Lesson 3.8
Direct, Inverse, and Joint Variations.
Direct Variation
y varies directly as
that
x
n
if there is some nonzero constant k such
y  kxn , n  0.
k is called the constant of variation.
Example 1.
Suppose y varies directly as x and y = 52
when x = 8.
a. Find the constant of variation and write an
equation of the form y = kxn.
y = kxn
The constant of variation is 6.5
The equation relating x and y is y = 6.5x
Example 1.
b. Use the equation to find the value of y
when x = -6.
y = 6.5x
y = 6.5(-6)
y = -39
When x = -6, the value of y is -39.
Joint Variation Variation
y varies jointly as
constant k such
that
x
n
and
zn
if there is some nonzero
y  kxn z n , where x  0, z  0, and n  0.
k is called the constant of variation.
Example 2.
If y varies jointly as x and z and y = 32 when x = 12
and z = 8, find y when x = 8 and z =18.
a. Find the constant of variation and write an
equation of the form y = kxnzn.
y = kxnzn
3 2  k(1 2 )(8 )
32  96k
1
k 
3
1
y  x nz n
3
Example 2.
b. Use the equation to find the value of y
when x = 8 and z = 18.
1 n n
y x z
3
1
y  (8 )(1 8 )
 48
3
Inverse Variation Variation
y varies inversely as
such that
x
n
if there is some nonzero constant k
k
y  n , n  0.
x
k is called the constant of variation.
Example 3.
If y varies inversely as x and y = 34 when x = 12,
find x when y = 8.
Use a proportion that relates the values.
Example 4.
Write a statement of variation relation the variables
of each equation. Then name the constant of
variation.
a.
A  lw
Answer: A varies jointly as l and w; k = 1
4
b. x  7
y
4
x
Solve for y 
7
Answer: y varies directly as x 4; k = 1/ 7
Example 4.
3
c. x 
y
Solve for y   3
x
Answer: y varies inversely as x, k = -3