9-9A. Graphing Proportional Relationships. Vocabulary. Activity 1. Lesson

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1 Chapter 9 Lesson 9-9A Graphing Proportional Relationships Vocabular unit rate BIG IDEA The graph of the pairs of positive numbers in a proportional relationship is a ra starting at (, ) and passing through (1, r), where r_ is the unit rate. 1 Consider the pairs of numbers in a true proportion, such as 9_ 5 = _ 27. If the 15 ordered pairs (9, 5) and (27, 15) are graphed, then the line containing the two points will contain the origin (, ) (9, 5) 6 (27, 15) A graph like this one can be found b eamining real-world situations involving rates. Activit 1 is about the rate at which a particular car uses gasoline. Activit 1 Step 1 Graph the data from the table at the right. Put the fuel used on the horizontal ais and distance traveled on the vertical ais. Step 2 Choose three points on the graph. For each point, write the miles traveled rate gallons used. Simplif each fraction. Identif the number of miles traveled per gallons used when the denominator is one gallon. This is called the unit rate. Step 3 Is the relationship between fuel used and distance traveled proportional? Support our answer. Fuel used (gallons) Distance traveled (miles) Patterns Leading to Division

2 Lesson 9-9A Ever rate calculated in Step 2 should be 5 miles per gallon, or 5 _ miles gallon. 5 miles This rate can also be written as _. When the denominator of a rate 1 gallon has the number 1, the rate is called a unit rate. In the graph of Activit 1, the unit rate is represented b the point (1, 5). 5 Distance traveled (miles) (2, 1) (8, 4) (1, 5) unit rate 5 miles 1 gallon Fuel used (gallons) Proportional Relationships and Equations The graph from Activit 1 has an equation that also describes this proportional relationship. Using for the number of miles traveled, for the number of gallons, and the point (8, 4), _ miles gallons = _ 4 8. To rewrite this equation in the usual form for graphing, use the Means-Etremes Propert. _ = _ = 4 Means-Etremes Propert _ 8 8 = _ 4 8 = 5 Division Propert of Equalit Simplif You can check other points in the table using this equation. For eample, using (6, 3), 3 = 5 6. You could also use the equation to find additional values that work in this proportional relationship. For an pair of points (, ) and (a, b) on a graph of the equation, _ = b_ a is a true proportion. For eample, from the equation above, the points (1.5, 75) and (2.5, 125) are both on the line so _ 75 miles = _ 125 miles. The are all equal to the unit rate _ 5 miles 1.5 gallons 2.5 gallons 1 gallon. Graphing Proportional Relationships 2

3 Chapter 9 Caution! Some pairs of values on the graph of a proportional relationship cannot be applied to the proportional situation. Consider the point ( 2, 1) on the graph of the line on the previous page. A negative number of gallons used is not possible in this situation. A negative number of miles driven is also not possible, so we must onl consider the first quadrant of this graph. As ou have seen above, the table, the graph, the unit rate, and the equation are all was of representing the same proportional relationship. QY QY If the car has used 1.5 gallons of gasoline, how far has it traveled? Eample The graph at the right shows a proportional relationship between energ used (in joules) and time (in seconds) of a certain brand of microwave. a. Interpret the point (2, 3,) on the graph. b. Find the unit rate and eplain what it means in terms of the situation. Solution a. The point (2, 3,) indicates that when the microwave runs for 2 seconds it uses 3, joules of energ. b. The unit rate is the rate when the denominator is 1. Using the point (2, 3,), energ used the rate _ is 3, joules, which can be seconds 2 seconds 15 joules simplified to _ = 15 joules per second. 1 second This is the unit rate. This means the microwave uses Energ (joules) 5, 4, 3, 2, 1, (2, 3,) Time (seconds) 15 joules of energ for ever 1 second it runs. The Origin and (1, r) There are two points of special significance on the graph of a proportional relationship. On the graph in the Eample, the point (, ) means running the microwave for seconds will use joules of energ. The point (, ) appears on the graph of ever proportional relationship. The value of when = 1 is the numerator of the unit rate, r. Because ou know the denominator of the unit rate is 1, ou can use this pair of values to easil find the unit rate. On a graph, this point is (1, r), indicating that the unit rate is r_ 1. In the Eample the unit rate is 15 joules per second, and the point (1, 15) is found on the graph. It is also possible to find the unit rate from an point on the graph b simplifing _ Energ (joules) (1, 15) (, ) Time (seconds) 3 Patterns Leading to Division

4 Lesson 9-9A as ou did in Part b of the Eample. For eample, the point (22, 33,) 33, joules would give the rate = 15 joules = 15 joules per second, 22 seconds 1 second the unit rate. Using a Graphing Utilit Activit 2 MATERIALS graphing utilit On September 2, 21, 1 U.S. dollar was equivalent to about.78 euro ( ), or _ dollars euro = 1_.78. Step 1 Cop and complete the table at the right. Step 2 For each nonzero pair of values in the chart, write the value as the fraction _ dollars euro. Rewrite each fraction in lowest terms. Step 3 Find the unit rate to the nearest thousandth. Step 4 Let be the dollar cost and be the price in euro. Write a proportion for this situation. Rewrite our proportion in the form =. Round our coeffi cient to the nearest thousandth. Euro Dollars? ? 8.5? 1? Step 5 On the graphing utilit, enter our equation from Step 4 in!. Step 6 Pick a window that will allow ou to trace up through 5. For each value of given in Parts a c below, use the $ feature to fi nd the dollar value of an item that costs euro. Remember that once ou have selected $, ou can enter a value for, press e, and the corresponding value will appear on the screen. Round the dollar amount to the nearest cent. Questions a. 15. b c COVERING THE IDEAS 1. Jon rides his bike at an average rate of 12 miles per hour. a. Cop and complete the table at the right. b. Graph the pairs of numbers in the table. c. What is the unit rate? d. What point on the graph corresponds to this unit rate? e. What does the point (, ) mean in this situation? Hours Miles? 2? 3 1_ 4? 1_ 2? Graphing Proportional Relationships 4

5 Chapter 9 In 2 and 3, simplif and rewrite the equation as =. 2. 5_ 8 = _ 3. _ = _ In 4 and 5, solve each problem using the following steps. a. Write a proportion. b. Simplif and rewrite the equation as =. c. Graph our equation from Part b. d. Eplain the meaning of (, ) in the contet of the situation. e. Find the value for r, and eplain the meaning of (1, r) in the contet of the situation. f. Use the graph to answer the question. 4. Alicia can do 3 math problems in two minutes. Assume she completes the rest of the assignment at the same rate. How long will it take her to finish the 42-problem assignment? Use for the number of minutes and for number of problems. 5. Larr has a dog-walking business. He charges $7.5 per hour. During one week last summer, he worked 22.5 hours. How much did he earn that week? Use for the total amount earned and for the number of hours worked. APPLYING THE MATHEMATICS In 6 and 7, simplif and rewrite the equation as =. 2_ 6. 3_ 4 = _ 7. _ = 1_ 2_ The scale on a map of Australia is 1.1 cm = 5 km. a. Write a proportion to show the actual distance for a distance of cm on the map. b. Solve for. Round to the nearest hundredth. c. Use a graphing utilit to graph the equation. Set the window to include measures on the map up to 5 cm. d. On the map it is 4.6 cm from Uluru, Aers Rock, located in central Australia, to Sdne. Use the $ feature to determine the actual distance from Uluru to Sdne. e. Eplain the meaning of (, ) in the contet of the situation. f. Eplain the meaning of (1, ) in the contet of the situation. 5 Patterns Leading to Division

6 Lesson 9-9A 9. Use the graph at the right to answer the following questions. a. What is represented b? b. What is represented b? c. What situation could be described b the graph? d. What is Mahesh s unit rate? e. How man pages does Anna read in an hour? f. Who reads faster? 1. A 36-sheet manuscript weighs 1.62 kg. a. Write an equation for this situation with number of pages as the independent variable and weight as the dependent variable. b. Graph our equation from Part a. c. Find the unit rate associated with this situation. d. Another manuscript printed on the same stock weighs 252 g. How man sheets are in this manuscript? Pages Read 8 Mahesh 6 4 Anna Number of minutes QY ANSWER 525 miles Graphing Proportional Relationships 6