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Direct Variation: Learn and Apply Concepts Through Examples

Gain a thorough understanding of direct variation through graphing, ratios, and equations with practical examples in this comprehensive course. Explore how to recognize, analyze, and apply direct variation in various scenarios.

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Direct Variation: Learn and Apply Concepts Through Examples

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  1. 12-5 Direct Variation Course 3 Warm Up Problem of the Day Lesson Presentation

  2. Direct Variation 12-5 1 2 1 2 1 1 4 7 3 7 3 4 (9, 3), – (5, –2), (–7, 5), – Course 3 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. Direct Variation 12-5 Course 3 Problem of the Day Where do the lines defined by the equations y = –5x + 20 and y = 5x – 20 intersect? (4, 0)

  4. Direct Variation 12-5 Course 3 Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

  5. Direct Variation 12-5 Course 3 Insert Lesson Title Here Vocabulary direct variation constant of proportionality

  6. Direct Variation 12-5 Course 3

  7. Direct Variation 12-5 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. Course 3

  8. Direct Variation 12-5 Course 3 Additional Example 1A: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation.

  9. Direct Variation 12-5 Course 3 Additional Example 1A Continued Make a graph that shows the relationship between Adam’s age and his length. The graph is not linear.

  10. Direct Variation 12-5 27 22 12 3 ? = Course 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.

  11. Direct Variation 12-5 Course 3 Additional Example 1B: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation.

  12. Direct Variation 12-5 Course 3 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.

  13. Direct Variation 12-5 25 10 75 100 50 30 40 20 Course 3 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.

  14. Direct Variation 12-5 Course 3 Check It Out: Example 1A Determine whether the data set shows direct variation.

  15. Direct Variation 12-5 Course 3 Check It Out: Example 1A Continued Make a graph that shows the relationship between number of baskets and distance. The graph is not linear. 5 4 3 Number of Baskets 2 1 20 30 40 Distance (ft)

  16. Direct Variation 12-5 3 5 30 20 ? = Course 3 Check It Out: Example 1A Continued 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.

  17. Direct Variation 12-5 Course 3 Check It Out: Example 1B Determine whether the data set shows direct variation.

  18. Direct Variation 12-5 4 3 Number of Cups 2 1 32 8 16 24 Number of Ounces Course 3 Check It Out: 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.

  19. Direct Variation 12-5 1 = = = 8 3 4 2 24 32 16 Course 3 Check It Out: 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.

  20. Direct Variation 12-5 Course 3 Additional Example 2A: Finding Equations of Direct Variation Find each equation of direct variation, given that y varies directly with x. 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

  21. Direct Variation 12-5 = k y = x Substitute for k in the original equation. 5 5 5 4 4 4 Course 3 Additional Example 2B: Finding Equations of Direct Variation 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. Direct Variation 12-5 Course 3 Check It Out: Example 2A Find each equation of direct variation, given that y varies directly with x. 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 6 for k in the original equation. y = 6x

  23. Direct Variation 12-5 = k y = x Substitute for k in the original equation. 1 1 1 2 2 2 Course 3 Check It Out: Example 2B 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.

  24. Direct Variation 12-5 Course 3 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.

  25. Direct Variation 12-5 interest from CD 34 = = = 17 = = 17 time 2 68 51 34 4 3 2 interest from CD interest from CD 17 17 = = = time time 1 1 Course 3 Additional Example 3 Continued 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.

  26. Direct Variation 12-5 37 19 2 1 interest from money market interest from money market = =18.5 = = 19 time time Course 3 Additional Example 3 Continued 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.

  27. Direct Variation 12-5 Course 3 Check It Out: 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.

  28. Direct Variation 12-5 interest from CD 30 = = 15 time 2 interest from CD 12 = time 1 Course 3 Check It Out: 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.

  29. Direct Variation 12-5 40 15 2 1 interest from money market interest from money market = =20 = = 15 time time Course 3 Check It Out: 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.

  30. Direct Variation 12-5 y = x y = x 1 6 9 5 Course 3 Insert Lesson Title Here Lesson Quiz: Part I 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

  31. Direct Variation 12-5 Course 3 Insert Lesson Title Here Lesson Quiz: Part II 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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