1.2 Visualizing and Graphing Data

Transcription

1 6360_ch01pp qd 10/16/08 4:8 PM Page 1 1 CHAPTER 1 Introduction to Functions and Graphs 9. Volume of a Cone The volume V of a cone is given b V = 1 3 pr h, where r is its radius and h is its height. Find V when r = 4 inches and h = 1 foot. Round our answer to the nearest hundredth. 93. Size of a Soda Can (Refer to Eample 8.) The volume V of a clindrical soda can is given b V = pr h, where r is its radius and h is its height. (a) If r = 1.3 inches and h = 4.4 inches, find the volume of the can in cubic inches. (b) Could this can hold 1 fluid ounces? (Hint: 1 cubic inch equals about 0.55 fluid ounce.) 94. Volume of a Sphere The volume of a sphere is given b V = 4 3 pr3, where r is the radius of the sphere. Calculate the volume if the radius is 3 feet. Approimate our answer to the nearest tenth. r h 95. Thickness of Cement (Refer to Eample 9.) A 100- foot-long sidewalk is 5 feet wide. If 15 cubic feet of cement are evenl poured to form the sidewalk, find the thickness of the sidewalk. 96. Depth of a Lake (Refer to Eample 9.) A lake covers.5 * 10 7 square feet and contains 7.5 * 10 8 cubic feet of water. Find the average depth of the lake. Writing about Mathematics 97. Describe some basic sets of numbers that are used in mathematics. 98. Suppose that a positive number a is written in scientific notation as a = b * 10 n, where n is an integer and 1 b 10. Eplain what n indicates about the size of a. EXTENDED AND DISCOVERY EXERCISE 1. If ou have access to a scale that weighs in grams, find the thickness of regular and heav-dut aluminum foil. Is heav-dut foil worth the price difference? (Hint: Each.7 grams of aluminum equals 1 cubic centimeter.) 1. Visualizing and Graphing Data Analze one-variable data Find the domain and range of a relation Graph in the -plane Calculate distance Find the midpoint Learn the standard equation of a circle Learn to graph equations with a calculator (optional) Introduction Technolog is giving us access to huge amounts of data. For eample, space telescopes, such as the Hubble telescope, are providing a wealth of information about the universe. The challenge is to convert the data into meaningful information that can be used to solve important problems. Before conclusions can be drawn, data must be analzed. A powerful tool in this step is visualization, as pictures and graphs are often easier to understand than words. This section discusses how different tpes of data can be visualized b using various mathematical techniques. One-Variable Data Data often occur in the form of a list. A list of test scores without names is an eample; the onl variable is the score. Data of this tpe are referred to as one-variable data. If the values in a list are unique, the can be represented visuall on a number line. Means and medians can be found for one-variable data sets. To calculate the mean (or average) of a set of n numbers, we add the n numbers and then divide the sum b n. The median is equal to the value that is located in the middle of a sorted list. If there is an odd

2 6360_ch01pp qd 10/16/08 4:8 PM Page Visualizing and Graphing Data 13 number of data items, the median is the middle data item. If there is an even number of data items, the median is the average of the two middle items. EXAMPLE 1 Analzing a list of data Table 1. lists the monthl average temperatures in degrees Fahrenheit at Mould Ba, Canada. Table 1. Monthl Average Temperatures at Mould Ba, Canada Temperature ( F) Source: A. Miller and J. Thompson, Elements of Meteorolog (a) Plot these temperatures on a number line. (b) Find the maimum and minimum temperatures. (c) Determine the mean of these 1 temperatures. (d) Find the median and interpret the result. (a) In Figure 1.9 the numbers in Table 1. are plotted on a number line Figure 1.9 Monthl Average Temperatures CLASS DISCUSSION In Eample 1(c), the mean of the temperatures is approimatel 0.6 F. Interpret this temperature. Eplain our reasoning. (b) The maimum temperature of 39 F is plotted farthest to the right in Figure 1.9. Similarl, the minimum temperature of -31 F is plotted farthest to the left. (c) The sum of the 1 temperatures in Table 1. equals 7. The mean, or average, of these 7 temperatures is 1 L 0.6 F. (d) Because there is an even number of data items, the median is the average of the middle two values. From the number line we see that the middle two values are -9 F and F. Thus the median is = -4 F. This result means that half the months have an average temperature that is greater than -4 F and half the months have an average temperature that is below -4 F. Now Tr Eercises 1 and 5 Two-Variable Data Sometimes a relationship eists between two lists of data. Table 1.3 lists the monthl average precipitation in inches for Portland, Oregon. In this table, 1 corresponds to Januar, to Februar, and so on, until 1 represents December. We show the relationship between a month and its average precipitation b combining the two lists so that corresponding months and precipitations are visuall paired. Table 1.3 Average Precipitation for Portland, Oregon Month Precipitation (inches)

3 6360_ch01pp qd 10/16/08 4:8 PM Page CHAPTER 1 Introduction to Functions and Graphs If is the month and is the precipitation, then the ordered pair (, ) represents the average amount of precipitation during month. For eample, the ordered pair (5,.0) indicates that the average precipitation in Ma is.0 inches, whereas the ordered pair (, 3.9) indicates that the average precipitation in Februar is 3.9 inches. Order is important in an ordered pair. Since the data in Table 1.3 involve two variables, the month and precipitation, we refer to them as two-variable data. It is important to realize that a relation established b twovariable data is between two lists rather than within a single list. Januar is not related to August, and 6. inches of precipitation is not associated with 1.1 inches of precipitation. Instead, Januar is paired with 6. inches, and August is paired with 1.1 inches. We now define the mathematical concept of a relation. Relation A relation is a set of ordered pairs. If we denote the ordered pairs in a relation b (, ), then the set of all -values is called the domain of the relation and the set of all -values is called the range. The relation shown in Table 1.3 has domain and range D = {1,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1} R = {0.5, 1.1, 1.5, 1.6,.0,.3, 3.1, 3.6, 3.9, 5., 6., 6.4}. -values -values EXAMPLE Finding the domain and range of a relation A phsics class measured the time that it takes for an object to fall feet, as shown in Table 1.4. The object was dropped twice from each height. Table 1.4 Falling Object (feet) (seconds) (a) Epress the data as a relation S. (b) Find the domain and range of S. (a) A relation is a set of ordered pairs, so we can write S = {(0, 1.), (0, 1.1), (40, 1.5), (40, 1.6)}. (b) The domain is the set of -values of the ordered pairs, or D = {0, 40}. The range is the set of -values of the ordered pairs, or R = {1.1, 1., 1.5, 1.6}. Now Tr Eercise 47 To visualize a relation, we often use the Cartesian (rectangular) coordinate plane, or -plane. The horizontal ais is the -ais and the vertical ais is the -ais. The aes

4 6360_ch01pp qd 10/16/08 4:8 PM Page Visualizing and Graphing Data 15 Quadrant II Quadrant I ( 3, 0) ( 4, 4) (, 3) 1 (1, ) (1, ) (0, 3) intersect at the origin and determine four regions called quadrants, numbered I, II, III, and IV, counterclockwise, as shown in Figure We can plot the ordered pair (, ) using the - and -aes. The point (1, ) is located in quadrant I, (-, 3) in quadrant II, (-4, -4) in quadrant III, and (1, -) in quadrant IV. A point ling on a coordinate ais does not belong to an quadrant. The point (-3, 0) is located on the -ais, whereas the point (0, -3) lies on the -ais. The term scatterplot is given to a graph in the -plane where distinct points are plotted. Figure 1.10 is an eample of a scatterplot. Quadrant III Figure 1.10 Quadrant IV The -plane EXAMPLE 3 Graphing a relation Complete the following for the relation S = {(5, 10), (5, -5), (-10, 10), (0, 15), (-15, -10)}. (a) Find the domain and range of the relation. (b) Determine the maimum and minimum of the -values and then of the -values. (c) Label appropriate scales on the -aes. (d) Plot the relation. (a) The elements of the domain correspond to the first number in each ordered pair. Thus D = {-15, -10, 0, 5}. Similarl, the elements of the range correspond to the second number in each ordered pair. Thus R = {-10, -5, 10, 15}. (b) -minimum: -15; -maimum: 5; -minimum: -10; -maimum: 15 (c) An appropriate scale for both the -ais and the -ais might be -0 to 0, with each tick mark representing a distance of 5. This scale is shown in Figure (0, 15) ( 10, 10) (5, 10) (5, 5) ( 15, 10) Figure 1.11 Figure 1.1 (d) The points (5, 10), (5, -5), (-10, 10), (0, 15), and (-15, -10) have been plotted in Figure 1.1. Now Tr Eercise 51 Sometimes it is helpful to connect consecutive data points in a scatterplot with straight-line segments. This tpe of graph, which visuall emphasizes changes in the data, is called a line graph.

5 6360_ch01pp qd 10/16/08 4:8 PM Page CHAPTER 1 Introduction to Functions and Graphs EXAMPLE 4 Making a scatterplot and a line graph Use Table 1.3 on page 13 to make a scatterplot of average monthl precipitation in Portland, Oregon. Then make a line graph. Use the -ais for the months and the -ais for the precipitation amounts. To make a scatterplot, simpl graph the ordered pairs (1, 6.), (, 3.9), (3, 3.6), (4,.3), (5,.0), (6, 1.5), (7, 0.5), (8, 1.1), (9, 1.6), (10, 3.1), (11, 5.), and (1, 6.4) in the -plane, as shown in Figure Then connect consecutive data points to make a line graph, as shown in Figure Average precipitation (inches) Month Average precipitation (inches) Month Geometr Review To review the Pthagorean theorem, see Chapter R (page R-). Figure 1.13 Precipitation Monthl Average Figure 1.14 A Line Graph Now Tr Eercise 57 (, ) d ( 1, 1 ) The Distance Formula In the -plane, the length of a line segment with endpoints ( 1, 1 ) and (, ) can be calculated b using the Pthagorean theorem. See Figure The lengths of the legs of the right triangle are ƒ - 1 ƒ and ƒ - 1 ƒ. The distance d is the hpotenuse of the right triangle. Appling the Pthagorean theorem to this triangle gives d = ( - 1 ) + ( - 1 ). Because distance is nonnegative, we can solve this equation for d to get d = ( - 1 ) + ( - 1 ). Figure 1.15 ( 1, 3) Distance Formula The distance d between the points ( 1, 1 ) and (, ) in the -plane is d = ( - 1 ) + ( - 1 ) (4, 3) Figure 1.16 shows a line segment connecting the points (-1, 3) and (4, -3). Its length is d = (4 - (-1)) + (-3-3) = 61 (eact length) Figure 1.16 L 7.81.

6 6360_ch01pp qd 10/7/08 10:05 AM Page Visualizing and Graphing Data 17 EXAMPLE 5 Finding the distance between two points Find the eact distance between (3, -4) and (-, 7). Then approimate this distance to the nearest hundredth. In the distance formula, let ( 1, 1 ) be (3, -4) and (, ) be (-, 7). ( - 1 ) + ( - 1 ) = (- - 3) + (7 - (-4)) = (-5) + 11 = 146 L 1.08 Distance formula Subtract. Simplif. Approimate. The eact distance is 146, and the approimate distance, rounded to the nearest hundredth, is Note that we would obtain the same answer if we let ( 1, 1 ) be (-, 7) and (, ) be (3, -4). Now Tr Eercise 15 In the net eample the distance between two moving cars is found. W (0, 80) (0, 50) N (0, 0) Car B (60, 0) S Figure Figure 1.18 Car A d M = 1 + M E EXAMPLE 6 Finding the distance between two moving cars Suppose that at noon car A is traveling south at 0 miles per hour and is located 80 miles north of car B. Car B is traveling east at 40 miles per hour. (a) Let (0, 0) be the initial coordinates of car B in the -plane, where units are in miles. Plot the location of each car at noon and at 1:30 P.M. (b) Find the distance between the cars at 1:30 P.M. (a) If the initial coordinates of car B are (0, 0), then the initial coordinates of car A are (0, 80), because car A is 80 miles north of car B. After 1 hour and 30 minutes, or 1.5 hours, car A has traveled 1.5 * 0 = 30 miles south, and so it is located 50 miles north of the initial location of car B. Thus its coordinates are (0, 50) at 1:30 PM. Car B traveled 1.5 * 40 = 60 miles east, so its coordinates are (60, 0) at 1:30 PM. See Figure 1.17, where these points are plotted. (b) To find the distance between the cars at 1:30 P.M. we must find the distance d between the points (0, 50) and (60, 0). The Midpoint Formula d = (60-0) + (0-50) L 78.1 miles Now Tr Eercise 31 A common wa to make estimations is to average data values. For eample, in 1980 the average cost of tuition and fees at public colleges and universities was $800, whereas in 198 it was $1000. One might estimate the cost of tuition and fees in 1981 to be $900. This tpe of averaging is referred to as finding the midpoint. If a line segment is drawn between two data points, then its midpoint is the unique point on the line segment that is equidistant from the endpoints. On a real number line, the midpoint M of two data points 1 and is calculated b averaging their coordinates, as shown in Figure For eample, the midpoint of -3 and 5 is M = = 1.

7 6360_ch01pp qd 10/16/08 4:8 PM Page CHAPTER 1 Introduction to Functions and Graphs ( 1, 1 ) 1 M (, ) The midpoint formula in the -plane is similar to the formula for the real number line, ecept that both coordinates are averaged. Figure 1.19 shows midpoint M located on the line segment connecting the two data points ( 1, 1 ) and (, ). The -coordinate of the midpoint M is located halfwa between and and is Similarl, the -coordinate of M is the average of 1 and. For eample, if we let ( 1, 1 ) be (-3, 1) and (, ) be (-1, 3) in Figure 1.19, then the midpoint is computed b a , b = (-, ). Figure 1.19 Midpoint Formula in the -Plane The midpoint of the line segment with endpoints ( 1, 1 ) and (, ) in the -plane is a 1 +, 1 + b. EXAMPLE 7 Finding the midpoint Find the midpoint of the line segment connecting the points (6, -7) and (-4, 3). In the midpoint formula, let ( 1, 1 ) be (6, -7) and (, ) be (-4, 3). Then the midpoint M can be found as follows. M = a 1 +, 1 + b = a 6 + (-4), b Midpoint formula Substitute. = (1, -) Simplif. Now Tr Eercise 37 The population of the United States from 1800 to 1990 is shown b the blue curve in Figure 1.0. The red line segment connecting the points (1800, 5) and (1990, 49) does not accuratel describe the population over the time period from 1800 to However, over a shorter time interval this tpe of line segment ma be more accurate. Population (millions) 300 (1990, 49) (1800, 5) Year Figure 1.0 U.S. Population

8 6360_ch01pp qd 10/16/08 4:8 PM Page Visualizing and Graphing Data 19 Population (millions) 300 (1990, 49) 50 (1970, 03) M Year Figure 1.1 A Linear Approimation EXAMPLE 8 Using the midpoint formula In 1970 the population of the United States was 03 million, and b 1990 it had increased to 49 million. (Source: Bureau of the Census.) (a) Use the midpoint formula to estimate the population in (b) Describe this approimation graphicall. (a) The U.S. populations in 1970 and 1990 are given b the data points (1970, 03) and (1990, 49). The midpoint M of the line segment connecting these points is M = a , b = (1980, 6). The midpoint formula estimates a population of 6 million in (The actual population was 8 million.) (b) Figure 1.1 shows a blue curve for the U.S. population and a red line segment with midpoint M connecting the data points (1970, 03) and (1990, 49). The line segment appears to be an accurate approimation over a relativel short period of 0 ears. Now Tr Eercise 33 (h, k) (, ) Circles A circle consists of the set of points in a plane that are equidistant from a fied point. The distance is called the radius of the circle, and the fied point is called the center. If we let the center of the circle be (h, k), the radius be r, and (, ) be an point on the circle, then the distance between (, ) and (h, k) must equal r. See Figure 1.. B the distance formula we have Squaring each side gives ( - h) + ( - k) = r. ( - h) + ( - k) = r. Figure 1. Standard Equation of a Circle The circle with center (h, k) and radius r has equation ( - h) + ( - k) = r. NOTE If the center of a circle is (0, 0), then the equation simplifies to + = r. For eample, the equation of the circle with center (0, 0) and radius 7 is + = 49. EXAMPLE 9 Finding the center and radius of a circle Find the center and radius of the circle with the given equation. Graph each circle. (a) + = 9 (b) ( - 1) + ( + ) = 4

9 6360_ch01pp qd 10/16/08 4:8 PM Page 0 0 CHAPTER 1 Introduction to Functions and Graphs (a) Because the equation + = 9 can be written as ( - 0) + ( - 0) = 3, the center is (0, 0) and the radius is 3. The graph of this circle is shown in Figure 1.3. (b) For ( - 1) + ( + ) = 4, the center is (1, -) and the radius is. The graph is shown in Figure (0, 0) (1, ) 4 Figure 1.3 Figure 1.4 Now Tr Eercises 59 and 63 EXAMPLE 10 Finding the equation of a circle Find the equation of the circle that satisfies the conditions. Graph each circle. (a) Radius 4, center (-3, 5) (b) Center (6, -3) with the point (1, ) on the circle (a) Let r = 4 and (h, k) = (-3, 5). The equation of this circle is ( - (-3)) + ( - 5) = 4 or ( + 3) + ( - 5) = 16 A graph of the circle is shown in Figure 1.5. (b) First we must find the distance between the points (6, -3) and (1, ) to determine r. r = (6-1) + (-3 - ) = 50 L 7.1 Since r = 50, the equation of the circle is ( - 6) + ( + 3) = 50. Its graph is shown in Figure 1.6. ( 3, 5) 4 (1, ) (6, 3) Figure 1.5 Figure 1.6 Now Tr Eercises 71 and 75

10 6360_ch01pp qd 10/16/08 4:8 PM Page 1 1. Visualizing and Graphing Data 1 EXAMPLE 11 Finding the equation of a circle The endpoints of a diameter of a circle are (-3, 4) and (5, 6). Find the standard equation of the circle. Getting Started First find the center of the circle, which is the midpoint of a diameter. Then find the radius b calculating the distance between the center and one of the given endpoints of the diameter. Find the center C of the circle b appling the midpoint formula to the endpoints of the diameter (-3, 4) and (5, 6). C = a , b = (1, 5) Use the distance formula to find the radius, which equals the distance from the center (1, 5) to the endpoint (5, 6). r = (5-1) + (6-5) = 17 Thus r equals 17, and the standard equation is ( - 1) + ( - 5) = 17. Graphing with a Calculator (Optional) Now Tr Eercise 77 Graphing calculators can be used to create tables, scatterplots, line graphs, and other tpes of graphs. The viewing rectangle, or window, on a graphing calculator is similar to the view finder in a camera. A camera cannot take a picture of an entire scene; it must be centered on a portion of the available scener and then it can capture different views of the same scene b zooming in and out. Graphing calculators have similar capabilities. The calculator screen can show onl a finite, rectangular region of the -plane, which is infinite. The viewing rectangle must be specified b setting minimum and maimum values for both the - and -aes before a graph can be drawn. We will use the following terminolog to describe a viewing rectangle. Xmin is the minimum -value and Xma is the maimum -value along the -ais. Similarl, Ymin is the minimum -value and Yma is the maimum -value along the -ais. Most graphs show an -scale and a -scale using tick marks on the respective aes. The distance represented b consecutive tick marks on the -ais is called Xscl, and the distance represented b consecutive tick marks on the -ais is called Yscl. See Figure 1.7. This information can be written concisel as [Xmin, Xma, Xscl] b [Ymin, Yma, Yscl]. For eample, [-10, 10, 1] b [-10, 10, 1] means that Xmin = -10, Xma = 10, Xscl = 1, Ymin = -10, Yma = 10, and Yscl = 1. This setting is referred to as the standard viewing rectangle. Yma }Yscl Xmin } Xscl Xma Figure 1.7 Ymin

11 6360_ch01pp qd 10/16/08 4:8 PM Page CHAPTER 1 Introduction to Functions and Graphs EXAMPLE 1 Setting the viewing rectangle Show the standard viewing rectangle and the viewing rectangle given b [-30, 40, 10] b [-400, 800, 100] on our calculator. The required window settings and viewing rectangles are displaed in Figures Notice that in Figure 1.9, there are 10 tick marks on the positive - ais, since its length is 10 and the distance between consecutive tick marks is 1. Calculator Help To set a viewing rectangle or window, see Appendi A (page AP-3). WINDOW Xmin 10 Xma 10 Xscl1 Ymin 10 Yma 10 Yscl1 Xres1 [-10, 10, 1] b [-10, 10, 1] Figure 1.8 Figure 1.9 WINDOW Xmin 30 Xma 40 Xscl10 Ymin 400 Yma 800 Yscl100 Xres1 Figure 1.30 Figure 1.31 [-30, 40, 10] b [-400, 800, 100] Now Tr Eercise 81 EXAMPLE 13 Making a scatterplot with a graphing calculator Plot the points (-5, -5), (-, 3), (1, -7), and (4, 8) in the standard viewing rectangle. The standard viewing rectangle is given b [-10, 10, 1] b [-10, 10, 1]. The points (-5, -5), (-, 3), (1, -7), and (4, 8) are plotted in Figure 1.3. [-10, 10, 1] b [-10, 10, 1] Calculator Help To make the scatterplot in Figure 1.3, see Appendi A (page AP-3). Figure 1.3 Now Tr Eercise 89

12 6360_ch01pp qd 10/16/08 4:8 PM Page 3 1. Visualizing and Graphing Data 3 EXAMPLE 14 Creating a line graph with a graphing calculator Table 1.5 lists the percentage of music sales accounted for b compact discs (CDs) from 1990 to 005. Make a line graph of these sales in [1988, 007, ] b [0, 100, 0]. Table 1.5 Year CDs (% share) Source: Recording Industr Association of America. Enter the data points (1990, 31), (1995, 65), (000, 89), and (005, 90). A line graph can be created b selecting this option on our graphing calculator. The graph is shown in Figure Calculator Help To make a line graph, see Appendi A (page AP-3). [1988, 007, ] b [0, 100, 0] Figure 1.33 Now Tr Eercise Putting It All Together The following table lists basic concepts in this section. Concept Eplanation Eamples Mean, or average To find the mean, or average, of n numbers, divide their sum b n. The mean of the four numbers -3, 5, 6, 9 is = Median Relation, domain, and range The median of a sorted list of numbers equals the value that is located in the middle of the list. Half the data are greater than or equal to the median, and half the data are less than or equal to the median. A relation is a set of ordered pairs (, ). The set of -values is called the domain, and the set of -values is called the range. The median of, 3, 6, 9, 11 is 6, the middle data item. The median of, 3, 6, 9 is the average of the two middle values: 3 and 6. Therefore the median is = 4.5. The relation S = {(1, 3), (, 5), (1, 6)} has domain D = {1, } and range R = {3, 5, 6}. continued on net page

13 6360_ch01pp qd 10/16/08 4:8 PM Page 4 4 CHAPTER 1 Introduction to Functions and Graphs continued from previous page Concept Eplanation Eamples Distance formula The distance between ( 1, 1 ) and (, ) is d = ( - 1 ) + ( - 1 ). The distance between (, -1) and (-1, 3) is d = (-1 - ) + (3 - (-1)) = 5. Midpoint formula The midpoint of the line segment connecting ( 1, 1 ) and (, ) is M = a 1 +, 1 + b. The midpoint of the line segment connecting (4, 3) and (-, 5) is M = a 4 + (-), b = (1, 4). Standard equation of a circle The circle with center (h, k) and radius r The circle with center (-3, 4) and radius 5 has the has the equation equation ( - h) + ( - k) = r. ( + 3) + ( - 4) = 5. The net table summarizes some basic concepts related to one-variable and two-variable data. Tpe of Data Methods of Visualization Comments One-variable data Number line, list, one-column or one-row table The data items are the same tpe and can be described using -values. Computations of the mean and median are performed on one-variable data. Two-variable data Two-column or two-row table, scatterplot, line graph or other tpe of graph in the -plane Two tpes of data are related and can be described b using ordered pairs (, ). 1. Eercises Data Involving One Variable Eercises 1 4: For the table of data, complete the following. (a) Plot the numbers on a number line. (b) Find the maimum and minimum of the data. (c) Determine the mean of the data Eercises 5 8: Sort the list of numbers from smallest to largest and displa the result in a table. (a) Determine the maimum and minimum values. ( b) Calculate the mean and median. Round each result to the nearest hundredth when appropriate , 5, 15, -30, 55, 61, -30, 45, , 4.75, -3.5, 1.5,.5, 4.75, ,.3, 13 69, p, p, , 3.14, 3 8, 19.4, 4 0.9, 3 1.