1. Pre-Algebra 11-5 Direct Variation 11-5 Direct Variation Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Pre-Algebra 11-5 Direct Variation Warm Up Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. 1. y – 3 = – (x – 9) 2. y + 2 = (x – 5) 3. y – 9 = –2(x + 4) 4. y – 5 = – (x + 7) (–4, 9), –2 Pre-Algebra 11-5 Direct Variation 1 7 2 3 1 4 (9, 3), – 1 7 (5, –2), 2 3 (–7, 5), – 1 4

3. Pre-Algebra 11-5 Direct Variation Problem of the Day Where do the lines defined by the equations y = -5x + 20 and y = 5x – 20 intersect? (4, 0)

4. Pre-Algebra 11-5 Direct Variation Today’s Learning Goal Assignment Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

5. Pre-Algebra 11-5 Direct Variation Vocabulary direct variation constant of proportionality

6. Pre-Algebra 11-5 Direct Variation

7. Pre-Algebra 11-5 Direct Variation The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together. Helpful Hint

8. Pre-Algebra 11-5 Direct Variation Determine whether the data set shows direct variation. A. Additional Example 1A: Determining Whether a Data Set Varies Directly

9. Pre-Algebra 11-5 Direct Variation Make a graph that shows the relationship between Adam’s age and his length. Additional Example 1A Continued

10. Pre-Algebra 11-5 Direct Variation You can also compare ratios to see if a direct variation occurs. 22 3 27 12 = ? 81 264 81 ≠ 264 The ratios are not proportional. The relationship of the data is not a direct variation. Additional Example 1A Continued

11. Pre-Algebra 11-5 Direct Variation Determine whether the data set shows direct variation. A. Try This: Example 1A Kyle's Basketball Shots Distance (ft)203040 Number of Baskets530

12. Pre-Algebra 11-5 Direct Variation Make a graph that shows the relationship between number of baskets and distance. Try This: Example 1A Continued Number of Baskets Distance (ft) 2 3 4 203040 1 5

13. Pre-Algebra 11-5 Direct Variation You can also compare ratios to see if a direct variation occurs. Try This: Example 1A 5 20 3 30 = ? 60 150 150  60. The ratios are not proportional. The relationship of the data is not a direct variation.

14. Pre-Algebra 11-5 Direct Variation Determine whether the data set shows direct variation. B. Additional Example 1B: Determining Whether a Data Set Varies Directly

15. Pre-Algebra 11-5 Direct Variation Make a graph that shows the relationship between the number of minutes and the distance the train travels. Additional Example 1B Continued Plot the points. The points lie in a straight line. (0, 0) is included.

16. Pre-Algebra 11-5 Direct Variation You can also compare ratios to see if a direct variation occurs. The ratios are proportional. The relationship is a direct variation. 25 10 50 20 75 30 100 40 === Compare ratios. Additional Example 1B Continued

17. Pre-Algebra 11-5 Direct Variation Determine whether the data set shows direct variation. B. Try This: Example 1B Ounces in a Cup Ounces (oz)8162432 Cup (c)1234

18. Pre-Algebra 11-5 Direct Variation Make a graph that shows the relationship between ounces and cups. Try This: 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.

19. Pre-Algebra 11-5 Direct Variation You can also compare ratios to see if a direct variation occurs. Try This: Example 1B Continued The ratios are proportional. The relationship is a direct variation. Compare ratios. = 1 8 == 2 16 3 24 4 32

20. Pre-Algebra 11-5 Direct Variation Find each equation of direct variation, given that y varies directly with x. A. y is 54 when x is 6 Additional Example 2A: Finding Equations of Direct Variation y = kx 54 = k  6 9 = k y = 9x y varies directly with x. Substitute for x and y. Solve for k. Substitute 9 for k in the original equation.

21. Pre-Algebra 11-5 Direct Variation Find each equation of direct variation, given that y varies directly with x. A. y is 24 when x is 4 Try This: Example 2A y = kx 24 = k  4 6 = k y = 6x y varies directly with x. Substitute for x and y. Solve for k. Substitute 9 for k in the original equation.

22. Pre-Algebra 11-5 Direct Variation B. x is 12 when y is 15 Additional Example 2B: Finding Equations of Direct Variation y = kx 15 = k  12 y varies directly with x. Substitute for x and y. Solve for k. = k 5 4 Substitute for k in the original equation. 5 4 y = k 5 4

23. Pre-Algebra 11-5 Direct Variation B. x is 28 when y is 14 Try This: Example 2B y = kx 14 = k  28 y varies directly with x. Substitute for x and y. Solve for k. = k 1 2 Substitute for k in the original equation. 1 2 y = k 1 2

24. Pre-Algebra 11-5 Direct Variation C. y is 8 when x is 5 Additional Example 2C: Finding Equations of Direct Variation y = kx 8 = k  5 y varies directly with x. Substitute for x and y. Solve for k. = k 8 5 Substitute for k in the original equation. 8 5 y = k 8 5

25. Pre-Algebra 11-5 Direct Variation C. y is 7 when x is 3 Try This: Example 2C y = kx 7 = k  3 y varies directly with x. Substitute for x and y. Solve for k. = k 7 3 Substitute for k in the original equation. 7 3 y = k 7 3

26. Pre-Algebra 11-5 Direct 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

27. Pre-Algebra 11-5 Direct Variation Additional Example 3 Continued A. interest from CD and time interest from CD time = 17 1 interest from CD time = = 17 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, and (0, 0) is included. The equation of direct variation is y = 17x, where x is the time, y is the interest from the CD, and 17 is the constant of proportionality. = = = 17 interest from CD time = 17 1 34 2 51 3 68 4

28. Pre-Algebra 11-5 Direct 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.

29. Pre-Algebra 11-5 Direct 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. Try This: Example 3

30. Pre-Algebra 11-5 Direct Variation Try This: Example 3 Continued InterestInterest from Time (mo)from CD ($)Money Market ($) 000 11215 23040 3 45 450

31. Pre-Algebra 11-5 Direct Variation Try This: 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

32. Pre-Algebra 11-5 Direct Variation Try This: 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.

33. Pre-Algebra 11-5 Direct Variation Lesson Quiz: Part 1 Find each equation of direct variation, given that y varies directly with x. 1. y is 78 when x is 3. 2. x is 45 when y is 5. 3. y is 6 when x is 5. y = 26x y = x 1 9 6 5

34. Pre-Algebra 11-5 Direct Variation Lesson Quiz: Part 2 4. The table shows the amount of money Bob makes for different amounts of time he works. Determine whether there is a direct variation between the two sets of data. If so, find the equation of direct variation. direct variation; y = 12x