16. y = m(x - x 1 ) + y 1, m y = mx, m y = mx + b, m 6 0 and b 7 0 (3, 1) 25. y-intercept 5, slope -7.8

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1 660_ch0pp qd 10/7/08 10:10 AM Page 107. Equations of Lines 107. Eercises Equations of Lines Eercises 1 4: Find the point-slope form of the line passing through the given points. Use the first point as ( 1, 1 ). Plot the points and graph the line b hand. 1. (1, ), (, -). (-, ), (1, 0). (-, -1), (1, ) 4. (-1, ), (-, -) Eercises 5 10: Find a point-slope form of the line satisfing the conditions. Use the first point given for ( 1, 1 ). Then convert the equation to slope-intercept form. 5. Slope -.4, passing through (4, 5) 6. Slope 1.7, passing through (-8, 10) = m( - 1 ) + 1, m 6 0 = m, m 7 0 = m + b, m 6 0 and b = k, k = b, b 6 0 a. b. 7. Passing through (1, -) and (-9, ) 8. Passing through (-6, 10) and (5, -1) 9. -intercept 4, -intercept intercept -, -intercept 5 c. d. ( 1, 1 ) Eercises 11 14: Find the slope-intercept form for the line in the figure e. f ( 1, 1 ) (, 1.8) (1, 0) ( 4, ) 4 (, 1) Eercises 15 0: Concepts Match the equation to its graph (a f ) shown in the net column. 15. = m( - 1 ) + 1, m 7 0 Eercises 1 48: Find the slope-intercept form for the line satisfing the conditions. 1. Passing through (-1, -4) and (1, ). Passing through (-1, 6) and (, -). Passing through (4, 5) and (1, -) 4. Passing through (8, -) and (-, ) 5. -intercept 5, slope intercept -155, slope intercept 45, -intercept 90

2 660_ch0pp qd 10/16/08 4:1 PM Page CHAPTER Linear Functions and Equations 8. -intercept -6, -intercept Perpendicular to = 15, passing through (4, -9) 9. Slope -, passing through (0, 5) 1 0. Slope passing through A 1,, -B 1. Passing through (0, -6) and (4, 0). Passing through A 5 4, -1 4 B and A 4, 7 4 B. Passing through A 1 1, 4 B and A 5, B 4. Passing through A-7, 5 B and A 5 6, -7 6 B 5. Parallel to = , passing through (-4, -7) 6. Parallel to the line = - 4 ( - 100) - 99, passing through (1, ) 7. Perpendicular to the line = - ( ) + 5, passing through (1980, 10) 8. Perpendicular to = 6-10, passing through (15, -7) 54. Perpendicular to = 15, passing through (1.6, -9.5) 55. Parallel to = 4.5, passing through (19, 5.5) 56. Parallel to = -.5, passing through (1985, 67) Finding Intercepts Eercises 57 68: Determine the - and -intercepts on the graph of the equation. Graph the equation = = = = = = = = = = 1 9. Parallel to = +, passing through (0, -.1) 67. = = Parallel to = , passing through (, -5) 41. Perpendicular to = -, passing through (-, 5) 4. Perpendicular to = , passing through (, 8) 4. Perpendicular to + = 4, passing through (15, -5) 44. Parallel to - = -6, passing through (4, -9) 45. Passing through (5, 7) and parallel to the line passing through (1, ) and (-, 1) 46. Passing through (1990, 4) and parallel to the line passing through (1980, ) and (000, 8) 47. Passing through (-, 4) and perpendicular to the line passing through A-5, 1 B and A-, B 48. Passing through A 4, 1 4 B and perpendicular to the line passing through (-, -5) and (-4, 0) Eercises 49 56: Find an equation of the line satisfing the conditions. 49. Vertical, passing through (-5, 6) 50. Vertical, passing through (1.95, 10.7) 51. Horizontal, passing through (-5, 6) Eercises 69 7: The intercept form of a line is a + b = 1. Determine the - and -intercepts on the graph of the equation. Draw a conclusion about what the constants a and b represent in this form = = 1 Eercises 7 and 74: (Refer to Eercises 69 7.) Write the intercept form for the line with the given intercepts. 7. -intercept 5, -intercept intercept -intercept - 5, 4 Interpolation and Etrapolation + = = 1 Eercises 75 78: The table lists data that are eactl linear. (a) Find the slope-intercept form of the line that passes through these data points. (b) Predict when = -.7 and 6.. Decide if these calculations involve interpolation or etrapolation. 5. Horizontal, passing through (1.95, 10.7)

3 660_ch0pp qd 10/16/08 4:1 PM Page 109. Equations of Lines Air Safet Inspectors The number of air safet inspectors for selected ears is shown in the table. Year Inspectors Source: Federal Aviation Administration. (a) Find a linear function ƒ that models these data. Is ƒ eact or approimate? (b) Use ƒ to estimate the number of inspectors in 005. Compare our answer to the actual value of 450. Did our estimate involve interpolation or etrapolation? (c) Eplain the difficult with tring to model these data with a linear function. 80. Deaths on School Grounds Deaths on school grounds nationwide for school ears ending in ear are shown in the table. (ear) (deaths) Source: FBI. (a) Find a linear function ƒ that models these data. Is ƒ eact or approimate? (b) Use ƒ to estimate the number of deaths on school grounds in 00. Compare our answer to the actual value of 49. Did our estimate involve interpolation or etrapolation? (c) Eplain the difficult with tring to model these data with a linear function. Applications 81. Projected Cost of College In 00 the average annual cost of attending a private college or universit, including tuition, fees, room, and board, was $5,000. This cost is projected to rise to $7,000 in 010, as illustrated in the figure. (Source: Cerulli Associates.) Cost ($ thousands) (a) Find a point-slope form of the line passing through the points (00, 5000) and (010, 7000). Interpret the slope. (b) Use the equation to estimate the cost of attending a private college in 007. Did our estimate involve interpolation or etrapolation? (c) Find the slope-intercept form of this line. 8. Distance A person is riding a biccle along a straight highwa. The graph shows the rider s distance in miles from an interstate highwa after hours. Distance (miles) Year (1, 18) (4, 161) Time (hours) (a) How fast is the bicclist traveling? (b) Find the slope-intercept form of the line. (c) How far was the bicclist from the interstate highwa initiall? (d) How far was the bicclist from the interstate highwa after 1 hour and 15 minutes?

4 660_ch0pp qd 10/16/08 4:1 PM Page CHAPTER Linear Functions and Equations 8. Music on the Internet In 00 sales of premium online music totaled $1.6 billion. In 005 this revenue reached $.6 billion. (Source: Jupiter Research.) (a) Find a point-slope form of the line passing through (00, 1.6) and (005,.6). Interpret the slope. (b) Use the equation to estimate projected sales in 008. Did ou use interpolation or etrapolation? (c) Find the slope-intercept form of this line. 84. Water in a Tank The graph shows the amount of water in a 100-gallon tank after minutes have elapsed. Water (gallons) Time (minutes) (a) Is water entering or leaving the tank? How much water is in the tank after minutes? (b) Find the - and -intercepts. Interpret each intercept. (c) Find the slope-intercept form of the equation of the line. Interpret the slope. (d) Estimate the -coordinate of the point (, 50) that lies on the line Spam The table lists the average number of worldwide spam messages dail in billions for selected ears. Year Messages Source: IDC. (a) Make a scatterplot of the data. (b) Find a linear function ƒ that models these data. (Answers ma var.) Interpret the slope m. (c) Use our function to estimate the average number of worldwide spam messages dail during Tuition and Fees The table in the net column lists average tuition and fees at public 4-ear colleges. Year Tuition and fees $804 $118 $1908 Year Tuition and fees $811 $487 $5491 Source: The College Board. (a) Make a scatterplot of the data. (b) Find a linear function that models the data. Interpret the slope m. (c) Use this function to estimate tuition in 199. Compare it to the actual value of $4. (d) Of the si data values, which one would ou remove to make the data more linear? Eplain. 87. Toota Vehicles Sold The table lists the U.S. sales of Toota vehicles in millions. Year Vehicles Source: Autodata. (a) Make a scatterplot of the data. (b) Find ƒ() = m( - 1 ) + 1 so that ƒ() models these data. Interpret the slope m. (c) Is ƒ() an eact or approimate model for the data listed in the table? 88. Farm Pollution In 1988 the number of farm pollution incidents reported in England and Wales was This number had increased at a rate of 80 per ear since (Source: C. Mason, Biolog of Freshwater Pollution.) (a) Find an equation = m( - 1 ) + 1 that models these data, where represents the number of pollution incidents during the ear. (b) Estimate the number of incidents in Cost of Driving The cost of driving a car includes both fied costs and mileage costs. Assume that insurance and car paments cost $50 per month and gasoline, oil, and routine maintenance cost $0.9 per mile. (a) Find a linear function ƒ that gives the annual cost of driving this car miles. (b) What does the -intercept on the graph of ƒ represent?

5 660_ch0pp qd 10/16/08 4:1 PM Page 111. Equations of Lines Average Wages The average hourl wage (adjusted to 198 dollars) was $8.46 in 1970 and $8.18 in 005. (Source: Department of Commerce.) (a) Find an equation of a line that passes through the points (1970, 8.46) and (005, 8.18). (b) Interpret the slope. (c) Approimate the hourl wage in 000. Compare the estimate to the actual value of $8.04. Eercises 91 and 9: Modeling Real Data The table contains data that can be modeled b a linear function ƒ. (a) Make a scatterplot of the data. (Do not tr to plot the undetermined point in the table.) (b) Find a formula for ƒ. Graph ƒ together with the data. (c) Interpret the slope m. (d) Use ƒ to approimate the undetermined value. 91. Asian-American population in millions Year Population ? Source: Bureau of the Census. 9. Population of the western states in millions Year Population ? Source: Bureau of the Census. Perspectives and Viewing Rectangles 9. Graph = in [0,, 1] b [-,, 1]. (a) Is the graph a horizontal line? (b) Wh does the calculator screen appear as it does? 94. Graph = in the standard window. (a) Is the graph a vertical line? (b) Eplain wh the calculator screen appears as it does. 95. Square Viewing Rectangle Graph the lines = and = - 1 in the standard viewing rectangle. (a) Do the lines appear to be perpendicular? (b) Graph the lines in the following viewing rectangles. i. [-15, 15, 1] b [-10, 10, 1] ii. [-10, 10, 1] b [-,, 1] iii. [-,, 1] b [-,, 1] Do the lines appear to be perpendicular in an of these viewing rectangles? (c) Determine the viewing rectangles where perpendicular lines will appear perpendicular. (Answers ma var.) 96. Square Viewing Rectangle Continuing with Eercise 95, make a conjecture about which viewing rectangles result in the graph of a circle with radius 5 and center at the origin appearing circular. i. [-9, 9, 1] b [-6, 6, 1] ii. [-5, 5, 1] b [-10, 10, 1] iii. [-5, 5, 1] b [-5, 5, 1] iv. [-18, 18, 1] b [-1, 1, 1] Test our conjecture b graphing this circle in each viewing rectangle. (Hint: Graph 5-1 = and = -5 - to create the circle.) Graphing a Rectangle Eercises : (Refer to Eample 9.) A rectangle is determined b the stated conditions. Find the slope-intercept form of the four lines that outline the rectangle. 97. Vertices (0, 0), (, ), and (1, ) 98. Vertices (1, 1), (5, 1), and (5, 5) 99. Vertices (4, 0), (0, 4), (0, -4), and (-4, 0) 100. Vertices (1, 1) and (, ); the point (.5, 1) lies on a side of the rectangle. Direct Variation Eercises : Let be directl proportional to. Complete the following Find when = 5, if = 7 when = Find when =.5, if = 1 when = Find when = 1 if = when =, Find when = 1., if = 7. when = 5.. Eercises : Find the constant of proportionalit k and the undetermined value in the table if is directl proportional to. Support our answer b graphing the equation = k and the data points ? ?

6 660_ch0pp qd 10/16/08 4:1 PM Page CHAPTER Linear Functions and Equations 107. Sales ta on a purchase of dollars $5 $55? $1.50 $.0 $ Cost of buing compact discs having the same price 4 5 $41.97 $55.96? 109. Cost of Tuition The cost of tuition is directl proportional to the number of credits taken. If 11 credits cost $70.50, find the cost of taking 16 credits. What is the constant of proportionalit? 110. Strength of a Beam The maimum load that a horizontal beam can carr is directl proportional to its width. If a beam 1.5 inches wide can support a load of 50 pounds, find the load that a beam of the same tpe can support if its width is.5 inches Antarctic Ozone Laer Stratospheric ozone occurs in the atmosphere between altitudes of 1 and 18 miles. Ozone in the stratosphere is frequentl measured in Dobson units, where 00 Dobson units corresponds to an ozone laer millimeters thick. In 1991 the reported minimum in the Antarctic ozone hole was about 110 Dobson units. (Source: R. Huffman, Atmospheric Ultraviolet Remote Sensing.) (a) The thickness of the ozone laer is directl proportional to the number of Dobson units. Find the constant of proportionalit k. (b) How thick was the ozone laer in 1991? 11. Weight on Mars The weight of an object on Earth is directl proportional to the weight of an object on Mars. If a 5-pound object on Earth weighs 10 pounds on Mars, how much would a 195-pound astronaut weigh on Mars? 11. Hooke s Law Suppose a 15-pound weight stretches a spring 8 inches, as shown in the figure. (a) Find the spring constant. (b) How far will a 5-pound weight stretch this spring? 114. Hooke s Law If an 80-pound force compresses a spring inches, how much force must be applied to compress the spring 7 inches? 115. Force of Friction The table lists the force F needed to push a cargo bo weighing pounds on a wood floor. (lb) F (lb) F (a) Compute the ratio for each data pair in the table. Interpret these ratios. (b) Approimate a constant of proportionalit k satisfing F = k. (k is the coefficient of friction.) (c) Graph the data and the equation together. (d) Estimate the force needed to push a 75-pound cargo bo on the floor Electrical Resistance The electrical resistance of a wire varies directl with its length. If a 55-foot wire has a resistance of 1. ohms, find the resistance of 15 feet of the same tpe of wire. Interpret the constant of proportionalit in this situation. Writing about Mathematics 117. Compare the slope-intercept form with the point-slope form. Give eamples of each Give an eample of two quantities in real life that var directl. Eplain our answer. Use an equation to describe the relationship between the two quantities The graph of ƒ() = - passes through (-1, 5) and (, -). Evaluate ƒ(1) and find the midpoint of the two points. Compare and eplain our results. 15 lb 8 in. 10. Eplain how ou would find the equation of a line passing through two points. Give an eample. EXTENDED AND DISCOVERY EXERCISES Eercises 1 and : Estimating Populations Biologists sometimes use direct variation to estimate the number of fish in small lakes. The start b tagging a small number of fish and then releasing them. The assume that over a period of time, the tagged fish distribute themselves evenl throughout the lake. Later, the collect a second sample. The total

7 660_ch0pp qd 10/16/08 4:1 PM Page 11. Linear Equations 11 number of fish and the number of tagged fish in the second sample are counted. To determine the total population of fish in the lake, biologists assume that the proportion of tagged fish in the second sample is equal to the proportion of tagged fish in the entire lake. This technique can also be used to count other tpes of animals, such as birds, when the are not migrating. 1. Eight-five fish are tagged and released into a pond. A later sample of 94 fish from the pond contains 1 tagged fish. Estimate the number of fish in the pond.. Sit-three blackbirds are tagged and released. Later it is estimated that out of a sample of blackbirds, onl 8 are tagged. Estimate the population of blackbirds in the area. CHECKING BASIC CONCEPTS FOR SECTIONS.1 AND. 1. Graph ƒ() = 4 - b hand. Identif the slope, the -intercept, and the -intercept.. The death rate from heart disease for people ages 15 through 4 is.7 per 100,000 people. (a) Write a function ƒ that models the number of deaths in a population of million people 15 to 4 ears old. (b) There are about 9 million people in the United States who are 15 to 4 ears old. Estimate the number of deaths from heart disease in this age group.. A driver of a car is initiall 50 miles south of home, driving 60 miles per hour south. Write a function ƒ that models the distance between the driver and home. 4. Find an equation of the line passing through the points (-, 4) and (5, -). Give equations of lines that are parallel and perpendicular to this line. 5. Find equations of horizontal and vertical lines that pass through the point (-4, 7). 6. Write the slope-intercept form of the line Find the - and -intercepts of the graph of the equation - + = Linear Equations Learn about equations and recognize a linear equation Solve linear equations smbolicall Solve linear equations graphicall and numericall Solve problems involving percentages Appl problem-solving strategies Introduction In Eample 4 of Section., we modeled ipod sales in millions during ear with the equation of a line, or linear function, given b ƒ() = 1( - 004) To predict the ear when ipod sales might be 10 million, we could set the formula for equal to 10 and solve the linear equation 10 = 1( - 004) for. This section discusses linear equations and their solutions. See Eample 4 in this section. ƒ() Equations An equation is a statement that two mathematical epressions are equal. Equations alwas contain an equals sign. Some eamples of equations include , - + 1, z + 5 0, + +, and 1 +.