1.4 Direct Variation and Proportion Objectives: Write and apply direct variation equations. Write and solve proportions. Standard: 2.8.11.P Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically.

I. Determine whether each equation describes a direct variation. Direct Variation - The variable y varies directly as x if there is a nonzero constant k such that y = kx. The equation y = kx is called a direct-variation equation and the number k is called the constant of variation. y = 2x yes y = ½ x yes y = 2x + 1 no y = 3/x no

II. y varies directly as x II. y varies directly as x. Find the constant of variation, k, and write an equation of direct variation that relates the two variables, y = kx. Ex 1. y = -24 when x = 4 -24 = k (4) k = -6 equation: y = -6x Ex 2. y = -16 when x= 2 - 16 = k (2) k = -8 equation: y = -8x Ex 3. y = 1 when x = ½ 1 = k (1/2) k = 2 equation: y = 2x

III. y varies directly as x. Ex. 1 If y is 2.8 when x is 7, find y when x is -4. 2.8 = k (7) .4 = k y = .4 (-4) y = -1.6 Ex. 2 If y is 6.3 when x is 70, find y when x is 5.4. 6.3 = k (70) .09 = k y = .09 (5.4) Y = .486

III. y varies directly as x. Ex. 3 If y is -5 when x is 2.5, find y when x is 6. -5 = k (2.5) -2 = K Y = -2 (6) Y = -12

IV. Use a direct variation equation to solve each word problem. Ex 1. If 6 tickets cost $72, find the cost of 10 tickets. 72 = k (6) 12 = k Y = 12 (10) Y = 120 Ex 2. If 3 CDs on sale cost $18, find the cost of 12 CDs. 18 = k (3) 6 = k Y = 6 (12) y = 72 Ex 3. If 8 sodas cost $3.20, find the cost of 20 sodas. Y = 8

IV. Use a direct variation equation to solve each word problem. Ex 4. Each day Jon rides his bicycle for exercise. When traveling a constant rate, he rides 4 miles in about 20 minutes. At this rate, how long would it take Jon to travel 7 miles? Recall that distance, d, rate, r, and elapsed time, t, are related by the equation d = rt. Rate = 4 miles/20 minutes = 1/5 miles per minute D = 1/5 t 7 = 1/5 t 35 = t

V. Proportions It is said that if y varies directly with x, then y is proportional to x. A proportion is a statement that two ratios are equal. A ratio is the comparison of 2 quantities by division. A proportion of the form a = c can be rearranged as follows: b d a = c a  bd = c  bd b d ad = bc

V. Proportions * Cross-Product Property of Property of Proportion For b  0 and d  0: If a = c, then ad = bc. b d * In a proportion of the form a/b = c/d a and d are the extremes and b and c are the means. * By the Cross-Product Property, the product of the extremes equals the product of the means.

V. Proportions Ex 2. 3 = x Ex 1. w = 10 5 2 4 12 6 = 5x 12w = 40 5 2 Ex 1. w = 10 4 12 6 = 5x 12w = 40 X = 6/5 W = 40/12 W = 10/3

V. Proportions Ex 3. 3x –1 = x 5 2 2 (3x – 1) = 5x 6x -2 = 5x X = 2

Ex. 4 Proportions