5.4 Direct Variation - NOTES

Transcription

1 Name Class Date 5.4 Direct Variation - NOTES Essential Question: What is direct variation? Eplore A1.2.D write and solve equations involving direct variation Recognizing Direct Variation Recipes give the amount of ingredients that chefs need to make a certain number of servings. However, sometimes chefs need to make more or fewer servings than the recipe calls for. When this happens, chefs can use direct variation to determine the ingredients needed for the number of servings the need. A recipe for tomato soup calls for 1 cup of tomatoes to make 3 servings. In this relationship, the number of of. varies directl with the number Fill the values in the table. Cups of Tomatoes Servings Identif the independent and dependent variables from the table. Write the equation for the given relationship. Reflect 1. Discussion The equation = 3 represents a direct variation equation. Based on this one equation, draw some conclusions about direct variation equations.

2 Eplain 1 Identifing Direct Variation from Equations A direct variation is a special tpe of linear relationship that can be written in the form = k, where k is a nonzero constant called the constant of variation. Eample 1 Tell whether each equation represents a direct variation. If so, identif the constant of variation. = 6 This equation represents a direct variation because it is in the form = k. The constant of variation is = Solve the equation for = _ = _ Since is added to, add to both sides. _ = _ Since is multiplied b, divide both sides b. = This equation represent a direct variation because it be written as = k. The constant of variation is. Reflect 2. When does a linear equation written in standard form A + B = C represent a direct variation? Tell whether each equation represents a direct variation. If so, identif the constant of variation = = -

3 Eplain 2 What happens if ou solve = k for k? Identifing Direct Variation from Tables = k _ = k_ Divide both sides b ( ). _ = k So, in a direct variation, the ratio is equal to the constant of variation. Another wa to identif a direct variation is to check whether is the same for each ordered pair (ecept where = ). Eample 2 Tell whether each relationship is a direct variation. Eplain Find for each ordered pair. 5_ 1 = 5 _ 15 3 = 5 _ 35 7 = 5 This is a direct variation because is the same for each ordered pair Find for each ordered pair. 8_ 2 = _ = _ = 4 Tell whether each relationship is a direct variation. Eplain

4 Eplain 3 Solving Direct Variation Equations Man real-world situations can be modeled b direct variation equations. You can solve the direct variation equation that represents the situation to answer the problem associated with the situation. Eample 3 Write a direct variation equation to model the situation. Then solve it to answer the question. Two ards of fabric costs $15. Find the cost for 7 ards of fabric if the cost of fabric varies directl with the number of ards. Analze Information Two ards of fabric costs. The varies directl with the. Formulate a Plan Write the equation for a direct variation. Substitute the values of and from the known information and solve for k. Solve Substitute = and =. = k = k ( ) _ = _ Since k is multiplied b, divide both sides b. k = The equation is =. When =, = ( ) = dollars. Justif and Evaluate Each ard of fabric costs, so 7 ards of fabric would cost Write a direct variation equation to model the situation. Then solve it to answer the question. 7. The cost of 5 pounds of apples is $ What will be cost of 12 pounds apples if the cost varies directl with the number of pounds purchased? 8. Stella takes 36 minutes to biccle 2 miles. If the number of miles biccled b Stella varies directl with the amount of time, how man minutes does she need to biccle 5 miles?

5 Eplain 4 Graphing Direct Variation Given a real-world situation that involves direct variation, ou can graph its direct variation equation to model the situation. Then, ou can use the graph to solve the problem represented b the situation. Eample 3 Write and graph a direct variation equation for each situation. Use the graph to solve the problem. The three-toed sloth is an etremel slow animal. On the ground it travels at a speed of about 5 feet per minute. Write a direct variation equation for the distance a sloth will travel in minutes. Graph the equation and use the graph to find how far a sloth travels in 2.5 minutes. Distance = 5 feet per minute times number of minutes. = 5 = 5 (,) = 5 () = (, ) 1 = 5 (1) = 5 (1, 5) 2 = 5 (2) = 1 (2, 1) A sloth travels 12.5 feet in 2.5 minutes. Distance (ft) Speed of a Sloth Time (min) A car travels at a speed of 65 miles per hour. Write a direct variation equation for the distance the car will travel in hours. Graph the equation and use the graph to find how far the car travels in 3.5 hours. Distance = miles per hour times number of hours. = Distance (miles) Speed of a Car = 65 (, ) = 65 ( ) = (, ) 1 = 65 ( ) = (, ) 2 = 65 ( ) = (, ) Time (hours) Cost of Gas The car travels miles in 3.5 hours. 9. Jermaine bought gasoline for his car. The cost of the gasoline was $3.5 per gallon. Write a direct variation equation to describe the cost of gallons of gasoline. Graph the equation and use the graph to find the cost of 5 gallons of gasoline. Cost ($) Gas (gallons)