1. Direct Variation y = kx

2. A direct variation is… A linear function Equation can be written in the form; y = mx ; m  0 ory = 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)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= -1 y= x k= y= x k=

5. Which is a direct variation? 1. y = 3x + 2 2. y = 2x 3. 3x – y = 0 4. 9x + 3y = -3 5. 2x + 3y = 0 6. 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 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. This problem: distance varies directly as time d = k t Write an equation for the relationship between the time and distance. Find “k” “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 Example 1 (cont.)

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

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. 1. (4, 2) and (6, y) 2/4 2/4 = y/6y/6 2. (2, 7) and (x, 3) 7/2 7/2 = 3/x3/x 3. (x, 9) and (1.4, 2.8) 9/x 9/x = 2.8 / 1.4 4. Slope is constant: 1 st ) given two ordered pairs: (x,y) & 2 nd ) k = y/xy/x 3 rd ) y/x y/x = y/xy/x 4 th ) solve the proportion for the unknown

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

12. The end