1. Holt CA Course 1 7-9Direct Variation Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview

2. Holt CA Course 1 7-9Direct Variation Warm Up Tell whether the ratios are proportional. 1. = 2. = 3. = 4. = 6 9 24 36 yes 56 68 14 17 12 13 60 78 45 6 30 4 yes no yes ? ? ? ?

3. Holt CA Course 1 7-9Direct Variation AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. Also covered: AF3.3, AF3.4 California Standards

4. Holt CA Course 1 7-9Direct Variation Vocabulary direct variation constant of proportionality

5. Holt CA Course 1 7-9Direct Variation A direct variation is a linear function that can be written as y = kx, where k is a nonzero constant called the constant of variation.

6. Holt CA Course 1 7-9Direct Variation Determine whether the data set shows direct variation. A. Additional Example 1: Determining Whether a Data Set Varies Directly

7. Holt CA Course 1 7-9Direct Variation Method 1: Make a graph. Additional Example 1A Continued The graph is not linear.

8. Holt CA Course 1 7-9Direct Variation Method 2: Compare ratios. 22 3 27 12 = ? 81 264 81 ≠ 264 The ratios are not equivalent. Both methods show the relationship is not a direct variation. Additional Example 1A Continued

9. Holt CA Course 1 7-9Direct Variation Determine whether the data set shows direct variation. B. Additional Example 1: Determining Whether a Data Set Varies Directly

10. Holt CA Course 1 7-9Direct Variation Method 1: Make a graph. Additional Example 1B Continued Plot the points. The points lie in a straight line. (0, 0) is included.

11. Holt CA Course 1 7-9Direct Variation Method 2: Compare ratios. Both methods show the relationship is a direct variation. 25 10 50 20 75 30 100 40 === Additional Example 1B Continued The ratio is constant.

12. Holt CA Course 1 7-9Direct Variation Determine whether the data set shows direct variation. A. Check It Out! Example 1 Kyle's Basketball Shots Distance (ft)203040 Number of Baskets530

13. Holt CA Course 1 7-9Direct Variation Method 1: Make a graph. Check It Out! Example 1A Continued Number of Baskets Distance (ft) 2 3 4 203040 1 5

14. Holt CA Course 1 7-9Direct Variation Method 2: Compare ratios. Check It Out! Example 1A Continued 5 20 3 30 = ? 60 150 150  60. The ratios are not equivalent. Both methods show the relationship is not a direct variation.

15. Holt CA Course 1 7-9Direct Variation Determine whether the data set shows direct variation. B. Check It Out! Example 1 Ounces in a Cup Ounces (oz)8162432 Cup (c)1234

16. Holt CA Course 1 7-9Direct Variation Method 1: Make a graph. Check It Out! Example 1B Continued Number of Cups Number of Ounces 2 3 4 81624 1 32 Plot the points. The points lie in a straight line. (0, 0) is included.

17. Holt CA Course 1 7-9Direct Variation Method 2: compare ratios. Check It Out! Example 1B Continued Both methods show the relationship is a direct variation. The ratio is constant. = 1 8 == 2 16 3 24 4 32

18. Holt CA Course 1 7-9Direct Variation Rachel rents space in a salon to cut and style hair. She paid the salon owner $24 for 3 cut and styles. Write a direct variation function for this situation. If Rachel does 7 cut and styles, how much will she pay the salon owner? Additional Example 2: Finding Equations of Direct Variation y = kx 24 = k  3 8 = k y = 8x Think: The amount owed varies directly with the amount of cuts given. Substitute 24 for y and 3 for x. Solve for k. Substitute 8 for k in the original equation. x = 3 and y = 24 Step 1 Write the direct variation function.

19. Holt CA Course 1 7-9Direct Variation Step 2 Find how much Rachel will pay the salon owner for 7 cut and styles. Additional Example 2 Continued y = 8(7) y = 56 Substitute 7 for x in the direct variation function. Multiply. Rachel will pay the salon owner $56 for 7 cut and styles.

20. Holt CA Course 1 7-9Direct Variation Rinny cuts and styles hair in a salon. She earns $120 for 4 cut and styles. Write a direct variation function for this situation. If Rinny does 9 cut and styles, how much will she earn? Check It Out! Example 2 y = kx 120 = k  4 30 = k y = 30x Think: The amount owed varies directly with the amount of cuts given. Substitute 120 for y and 4 for x. Solve for k. Substitute 30 for k in the original equation. x = 4 and y = 120 Step 1 Write the direct variation function.

21. Holt CA Course 1 7-9Direct Variation Step 2 Find how much Rinny will earn for 9 cut and styles. Check It Out! Example 2 Continued y = 30(9) y = 270 Substitute 9 for x in the direct variation function. Multiply. Rinny will earn $270 for 9 cut and styles.

22. Holt CA Course 1 7-9Direct Variation Mrs. Perez has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation. Additional Example 3: Money Application

23. Holt CA Course 1 7-9Direct Variation Additional Example 3 Continued A. interest from CD and time interest from CD time = 17 1 = = 17 interest from CD time 34 2 The second and third pairs of data result in a common ratio. In fact, all of the nonzero interest from CD to time ratios are equivalent to 17. The variables are related by a constant ratio of 17 to 1. = = = 17 interest from CD time = 17 1 34 2 51 3 68 4

24. Holt CA Course 1 7-9Direct Variation Additional Example 3 Continued B. interest from money market and time interest from money market time = = 19 19 1 interest from money market time = =18.5 37 2 19 ≠ 18.5 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

25. Holt CA Course 1 7-9Direct Variation Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation. Check It Out! Example 3 InterestInterest from Time (mo)from CD ($)Money Market ($) 000 11215 23040 3 45 450

26. Holt CA Course 1 7-9Direct Variation Check It Out! Example 3 Continued interest from CD time = 12 1 interest from CD time = = 15 30 2 The second and third pairs of data do not result in a common ratio. If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included. A. interest from CD and time

27. Holt CA Course 1 7-9Direct Variation Check It Out! Example 3 Continued B. interest from money market and time interest from money market time = = 15 15 1 interest from money market time = =20 40 2 15 ≠ 20 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

28. Holt CA Course 1 7-9Direct Variation Lesson Quiz: Part I Determine whether the data sets show direct variation. 1. 2. direct variation Amount of Water in a Rain Gauge Time (h)12345 Rain (in)246810 Driving Time Speed (mi/h)3040506080 Time (h)107.5653.75 no direct variation

29. Holt CA Course 1 7-9Direct Variation Lesson Quiz: Part II 3. Roy’s income varies directly with the number of dogs that he walks. He earned $8.50 for walking 2 dogs. Write a direct variation function for this situation. If Roy walks 5 dogs, how much will he earn? y = 4.25x; $21.25