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5-5: Direct Variation

5-5: Direct Variation. What’s Direct Variation?. Direct variation is a function where y = kx, where k ≠ 0 The variables y and x are vary directly with each other, where k is the constant of variation. What’s Direct Variation?. To put simply:

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5-5: Direct Variation

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  1. 5-5: Direct Variation

  2. What’s Direct Variation? • Direct variation is a function where y = kx, where k ≠ 0 • The variables y and x are vary directly with each other, where k is the constant of variation

  3. What’s Direct Variation? • To put simply: • In a direct variation, when one value increases, the other also increases. (So in the equation y = kx, when y increases, x also increases)

  4. Identify • The equation is a direct variation when… - it can be written in the form of y = kx

  5. Example 1 • Is the equation a direct variation? If it is, find the constant of variation. y 7.5x = 0

  6. y – 7.5x = 0 y – 7.5x + 7.5x = 0 + 7.5x y = 7.5x y = 7.5x • YES, it is a direct variation because it can be written as the form y = kx, the constant of variation (k) = 7.5

  7. Quick Check 6 6 7 7 y = x y = x + 6 7 7

  8. Example 2 • Write an equation of the direct variation that includes the given point. (5,1) • Start with the function form • Substitute (5,1) with (x,y) • Solve for k • Substitute 1/5 for k y = kx 1 = k(5) k = 1/5 y = 1/5x

  9. Quick Check 4 4 3 3 = k 3 y = x 3

  10. Example 3 • Tony works at a Pizza store, his pay (n) varies directly with his hours of work (w). On Saturday, Tony worked for 3 hours at the store, and his hourly pay is 20$. Answer the following questions. • Write an equation of direct variation for Tony’s pay and his hours of work. • What is Tony’s pay on Saturday? • What will the graph of this problem look like?

  11. Example 3 (Answers) • a) y = kx  n = 20$w • b) n = 20$(3)  n = 60$ • c) The graph will be positive, since in a direct variation, if one variable increases, the other also increases.

  12. Example 3 • Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. Write an equation for the relationship between time and distance.

  13. Relate:The distance varies directly with the time. When x = 10, y = 2 • Define: Let x = the # of seconds between seeing lightning and hearing thunder Let y = distance in miles from the lightning y = kx 2 = k(10) 11 10 10 = k y = x 5 5

  14. Quick Check • Let x = the number of hours • Let y = the amount of money • If you work for 5 hours, you’ll get $90. Write a direct variation for the relationship between the number of hours and the amount of money. y = kx 90 = k(5) 18 = k y = 18x 5 5

  15. Example 4 • For each table, use the ratio y/x to tell whether y varies directly with x. If it does, write an equation for the direct variation No, the ratio y/x is not the same for all pairs of data

  16. Yes, the constant of variation is 2. The equation is y = 2x

  17. Quick Check Yes, the constant of variation is -3. The equation is y = -3x

  18. THE END

  19. …Or NotNext we’ll play Jeopardy

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