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11-5

11-5. Direct Variation. Warm Up. Problem of the Day. Lesson Presentation. Pre-Algebra. Direct Variation. 11-5. 1. 2. 1. 2. 1. 1. 4. 7. 3. 7. 3. 4. (9, 3), –. (5, – 2),. (–7, 5), –. Pre-Algebra. Warm Up

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11-5

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

  2. Direct Variation 11-5 1 2 1 2 1 1 4 7 3 7 3 4 (9, 3), – (5, –2), (–7, 5), – Pre-Algebra 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

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

  4. Today’s Learning Goal Assignment Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

  5. Vocabulary direct variation constant of proportionality

  6. Helpful Hint 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.

  7. Additional Example 1A: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. A.

  8. Additional Example 1A Continued Make a graph that shows the relationship between Adam’s age and his length.

  9. 27 22 12 3 ? = Additional Example 1A Continued You can also compare ratios to see if a direct variation occurs. 81 81 ≠ 264 The ratios are not proportional. 264 The relationship of the data is not a direct variation.

  10. Try This: Example 1A Determine whether the data set shows direct variation. A.

  11. Try This: Example 1A Continued Make a graph that shows the relationship between number of baskets and distance. 5 4 3 Number of Baskets 2 1 20 30 40 Distance (ft)

  12. 3 5 30 20 ? = Try This: Example 1A You can also compare ratios to see if a direct variation occurs. 60 150  60. The ratios are not proportional. 150 The relationship of the data is not a direct variation.

  13. Additional Example 1B: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. B.

  14. Additional Example 1B Continued Make a graph that shows the relationship between the number of minutes and the distance the train travels. Plot the points. The points lie in a straight line. (0, 0) is included.

  15. 25 10 75 100 50 30 40 20 Additional Example 1B Continued You can also compare ratios to see if a direct variation occurs. Compare ratios. = = = The ratios are proportional. The relationship is a direct variation.

  16. Try This: Example 1B Determine whether the data set shows direct variation. B.

  17. 4 3 Number of Cups 2 1 32 8 16 24 Number of Ounces Try This: Example 1B Continued Make a graph that shows the relationship between ounces and cups. Plot the points. The points lie in a straight line. (0, 0) is included.

  18. 1 = = = 8 3 4 2 24 32 16 Try This: Example 1B Continued You can also compare ratios to see if a direct variation occurs. Compare ratios. The ratios are proportional. The relationship is a direct variation.

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

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

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

  22. = k y = k Substitute for k in the original equation. 1 1 1 2 2 2 Try This: Example 2B B. x is 28 when y is 14 y = kx y varies directly with x. 14 = k28 Substitute for x and y. Solve for k.

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

  24. = k y = k Substitute for k in the original equation. 7 7 7 3 3 3 Try This: Example 2C C. y is 7 when x is 3 y = kx y varies directly with x. 7 = k3 Substitute for x and y. Solve for k.

  25. Additional Example 3: Money Application 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.

  26. = = = 17 51 68 34 4 2 3 interest from CD interest from CD interest from CD 17 34 17 = = 17 = = = time time time 1 2 1 Additional Example 3 Continued A. interest from CD and time 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.

  27. 19 37 1 2 interest from money market interest from money market = = 19 = =18.5 time time Additional Example 3 Continued B. interest from money market and time 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.

  28. Try This: Example 3 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.

  29. Try This: Example 3 Continued

  30. interest from CD interest from CD 12 30 = = = 15 time time 1 2 Try This: Example 3 Continued A. interest from CD and time 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.

  31. 15 40 1 2 interest from money market interest from money market = = 15 = =20 time time Try This: Example 3 Continued B. interest from money market and time 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.

  32. y = x y = x 1 6 9 5 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

  33. 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

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