1Write and graph. 2Solve problems. Now. Then. Why? New Vocabulary
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1 Direct Variation Then You found rates of change of linear functions. (Lesson -) Now Write and graph direct variation equations. Solve problems involving direct variation. Wh? Bianca is saving her mone to bu a designer purse that costs $9. To help raise the mone, she charges $8 per hour to babsit her neighbors child. The slope of the line that represents the amount of mone Bianca earns is 8, and the rate of change is constant. New Vocabular direct variation constant of variation constant of proportionalit Direct Variation Equations A direct variation is described b an equation of the form = k, where k 0. The equation = k illustrates a constant rate of change, and k is the constant of variation, also called the constant of proportionalit. Eample Slope and Constant of Variation Virginia i SOL A.8 The student, given a situation in a real-world contet, will analze a relation to determine whether a direct or inverse variation eists, and represent a direct variation algebraicall and graphicall and an inverse variation algebraicall. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. a. (-, ) = - b. (, ) = The constant of variation is -. m = _ - - = _ Slope Formula (, ) = (, ) = (-, ) = - The slope is -. The constant of variation is _. m = _ - - Slope Formula = _ = _ (, ) = (, ) = (, _ ) The slope is. A. Name the constant of variation for = _. Then find the slope of the line that passes through and (, ), two points on the line. B. Name the constant of variation for = -. Then find the slope of the line that passes through and (, -), two points on the line. The slope of the graph of = k is k. Since 0 = k(0), the graph of = k alwas passes through the origin. Therefore the - and -intercepts are zero. 80 Lesson -
2 Eample Graph a Direct Variation Stud Tip Constant of Variation A line with a positive constant of variation will go up from left to right and a line with a negative constant of variation will go down from left to right. Graph = -. Step Write the slope as a ratio. - = _ - _ rise run Step Graph. Step From the point, move down units and right unit. Draw a dot. Step Draw a line containing the points. = - A. = B. = _ C. = - D. = - _ The graphs of all direct variation equations share some common characteristics. Concept Summar Direct Variation Graphs Direct variation equations are of the form = k, where k 0. The graph of = k alwas passes through the origin. The slope is positive if k > 0. O k The slope is negative if k < 0. O k If the relationship between the values of and can be described b a direct variation equation, then we sa that varies directl as. Eample Write and Solve a Direct Variation Equation Suppose varies directl as, and = 7 when = 8. a. Write a direct variation equation that relates and. = k Direct variation formula 7 = k(8) Replace with 7 and with 8. 9 = k Divide each side b 8. Therefore, the direct variation equation is = 9. b. Use the direct variation equation to find when =. = 9 Direct variation formula = 9 Replace with. 7 = Divide each side b 9. Therefore, = 7 when =.. Suppose varies directl as, and = 98 when =. Write a direct variation equation that relates and. Then find when = -. connected.mcgraw-hill.com 8
3 Direct Variation Problems One of the most common applications of direct variation is the formula d = rt. Distance d varies directl as time t, and the rate r is the constant of variation. Real-World Eample Estimate Using Direct Variation TRAVEL The distance a jet travels varies directl as the number of hours it flies. A jet traveled 0 miles in hours. a. Write a direct variation equation for the distance d flown in time t. Real-World Link In 00, domestic airlines transported over 0 million passengers an average distance of 7 miles per flight. Source: Bureau of Transportation Statistics Problem-Solving Tip Write an Equation Sometimes the best wa to solve a problem is to write an equation from the given information and solve. Words Distance equals rate times time Variable Let r = rate. Equation 0 = r Solve for the rate. 0 = r() Original equation _ 0 = _ r() Divide each side b. 70 = r Simplif. Therefore, the direct variation equation is d = 70t. The airliner flew at a rate of 70 miles per hour. Distance Flown b. Graph the equation. The graph of d = 70t passes through the origin with slope 70. m = _ 70 _ rise run c. Estimate how man hours it will take for an airliner to fl 00 miles. d = 70t Original equation 00 = 70t Replace d with 00. _ 00 _ 70 = 70t Divide each side b t. Simplif. Distance (mi) (,0) d = 70t Time (h) It would take the airliner approimatel. hours to fl 00 miles.. HOT-AIR BALLOONS A hot-air balloon s height varies directl as the balloon s ascent time in minutes. A. Write a direct variation for the distance d ascended in time t. B. Graph the equation. C. Estimate how man minutes it would take to ascend 00 feet. D. About how man minutes would it take to ascend 00 feet? 0 ft. in min. 8 Lesson - Direct Variation
4 Check Your Understanding = Step-b-Step Solutions begin on page R. Eample Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points.. (-, ). = - = (, ) Eample Eample Eample Graph each equation.. = -. = _. = -8. = - 8 _ Suppose varies directl as. Write a direct variation equation that relates and. Then solve. 7. If = when =, find when =. 8. If = - when =, find when =. 9. MESSAGE BOARDS You find that the number of messages ou receive on our message board varies directl as the number of messages ou post. When ou post messages, ou receive messages in return. a. Write a direct variation equation relating our posts to the messages received. Then graph the equation. b. Find the number of messages ou need to post to receive 9 messages. Practice and Problem Solving Etra Practice begins on page 8. Eample Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. 0.. (, ) (-, ) (, ) = = - =. ( 0, ) 0 = 8 8. = (, 8) = - (, ) connected.mcgraw-hill.com 8
5 Eample Eample Graph each equation.. = 0 7. = = 9. = 7_ 0. = _. = _ 9. = _. = - _ Suppose varies directl as. Write a direct variation equation that relates and. Then solve.. If = when = 0, find when = 8. Eample If = when = 8, find when = -.. If = _ when = _, find when = _. 7. If = when = _, find when = SPORTS The distance a golf ball travels at an altitude of 7000 feet varies directl with the distance the ball travels at sea level, as shown. a. Write and graph an equation that relates the distance a golf ball travels at an altitude of 7000 feet with the distance at sea level. Hitting a Golf Ball Altitude (ft) 0 (sea level) 7000 Distance (d) 00 0 B 8 Lesson - Direct Variation b. What would be a person s average driving distance at 7000 feet if his average driving distance at sea level is 80 ards? 9. FINANCIAL LITERACY Depreciation is the decline in a car s value over the course of time. The table below shows the values of a car with an average depreciation. Age of Car (ears) Value (dollars),000 0, a. Write an equation that relates the age of the car to the value that it lost after each ear. b. Find the age of the car if the value is $00. Suppose varies directl as. Write a direct variation equation that relates and. Then solve. 0. If =. when =., find when = 9.. If = when = _, find when =.. If =. when =., find when =.. If = - when =., find when = 8. ENDANGERED SPECIES Certain endangered species eperience ccles in their populations as shown in the graph at the right. Match each animal below to one of the colored lines in the graph.. red grouse, 8 ears per ccle. voles, ears per ccle. lemmings, ears per ccle 7. ln, 0 ears per ccle. Years Population Ccles of Endangered Species Number of Ccles
6 In Eercises 8 0, write and graph a direct variation equation that relates the variables. 8. PHYSICAL SCIENCE The weight W of an object is 9.8 m/ s times the mass of the object m. 9 MUSIC Music downloads are $0.99 per song. The total cost of d songs is T. 0. GEOMETRY The circumference of a circle C is approimatel. times the diameter d. C. MULTIPLE REPRESENTATIONS In this problem, ou will investigate the famil of direct variation functions. a. Graphical Graph =, =, and = on the same coordinate plane. b. Algebraic Describe the relationship among the constant of variation, the slope of the line, and the rate of change of the graph. c. Verbal Make a conjecture about how ou can determine without graphing which of two direct variation equations has the steeper graph.. TRAVEL A map of North Carolina is scaled so that inches represents 9 miles. How far apart are Raleigh and Charlotte if the are.8 inches apart on the map?. INTERNET A compan will design and maintain a Web site for our compan for $9.9 per month. Write a direct variation equation to find the total cost C for having a Web page for n months.. BASEBALL Before their first game, high school student Todd McCormick warmed all 00 seats in a new minor league stadium. He started at :0 a.m. and finished around p.m. a. Write a direct variation equation relating the number of seats to time. What is the meaning of the constant of variation in this situation? b. About how man seats had Todd sat in b :00 p.m.? c. How long would ou epect it to take Todd to sit in all of the seats at a major league stadium with more than 0,000 seats? H.O.T. Problems Use Higher-Order Thinking Skills. WHICH ONE DOESN T BELONG? Identif the equation that does not belong. Eplain. 9 = rt 9a = 0 z = _ 9 w = _ 9 t. REASONING How are the constant of variation and the slope related in a direct variation equation? Eplain our reasoning. 7. OPEN ENDED Model a real-world situation using a direct variation equation. Graph the equation and describe the rate of change. 8. CHALLENGE Suppose varies directl as. If the value of is doubled, then the value of is also alwas, sometimes or never doubled. Eplain our reasoning. 9. ERROR ANALYSIS Edd sas the slope between an two points on the graph of a direct variation equation = k is _. Adelle sas the slope depends on the points k chosen. Is either of them correct? Eplain. 0. E WRITING IN MATH Describe the graph of a direct variation equation. connected.mcgraw-hill.com 8
7 Virginia SOL Practice A., A.8. Patricia pas $.9 each to download songs to her digital media plaer. If n is the number of downloaded songs, which equation represents the cost C in dollars? A C =.9n B n =.9C C C =.9 n D C = n +.9. Suppose that varies directl as, and = 8 when =. What is the value of when = 8? F G H 0 _ J. What is the relationship between the input () and output ()? A The output is two more than the input. B The output is two less than the input. C The output is twice the input. D The output is half the input.. SHORT RESPONSE A telephone compan charges $0 per month plus $0.07 per minute. How much would a month of service cost a customer if the customer talked for 00 minutes? Spiral Review. TELEVISION The graph shows the average number of television channels American households receive. What was the annual rate of change from 00 to 008? Eplain the meaning of the rate of change. (Lesson -) Year TV channels at home Solve each equation b graphing. (Lesson -). 0 = = = = = 0. - = 0 Evaluate each epression if a =, b = -, and c = -. (Lesson -). a + c +. a - b +. - a + + c. -a + - a. c - b b - 8 Skills Review Find each difference. (Lesson 0-) 8. - (-) (-) 7. - (-0) Lesson - Direct Variation
Student Exploration: Direct and Inverse Variation
Name: Date: Class Period: Student Eploration: Direct and Inverse Variation Vocabular: constant of proportionalit, direct variation, inverse variation Overview:. Michelle makes $0 an hour babsitting. A.
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What You ll Learn Identify direct variation. Use direct variation to solve problems.
AM_S_C_L_3.indd Page // 3: PM s-user /Volumes//GO/CORE_READING/TENNESSEE/ANCILLARY... Proportionalit and Linear Relationships Teach the Concept Lesson - Direct Variation Interactive Stud Guide See pages
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