Date: Click here to enter text.
Section: 5.1
Objective(s): Solve problems involving direct, inverse, joint, and combined variation
Direct variation
y varies directly as x
𝑦 = 𝑘𝑥
where k is the constant of variation
also read as “y is directly proportional to x”
A direct variation equation is a linear equation in the form y = mx +
b, where b = 0 and the constant of variation k is the slope. Because b =
0, the graph of a direct variation always passes through the origin.
𝑘
Inverse variation
y varies indirectly as x
1
𝑦 = 𝑥 or 𝑦 = 𝑘 (𝑥)
where k is the constant of variation
An inverse variation is a relationship between two variables x and y
𝑘
that can be written in the form 𝑦 = 𝑥 , where k ≠ 0. For the equation =
𝑘
𝑥
Joint variation
y varies jointly as x
, y varies inversely as x.
𝑦 = 𝑘𝑧𝑥
where k is the constant of variation
A joint variation is a relationship among three variables that can be
written in the form y = kxz, where k is the constant of variation. For
the equation y = kxz, y varies jointly as x and z.
Combined variation
y varies inversely as x and directly
as z
𝑘𝑧
𝑦 = 𝑥 ; Note: x is in the denominator because y varies inversely as x.
z is in the numerator because y varies directly as z.
A combined variation is a relationship that contains both direct and
inverse variation. Quantities that vary directly appear in the
numerator, and quantities that vary inversely appear in the
denominator.
An example of direct variation:
The cost of an item in euros e varies
directly as the cost of the item in
dollars d, and e = 3.85 euros when d =
$5.00. Find d when e = 10.00 euros.
1) Find k
𝑒 = 𝑘𝑑
3.85 = 𝑘(5.00)
3.85
𝑘 = 5.00 = .77
2) Substitute k into original problem:
10.00 = (. 77)𝑑
10.00
𝑑=
= 12.99
. 77
3) 10.00 euros = $12.99
An example of inverse variation:
Given: y varies inversely as x, and y = 4
when x = 5. Write the inverse variation
function and graph.
1) Find k
𝑘
𝑦=
𝑥
4=
𝑘
5
𝑘 = 20
2) Substitute k into original problem
20
𝑦 = 𝑥 ; this is the inverse variation function.
3) Using either a calculator graphing utility or a table of values:
An example of joint variation:
The volume V of a cone varies jointly
as the area of the base B and the height
h, and V = 12 ft3 when B = 9 ft3 and
h = 4 ft. Find B when V = 24 ft3 and h
= 9 ft.
1) Find k
𝑉 = 𝑘𝐵ℎ
12𝜋 = 𝑘(9𝜋)(4)
12𝜋 = 36𝜋𝑘
12𝜋
1
𝑘=
=
36𝜋
3
2) Substitute into original problem:
1
24𝜋 = (3)(B)(9)
24𝜋 = 3𝐵
𝐵 = 8𝜋
The base is 8𝜋 ft2.
An example of combined variation:
The change in temperature (∆𝑇) of an
aluminum wire varies inversely as its
mass m and directly as the amount of
heat energy E transferred. The
temperature of an aluminum wire with
a mass of 0.1 kg rises 5°C when 450
joules (J) of heat energy are transferred
to it. How much heat energy must be
transferred to an aluminum wire with a
mass of 0.2 kg raise its temperature
20°C?
1) Find k.
𝑘𝐸
∆𝑇 = 𝑚
5=
𝑘(450)
.1
. 5 = 450𝑘
1
𝑘= (
)
900
2) Substitute into original problem:
1
(900) (𝐸)
20 =
.2
1
4=(
)𝐸
900
𝐸 = (4)(900) = 3600
The amount of heat energy that must be transferred is 3600
Joules.