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Direct Variation

Direct Variation. y = kx. A direct variation is…. A linear function Equation can be written in the form; y = mx ; m  0 or y = kx ; k  0 The y-intercept must be zero!!!! Graphs a line passing through the origin. Constant of Variation: y = k x.

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Direct Variation

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  1. Direct Variation y = kx

  2. A direct variation is… • A linear function • Equation can be written in the form; y = mx ; m  0 or y = kx ; k  0 • The y-intercept must be zero!!!! • Graphs a line passing through the origin.

  3. Constant of Variation: y = kx • The constant of variation is the rate of change for data that describes the variation (slope). • The constant of variation is found by dividing “y” by “x” (k = y/x) • The constant of variation is the slope of a linear equation whose y-intercept is zero!!!!

  4. Constant of Variation: y = kx y= x k= 1 y= 2x k= 2 y= x k= y= x k= y= -x k= -1

  5. Which is a direct variation? For an equation to be that of a direct variation: #1) Put equation into “y=“ form. #2) Check out equation: y = kx + 0 or y= kx no • 1. y = 3x + 2 • 2. y = 2x • 3. 3x – y = 0 • 4. 9x + 3y = -3 • 5. 2x + 3y = 0 • 6. yes yes no yes yes

  6. Example 1 • The distance that you travel at a constant speed varies directly with the time spent traveling. It takes you 2 hours to travel 100 miles. How long will it take you to travel 400 miles? • Direct variation: y varies directly as x y = k x • This problem: distance varies directly as time d = k t

  7. Example 1 (cont.) This problem: distance varies directly as time d = k t Write an equation for the relationship between the time and distance. Find “k” first using the given ordered pair (time, distance) …. (2, 100) 100 = k (2) 50 = k Thus the equation for this problem: d = 50 t

  8. Example 1 (cont.) Now this equation can be used to find any number of distances or times. equation :d = 50 t How long will it take you to travel 400 miles? 400 = 50 t How long will it take you to travel 750 miles? 750 = 50 t How many miles could you travel in 3.5 hours? d = 50 (3.5)

  9. Example 2 • The money you earn varies directly with the number of lawns you mow. You earn $36 for mowing 3 lawns. • Write an equation for the relationship between money earned and lawns mowed. • How much money would you earn for mowing seven lawns? • How many lawns did you mow for $60? Equation: m = 12L Equation: m = 12(7) Equation: 60 = 12L

  10. Suppose the ordered pairs in each problem are for the same direct variation. Find the missing value. Slope is constant: 1st) given two ordered pairs: (x,y) & (x,y) 2nd)k = y/x 3rd)y/x = y/x 4th) solve the proportion for the unknown • 1. (4, 2) and (6, y) 2/4 = y/6 • 2. (2, 7) and (x, 3) 7/2 = 3/x • 3. (x, 9) and (1.4, 2.8) 9/x = 2.8/1.4 • 4.

  11. k = y = x k = Write a direct variation equation for the following: y = kx (2,5) (0,0)

  12. The end

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