How can you use a graph to show the relationship between two quantities that vary directly? How can you use an equation?

Transcription

1 .6 Direct Variation How can ou use a graph to show the relationship between two quantities that var directl? How can ou use an equation? ACTIVITY: Math in Literature Direct Variation In this lesson, ou will identif direct variation from graphs or equations. use direct variation models to solve problems. Gulliver s Travels was written b Jonathan Swift and published in 76. Gulliver was shipwrecked on the island Lilliput, where the people were onl 6 inches tall. When the Lilliputians decided to make a shirt for Gulliver, a Lilliputian tailor stated that he could determine Gulliver s measurements b simpl measuring the distance around Gulliver s thumb. He said Twice around the thumb equals once around the wrist. Twice around the wrist is once around the neck. Twice around the neck is once around the waist. Work with a partner. Use the tailor s statement to complete the table. Thumb, t Wrist, w Neck, n Waist, x in. in. in. in. in. in. 98 Chapter Ratios and Proportions

2 ACTIVITY: Drawing a Graph Work with a partner. Use the information from Activit. a. In our own words, describe the relationship between t and w. b. Use the table to write the ordered pairs (t, w). Then plot the ordered pairs. c. What do ou notice about the graph of the ordered pairs? d. Choose two points and find the slope of the line between them w t e. The quantities t and w are said to var directl. An equation that describes the relationship is w = t. ACTIVITY: Drawing a Graph and Writing an Equation Math Practice Label Axes How do ou know which labels to use for the axes? Explain. Work with a partner. Use the information from Activit to draw a graph of the relationship. Write an equation that describes the relationship between the two quantities. a. Thumb t and neck n (n = t) b. Wrist w and waist x (x = w) c. Wrist w and thumb t (t = w) d. Waist x and wrist w (w = x). IN YOUR OWN WORDS How can ou use a graph to show the relationship between two quantities that var directl? How can ou use an equation?. STRUCTURE How are all the graphs in Activit alike? 6. Give a real-life example of two variables that var directl. 7. Work with a partner. Use string to find the distance around our thumb, wrist, and neck. Do our measurements agree with the tailor s statement in Gulliver s Travels? Explain our reasoning. Use what ou learned about quantities that var directl to complete Exercises and on page. Section.6 Direct Variation 99

3 .6 Lesson Lesson Tutorials Ke Vocabular direct variation, p. constant of proportionalit, p. Direct Variation Words Two quantities x and show direct variation when = kx, where k is a number and k. The number k is called the constant of proportionalit. Graph The graph of = kx is a line with a slope of k that passes through the origin. So, two quantities that show direct variation are in a proportional relationship. x x EXAMPLE Identifing Direct Variation Tell whether x and show direct variation. Explain our reasoning. a. x b. x 6 6 Stud Tip Other was to sa that x and show direct variation are varies directl with x and x and are directl proportional. Plot the points. Draw a line through the points. 6 x Plot the points. Draw a line through the points. 6 6 x The line does not pass through the origin. So, x and do not show direct variation. The line passes through the origin. So, x and show direct variation. EXAMPLE Identifing Direct Variation Tell whether x and show direct variation. Explain our reasoning. a. + = x b. = x = x Solve for. = x Solve for. The equation cannot be written as = kx. So, x and do not show direct variation. The equation can be written as = kx. So, x and show direct variation. Chapter Ratios and Proportions

4 Exercises 6 7 Tell whether x and show direct variation. Explain our reasoning.. x. x. x x =. x = 6. + = x EXAMPLE Real-Life Application x 8 6 The table shows the area (in square feet) that a robotic vacuum cleans in x minutes. a. Graph the data. Tell whether x and are directl proportional. Graph the data. Draw a line through the points. The graph is a line through the origin. So, x and are directl proportional. Area (square feet) Robotic Vacuum (, 6), 8 ) ) ) (, ), ) Time (minutes) x b. Write an equation that represents the line. Choose an two points to find the slope of the line. change in slope = change in x = 6 = 6 The slope of the line is the constant of proportionalit, k. So, an equation of the line is = 6x. c. Use the equation to find the area cleaned in minutes. = 6x Write the equation. = 6 () Substitute for x. = 6 Multipl. So, the vacuum cleans 6 square feet in minutes. Exercise 9 7. WHAT IF? The batter weakens and the robot begins cleaning less and less area each minute. Do x and show direct variation? Explain. Section.6 Direct Variation

5 .6 Exercises Help with Homework. VOCABULARY What does it mean for x and to var directl?. WRITING What point is on the graph of ever direct variation equation?. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Do x and show direct variation? Are x and in a proportional relationship? Is the graph of the relationship a line? Does var directl with x? x 9+(-6)= +(-)= +(-9)= 9+(-)= Graph the ordered pairs in a coordinate plane. Do ou think that graph shows that the quantities var directl? Explain our reasoning.. (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ) Tell whether x and show direct variation. Explain our reasoning. If so, find k. 6. x 7. x x 9. x x =. x =. + = x + 6. = x. x =. x = 6. 8 = x 7. x = 8. ERROR ANALYSIS Describe and correct the error in telling whether x and show direct variation. 9. RECYCLING The table shows the profit for reccling x pounds of aluminum. Graph the data. Tell whether x and show direct variation. If so, write an equation that represents the line. x The graph is a line, so it shows direct variation. Aluminum (lb), x Profit, $. $9. $. $8. Chapter Ratios and Proportions

6 The variables x and var directl. Use the values to find the constant of proportionalit. Then write an equation that relates x and.. = 7; x =. = ; x =. = ; x =. cm in.. MEASUREMENT Write a direct variation equation that relates x inches to centimeters.. MODELING Design a waterskiing ramp. Show how ou can use direct variation to plan the heights of the vertical supports. Vertical supports Cost (dollars) 8 6 Concert (9, 7) (, 6) (, 6) x Tickets. REASONING Use = kx to show wh the graph of a proportional relationship alwas passes through the origin. 6. TICKETS The graph shows the cost of buing concert tickets. Tell whether x and show direct variation. If so, find and interpret the constant of proportionalit. Then write an equation and find the cost of tickets. 7. CELL PHONE PLANS Tell whether x and show direct variation. If so, write an equation of direct variation. Minutes, x 7 9 Cost, $ $ $6 $7 8. CHLORINE The amount of chlorine in a swimming pool varies directl with the volume of water. The pool has. milligrams of chlorine per liter of water. How much chlorine is in the pool? 9. Is the graph of ever direct variation equation a line? Does the graph of ever line represent a direct variation equation? Explain our reasoning. 8 gallons Write the fraction as a decimal. (Section.) MULTIPLE CHOICE Which rate is not equivalent to 8 feet per 8 seconds? (Section.) A ft sec B ft sec C ft 6 sec D 8 ft sec Section.6 Direct Variation