1. 1.4 Direct Variation and Proportion
Objectives: Write and apply direct variation equations. Write and solve proportions.
Standard: P Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically.

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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.

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

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

13. Ex. 4 Proportions

15. Homework
Pg #14 – 56 even