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Vocabulary

Vocabulary. direct variation constant of variation. Direct variation is a linear relationship between two variable that can be written in the form y = k x or k = , where k  0. The fixed number k in a direct variation equation is the constant of variation. y x. Reading Math.

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Vocabulary

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  1. Vocabulary direct variation constant of variation

  2. Direct variation is a linear relationship between two variable that can be written in the form y = kx or k = , where k 0. The fixed number k in a direct variation equation is the constant of variation. yx

  3. Reading Math You can read direct variation as “y varies directly as x” or “y is directly proportional to x” or “y varies with x.”

  4. Additional Example 1A: Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y + 8 = x Solve the equation for y. Subtract 8 from both sides. y + 8 = x – 8 = – 8 y = x – 8 The equation is not in the form y = kx, so y + 8 = x is not a direct variation.

  5. 3y = 2x 33 2x3 23 23 Write as x . y = x The equation is in the form y = kx, so the original equation 3y = 2x is a direct variation. The constant of variation is . 23 Additional Example 1B: Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 3y = 2x Solve the equation for y. Divide both sides by 3.

  6. Check It Out: Example 1 Tell whether each equation represents a direct variation. If so, identify the constant of variation. A. 3y + 4x = 0 4 3 yes; k = – y –12x = 2x B. no

  7. yx Find for each ordered pair. y 2 y 4 y 3 1 = = = = x 69 x 129 x 99 33 k is not the same for each ordered pair. The data do not represent a direct variation. Additional Example 2A: Identifying a Direct Variation from a Table Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

  8. Helpful Hint In a direct variation where k is positive, when x increases, y also increases; when x decreases, y also decreases.

  9. yx Find for each ordered pair. y 2.54 y 5.08 y 12.7 = = 2.54 = = 2.54 = = 2.54 x 1 x 2 x 5 k = 2.54 for each ordered pair. The data represent a direct variation where k = 2.54. The equation is y = 2.54x Additional Example 2B: Identifying a Direct Variation from a Table Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

  10. Yes; k = 7.5; y = 7.5x Check It Out: Example 2 Tell whether the set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

  11. y The graph is a line through (0, 0). This is a direct variation. The Slope of the line is – , so k = – . The equation is y = – x. 4 2 x 12 0 12 –2 2 4 –4 –2 12 –4 Additional Example 3: Identifying a Direct Variation from a Graph Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

  12. Helpful Hint In a direct variation, the slope, k, represents a constant rate of change.

  13. y 4 2 x 0 –2 2 4 –4 –2 14 Yes; k = ; y = x –4 Check It Out: Example 3A Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. 14

  14. y 4 2 x 0 –2 2 4 –4 –2 –4 Check It Out: Example 3B Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. no, not a direct variation

  15. Additional Example 4A: Application A truck travels at a speed of 55 miles per hour. a. Write a direct variation equation for the distance y the truck travels in x hours. distance = times number of hours 55 miles per hour Use the formula y = kx. k = 55 y = 55  x y = 55x

  16. 12 yes; k = Lesson Quiz: Part I Tell whether each of the following represents a direct variation. If so, identify the constant of variation. 1. 12y = 6x 2. no

  17. Lesson Quiz: Part II 3. A cheetah runs at a speed of 0.75 mile per minute. a. Write a direct variation equation for the distance y the cheetah runs in x minutes. b. Graph the data. c. How far does the cheetah run in 5 minutes? y = 0.75x 8 Distance (mi) 6 4 2 3.75 miles 2 4 6 8 Time (min)

  18. Lesson Quiz for Student Response Systems 1. Tell whether the equation represents a direct variation. If so, identify the constant of variation. y = 1.5x A. yes; k = 0 B. yes; k = 1 C. yes; k = 1.5 D. no

  19. Lesson Quiz for Student Response Systems 2. Tell whether the data set represents a direct variation. If so, identify the constant of variation. A. yes; k = 10 B. yes; k = 5 C. yes; k = 0 D. no

  20. Lesson Quiz for Student Response Systems 3. An employee’s pay is $8.50 per number of hours worked. Write a direct variation equation for the amount y the employee gets in x number of hours. Graph the data. How much does the employee get in 8 hours? A.y = 8.5x; $52 B.y = 8.5x; $68

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