Modeling Functions
Using Variation
Section 3.6
Direct Variation
Let x and y represent two quantities. The
following are equivalent statements:
 y = kx, where k is a nonzero constant.
 y varies directly with x.
 y is directly proportional to x.
The constant k is called the constant of
variation or the constant of
proportionality.
Write an equation that
describes variation.

V varies directly with x³
V=kx³


z varies directly with m
z=km
y is directly proportional to the square
root of x
y=k x
Write an equation to
represent the variation.

W is directly proportional to the cube of H.
W = 128 and H = 4
W  kH
3
128  k  4
3
128  64 k
2k
Plug the k-value back into original.
W  2H
3
Inverse Variation
Let x and y represent two quantities. The following
are equivalent statements:
k
 y=
, where k is a nonzero constant.
x
 y varies inversely with x.
 y is inversely proportional to x.
The constant k is called the constant of variation
or the constant of proportionality.
NOTE: For inverse variation we see that when the x
increases, the y decreases or visa versa.
Write an equation that
describes variation.

f varies inversely with c
f= k
c

A varies inversely with t
A= k
t

y is inversely proportional to the cube
of x
y= k
x3
Write an equation to
represent the variation.

y varies inversely with x. y = 400 and
x = 1000.
k
y
x
k
400 
1000
400,000  k
400 ,000
y
x
Joint Variation
When one quantity is proportional to the product of two or
more other quantities, it is called Joint Variation.
I is directly proportional to P, r and t.

For example: I = Prt which represents the simple
interest formula where:
I is the interest in dollars
P is the principal (initial) dollars
r is the interest rate (decimal form)
t is the time in years
Write an equation to
describe the variation.

A is directly proportional to both b and h.
A = 10 when b = 5 and h = 4.
A  kbh
10  k  5  4
10  20 k
1
k
2
1
A  bh
2
Write an equation to
represent the variation.

F varies inversely with both λ and L.
F = 20π when λ = 1 and L = 100
k
F
L
k
20 
1 100
2000   k
2000 
F
L
Combined Variation
When the variation is mixed, this is called
Combined Variation.
P is directly proportional to T and inversely
proportional to V.

T
V
For example: P = k
which represents the
combined gas law in chemistry.
P is pressure
T is temperature
V is volume
k is a gas constant
Example of Direct Variation
Find a mathematical model describing the monthly long distance bill if
the cost is directly proportional to the number of minutes, and a
customer that used 160 minutes is billed $40.
1.
Declare your variables:


2.
Set up your direct variation equation and solve for the constant
of variation:
C = km


3.
c = is the cost in dollars
m = the number minutes
40 = k(160)
¼ =k
Solution: c = ¼ m
This equation describes the relationship between number of minutes
and cost. They are directly proportional.
Example of Inverse Variation
In New York City, the number of potential buyers in the
housing market is inversely proportional to the price of
a house. If there are 12,500 potential buyers for a $2
million condominium, how many potential buyers are
there for a $5 million condominium?
1. Declare your variables:
 p = price of a house
 b = potential buyers
k
b
p
2. Set up your inverse variation equation and solve for
the constant of variation:
k
 12500 =
2000000

2.5 X 1010 = k
3. Inverse variation equation: b = 2.5 X 1010
p
How many potential buyers are
there for a $5,000,000 condo?
10
2
.
5
X
10
4. Solution: b =
5000000
b = 5000
There are 5000
potential buyers for
condos at
$5,000,000.
2.5 X 1010
b
p
Levi’s makes jeans in a variety of price ranges for men. The
Silver Tab Baggy jeans sell for about $30, and they are trying to
figure out how much to sell the Offender jeans for. The demand
for Levi’s jeans is inversely proportional to the price. If 400,000
pairs of the Silver Tab jeans were bought, and they have 150,000
pairs of Offender jeans to sell, what should they list the price at
for the Offenders?
k
d
p
k
400 ,000 
30
12,000,000  k
12,000,000
d
p
12,000,000
150,000 
p
150,000p  12,000,000
p  80
$80 for Offenders
The gas in the headspace of a soda bottle has a volume of 9.0mL,
pressure of 2 atm, and a temperature of 298 K. If the soda bottle
is stored in a refrigerator, the temperature drops to approximately
279 K. What is the pressure of the gas in the headspace once the
bottle is chilled?
T
Pk
V
298
2k
9
18
k
298
18 T
P

298 V
18 279
P

298 9
P  1.87
About 1.87 atm