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5-5: Direct Variation

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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

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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) What’s Direct Variation?

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

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

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y – 7.5x = 0 y – 7.5x + 7.5x = 0 + 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

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6y = 12x7y = 3x + 4 Quick Check 6y = 12x7y = 3x + 4 6y = 12x 12 y = 2x YES, it’s a direct variation, k = 2 7y = 3x + 4 3 4 NO, it’s NOT a direct variation 66 y = x 6 77 y = x + 7 7

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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

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(4, 16)(3, 2) Quick Check (4, 16)(3, 2) y = kx 16 = k(4) 4 = k y = 4k y = kx 2 = k(3) 2 44 33 3 = k y = x 3

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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. a)Write an equation of direct variation for Tony’s pay and his hours of work. b) What is Tony’s pay on Saturday? c)What will the graph of this problem look like?

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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.

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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.

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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) 1 10 5 = k y = x 5

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Quick Check 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. Let x = the number of hours Let y = the amount of money y = kx 90 = k(5) 18 = k y = 18x 55

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Example 4 xy -10-5 -42 12-4 xyy/x -5-10-10/-5 = 2 2-4-4/2 = 2 -41212/-4 = -3 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

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xyy/x 714 12 -4-8 y/x 14/7 = 2 2/1 = 2 -8/-4 = 2 Yes, the constant of variation is 2. The equation is y = 2x

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Quick Check xyy/x 7-21 22-66 -515 y/x -21/7 = -3 -66/22 = -3 15/-5 = -3 Yes, the constant of variation is -3. The equation is y = -3x

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THE END

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…Or Not Next we’ll play Jeopardy

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