Sect' 9'4 Direct, Joint, and Inverse Variation

1
Sect. 9.4 Direct, Joint, and Inverse Variation
Goal 1 Recognize and Solve Direct and Joint
Variation Problems Goal 2 Recognize and Solve
Inverse Variation Problems
2
Direct Variation Use y kx. Means y varies
directly with x. k is called the constant of
variation.
3
If y varies directly as x, we know
4
If y varies directly as x and y - 15 when x
5, find y when x 3.
y kx
- 15 k(5)
or
k -3
y - 3x
5y -45
y - 3(3)
y - 9
5
Direct Variation is a linear function.
For example d r t If time is constant,
as rate goes up, distance goes up. We can say,
distance varies directly with rate...
Example The value of y varies directly as x.
Write an equation that relates the variables if
x 3 when y 12.
y kx 12 k 3 k 4 y 4x

The constant of variation (k) is 4
6
The area A (in square inches) of a rectangle
varies directly with its width, W, (in inches).
When the width is 4 inches, the area is 12 square
inches. Write an equation that relates A and
W. Find the width of the rectangle when the area
is 24 square inches. A k W This is the
model for direct variation. 12 k 4
Substitute to solve for k.. 3 k A 3
W Find W when A is 24. 24 3 W W
8 inches
7
Joint Variation Use y kxz. Means y varies
jointly with x and z. k is called the constant
of variation.
8
If y varies jointly as x and z
9
Joint Variation
Occurs when a quantity varies directly as the
product of two or more other quantities. Example
The variable z varies jointly with the product
of x and y. Find an equation that relates the
variables if x 4 when y 3 and z 2. z
kxy 2 k(4)(3) 2 12k k 1/6 z
1/6 xy
10
Suppose y varies jointly as x and z. Find y when
x 10, and z 5, if y 12 when z 8 and x 3
y kxz
12 k(8)(3)
or
k .5
y .5x
24y 600
y .5(10)(5)
y 25
11
The Power P in watts of an electrical circuit
varies jointly as the resistance R and the square
of the current I.
For a 600-watt microwave oven that draws a
current of 5.0 amperes, the resistance is 24
ohms.
What is the resistance of a 200-watt refrigerator
that draws a current of 1.7 amperes?
P k R I 2 Joint variation model.
600 k (24)(5)2 Substitute to solve for k..
Find R when P is 200 and I is 1.7.
k 1 so P (1)R I2
200 (1)R(1.7)2
12
Inverse Variation y varies inversely with
x. k is the constant of variation.
As one Quantity increases, the other Quantity
decreases.
13
If y varies inversely as x
14
Inverse Variation
The variables x and y vary inversely if, for a
constant k, yx k or y
k x Example The variables x and y
vary inversely. Find an equation that relates
the variables if x 4 when y 3. yx
k (-3)(4) k k 12
15
If a varies inversely as b and a - 6 when b
2, find a when b - 7
abk
a1b1 a2b2
(- 6)(2) k
(- 6)(2) a(-7)
- 12 k
or
a(- 7) - 12
16
The variables x and y vary inversely, and
y 8 when x 3.

• Write an equation that relates x and y.
• Find y when x -4.

k 24
y -6
17
Tell whether x and y show Direct Variation,
Inverse Variation, or Neither.

• xy 4.8
• y x4

Inverse Variation
Hint Solve the equation for y and take notice
of the relationship.
Neither
Direct Variation
18
Write an equation for each statement.

• y varies directly with x and inversely with z2.
• y varies inversely with x3.
• y varies directly with x2 and inversely with z.
• z varies jointly with x2 and y.
• y varies inversely with x and z.