 sales b finding the midpoint of the line segment connecting the points 00, 9.7 and 00,.7. Midpoint , 00, 0.9 So, ou would estimate the 00 sales to have been about $0.9 billion, as shown in Figure P.. (The actual 00 sales were $.0 billion.) Now tr Eercise. Sales (in billions of dollars) Figure P. Kraft Foods Inc. Annual Sales (00,.7) (00, 0.9) Midpoint (00, 9.7) Year The Equation of a Circle The Distance Formula provides a convenient wa to define circles. A circle of radius r with center at the point h, k is shown in Figure P.. The point, is on this circle if and onl if its distance from the center h, k is r. This means that
6 7_0P0.qp /7/0 9: AM Page a circle in the plane consists of all points, that are a given positive distance r from a fied point h, k. Using the Distance Formula, ou can epress this relationship b saing that the point, lies on the circle if and onl if h k r. B squaring each side of this equation, ou obtain the standard form of the equation of a circle. Section P. The Cartesian Plane Center: (h, k) Figure P. Radius: r Standard Form of the Equation of a Circle Point on circle: (, ) The standard form of the equation of a circle is h k r. The point h, k is the center of the circle, and the positive number r is the radius of the circle. The standard form of the equation of a circle whose center is the origin, h, k 0, 0, is r. Activities. Set up a Cartesian plane and plot the points, 0 and,.. Find such that the distance between, and, is. Answer:,. Find the midpoint of the line segment joining the points, and,. Answer:,. Write the standard form of the equation of the circle with center at, and radius. Answer: Eample 7 Writing the Equation of a Circle The point, lies on a circle whose center is at,, as shown in Figure P.. Write the standard form of the equation of this circle. 8 The radius r of the circle is the distance between, and,. r Substitute for,, h, and k. Simplif. 0 Radius Using h, k, and r 0, the equation of the circle is h k r Equation of circle (, ) (, ) 0 Substitute for h, k, and r. Figure P. 0. Standard form Now tr Eercise.
7 7_0P0.qp /7/0 9:7 AM Chapter P Page Prerequisites Eample 8 Translating Points in the Plane The triangle in Figure P.7 has vertices at the points 共, 兲, 共, 兲, and 共, 兲. Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure P.8. (, ) (, ) 7 7 Paul Morrell Much of computer graphics, including this computer-generated goldfish tessellation, consists of transformations of points in a coordinate plane. One tpe of transformation, a translation, is illustrated in Eample 8. Other tpes of transformations include reflections, rotations, and stretches. (, ) Figure P.7 Figure P.8 To shift the vertices three units to the right, add to each of the -coordinates. To shift the vertices two units upward, add to each of the -coordinates. Original Point Translated Point 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 Plotting the translated points and sketching the line segments between them produces the shifted triangle shown in Figure P.8. Now tr Eercise 79. Eample 8 shows how to translate points in a coordinate plane. The following transformed points are related to the original points as follows. Original Point Transformed Point 共, 兲 共, 兲 共, 兲 is a reflection of the original point in the -ais. 共, 兲 共, 兲 共, 兲 is a reflection of the original point in the -ais. 共, 兲 共, 兲 共, 兲 is a reflection of the original point through the origin. The figures provided with Eample 8 were not reall essential to the solution. Nevertheless, it is strongl recommended that ou develop the habit of including sketches with our solutions, even if the are not required, because the serve as useful problem-solving tools.
8 7_0P0.qp /7/0 9:7 AM Page Section P. The Cartesian Plane P. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check. Match each term with its definition. (a) -ais (b) -ais (c) origin (d) quadrants (e) -coordinate (f) -coordinate (i) point of intersection of vertical ais and horizontal ais (ii) directed distance from the -ais (iii) horizontal real number line (iv) four regions of the coordinate plane (v) directed distance from the -ais (vi) vertical real number line In Eercises, fill in the blanks.. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate sstem or the plane.. The is a result derived from the Pthagorean Theorem.. Finding the average values of the respective coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the.. The standard form of the equation of a circle is, where the point h, k is the of the circle and the positive number r is the of the circle. In Eercises and, approimate the coordinates of the points... In Eercises, plot the points in the Cartesian plane..,,,, 0,,,.,, 0, 0,, 0,,.. D A B C, 8, 0.,,,,,.,,,,,,, In Eercises 7 0, find the coordinates of the point. 7. The point is located five units to the left of the -ais and four units above the -ais. 8. The point is located three units below the -ais and two units to the right of the -ais. 9. The point is located si units below the -ais and the coordinates of the point are equal. 0. The point is on the -ais and 0 units to the left of the -ais. D C B A In Eercises 0, determine the quadrant(s) in which, is located so that the condition(s) is (are) satisfied.. > 0 and < 0. < 0 and < 0. and > 0. > and. <. > 7. < 0 and > 0 8. > 0 and < 0 9. > 0 0. < 0 In Eercises and, sketch a scatter plot of the data shown in the table.. Sales The table shows the sales (in millions of dollars) for Apple Computer, Inc. for the ears (Source: Value Line) Year Sales, (in millions of dollars) 997 7,08 998,9 999, 000 7,98 00, 00,7 00, ,79 00,900 00,00
9 7_0P0.qp /7/0 9:7 AM Page Chapter P Prerequisites. Meteorolog The table shows the lowest temperature on record (in degrees Fahrenheit) in Duluth, Minnesota for each month, where represents Januar. (Source: NOAA) In Eercises, find the distance between the points algebraicall and verif graphicall b using centimeter graph paper and a centimeter ruler..,,,.,, 8,.,,,.,,, 7.,,, ,.,.9, 8. In Eercises, (a) find the length of each side of the right triangle and (b) show that these lengths satisf the Pthagorean Theorem... (, ) (0, ) Month, 8,, 0, 0,,,,,,.,.,.,.8 (, ) Temperature, (, 0) (, ) 8 (, 0).. (9, ) (9, ) (, ) 8 In Eercises 7, show that the points form the vertices of the polgon. 7. Right triangle:, 0,,,, 8. Right triangle:,,,,, 9. Isosceles triangle:,,,,, 0. Isosceles triangle:,,, 9,, 7. Parallelogram:,, 0, 9,, 0, 0,. Parallelogram: 0,,, 7,,,,. Rectangle:,, 0, 8,,,, (Hint: Show that the diagonals are of equal length.). Rectangle:,,,,,,, (Hint: Show that the diagonals are of equal length.) In Eercises, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points..,, 9, 7.,,, 0 7., 0,, ,,, 8,,, 0., 0, 0,.,,,.,,,..,.,.7,.8..8,.,.,.9 Revenue In Eercises and, use the Midpoint Formula to estimate the annual revenues (in millions of dollars) for Wend s Intl., Inc. and Papa John s Intl. in 00. The revenues for the two companies in 000 and 00 are shown in the tables. Assume that the revenues followed a linear pattern. (Source: Value Line). Wend s Intl., Inc. Year (, ) (, ) Annual revenue (in millions of dollars) (, )
10 7_0P0.qp /7/0 9:7 AM Page 7 Section P. The Cartesian Plane 7. Papa John s Intl. 7. Eploration A line segment has, as one endpoint and m, m as its midpoint. Find the other endpoint, of the line segment in terms of,, m, and m. Use the result to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectivel, (a),,, (b),,, 8. Eploration Use the Midpoint Formula three times to find the three points that divide the line segment joining, and, into four parts. Use the result to find the points that divide the line segment joining the given points into four equal parts. (a),,, (b),, 0, 0 In Eercises 9 7, write the standard form of the equation of the specified circle. 9. Center: 0, 0 ; radius: 0. Center: 0, 0 ; radius:. Center:, ; radius:. Center: 0, ; radius:. Center:, ; solution point: 0, 0. Center:, ; solution point:,. Endpoints of a diameter: 0, 0,, 8. Endpoints of a diameter:,,, 7. Center:, ; tangent to the -ais 8. Center:, ; tangent to the -ais 9. The circle inscribed in the square with vertices 7,,,,, 0, and 7, The circle inscribed in the square with vertices, 0, 8, 0, 8, 0, and, Year Annual revenue (in millions of dollars) In Eercises 7 78, find the center and radius, and sketch the circle In Eercises 79 8, the polgon is shifted to a new position in the plane. Find the coordinates of the vertices of the polgon in the new position Original coordinates of vertices: Shift: three units upward, one unit to the left 8. Original coordinates of vertices: Shift: two units downward, three units to the left Analzing Data In Eercises 8 and 8, refer to the scatter plot, which shows the mathematics entrance test scores and the final eamination scores in an algebra course for a sample of 0 students. Final eamination score 9 0,,,, (,,,,,,,, (, ) (, ) units (, ) Report Card Math.A English.A Science.B PhsEd.A (, ) units (9, 7) (8, 90) (8, 9) (, 79) (0, ) (, 7) units 9 (, ) 7 (, ) units 7 (, 0) (, ) (, 7) (7, 99) (, 8) Mathematics entrance test score 8. Find the entrance eam score of an student with a final eam score in the 80s. 8. Does a higher entrance eam score necessaril impl a higher final eam score? Eplain.
11 7_0P0.qp /7/0 9:7 AM Page 8 8 Chapter P Prerequisites 8. Rock and Roll Hall of Fame The graph shows the numbers of recording artists inducted into the Rock and Roll Hall of Fame from 98 to 00. Number inducted Year (a) Describe an trends in the data. From these trends, predict the number of artists that will be elected in 007. (b) Wh do ou think the numbers elected in 98 and 987 were greater than in other ears? 8. Fling Distance A jet plane flies from Naples, Ital in a straight line to Rome, Ital, which is 0 kilometers north and 0 kilometers west of Naples. How far does the plane fl? 87. Sports In a football game, a quarterback throws a pass from the -ard line, 0 ards from the sideline, as shown in the figure. The pass is caught on the 0-ard line, ards from the same sideline. How long is the pass? Distance (in ards) (0, ) (, 0) Distance (in ards) 88. Sports A major league baseball diamond is a square with 90-foot sides. Place a coordinate sstem over the baseball diamond so that home plate is at the origin and the first base line lies on the positive -ais (see figure). Let one unit in the coordinate plane represent one foot. The right fielder fields the ball at the point 00,. How far does the right fielder have to throw the ball to get a runner out at home plate? How far does the right fielder have to throw the ball to get a runner out at third base? (Round our answers to one decimal place.) Distance (in feet) Figure for Boating A acht named Beach Lover leaves port at noon and travels due north at miles per hour. At the same time another acht, The Fisherman, leaves the same port and travels west at miles per hour. (a) Using graph paper, plot the coordinates of each acht at P.M. and P.M. Let the port be at the origin of our coordinate sstem. (b) Find the distance between the achts at P.M. and P.M. Are the achts twice as far from each other at P.M. as the were at P.M.? 90. Make a Conjecture Plot the points,,,, and 7, on a rectangular coordinate sstem. Then change the signs of the indicated coordinate(s) of each point and plot the three new points on the same rectangular coordinate sstem. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the -coordinate is changed. (b) The sign of the -coordinate is changed. (c) The signs of both the - and -coordinates are changed. 9. Show that the coordinates,,, 0, and, 0 form the vertices of an equilateral triangle. 9. Show that the coordinates,,, 7, and, form the vertices of a right triangle. Snthesis (0, 90) (0, 0) (00, ) True or False? In Eercises 9 9, determine whether the statement is true or false. Justif our answer. 9. In order to divide a line segment into equal parts, ou would have to use the Midpoint Formula times. 9. The points 8,,, and, represent the vertices of an isosceles triangle. 9. If four points represent the vertices of a polgon, and the four sides are equal, then the polgon must be a square. 9. Think About It What is the -coordinate of an point on the -ais? What is the -coordinate of an point on the -ais? 97. Think About It When plotting points on the rectangular coordinate sstem, is it true that the scales on the - and -aes must be the same? Eplain.
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