8-1
Variation Functions
Vocabulary
direct variation
constant of variation
combined variation
Holt Algebra 2
inverse variation
joint variation
8-1
Variation Functions
Notes
1. The volume V of a pyramid varies jointly as the area of the
base B and the height h, and V = 24 ft3 when B = 12 ft2
and h = 6 ft. Find B when V = 54 ft3 and h = 9 ft.
2. The cost per person c of chartering a tour bus varies
inversely as the number of passengers n. If it costs $22.50
per person to charter a bus for 20 passengers, how much will
it cost per person to charter a bus for 36 passengers?
3. Determine whether each data set represents a direct variation,
an inverse variation, or neither.
3a.
x
y
Holt Algebra 2
3.75 15
12
3
5
9
3b.
x
y
1
0.2
40
8
26
5.2
8-1
Variation Functions
You can use algebra to write variation functions
in terms of k.
Notice that in direct variation, the ratio of the
two quantities is constant. In inverse variation,
the product of the two quantities is constant.
Holt Algebra 2
8-1
Variation Functions
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.
Holt Algebra 2
8-1
Variation Functions
Example 1: Solving Direct Variation Problems
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.
Use a proportion.
e1
e2
=
d1
d2
3.85
10.00
=
5.00
d
12.99 ≈ d
Holt Algebra 2
Substitute.
Solve for d.
8-1
Variation Functions
Example 2: Writing and Graphing Direct Variation
Given: y varies directly as x, and y = 27 when
x = 6. Write and graph the direct variation
function.
y = kx
y varies directly as x.
27 = k(6)
Substitute 27 for y and 6 for x.
k = 4.5
Solve for the constant of variation k.
y = 4.5x
Write the variation function in linear
form using the value of k as the slope.
Holt Algebra 2
8-1
Variation Functions
Example 2 Continued
Graph the direct
variation function.
The y-intercept is 0,
and the slope is 4.5.
Holt Algebra 2
8-1
Variation Functions
Helpful Hint
If k is positive in a direct variation, the value of y
increases as the value of x increases.
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.
Holt Algebra 2
8-1
Variation Functions
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.
Holt Algebra 2
8-1
Variation Functions
Example 3: Solving Joint Variation Problems
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.
Step 1 Find k.
V = kBh
12 = k(9)(4) Substitute.
1 =k
Solve for k.
3
Step 2 Use the variation
function.
1
1 for k.
V = Bh
Use
3
3
1
24 = B(9) Substitute.
3
8 = B
The base is 8 ft2.
Holt Algebra 2
Solve for B.
8-1
Variation Functions
A third type of variation
describes a situation in
which one quantity increases
and the other decreases. For
example, the table shows
that the time needed to
drive 600 miles decreases
as speed increases.
This type of variation is an inverse variation. An
inverse variation is a relationship between two
variables x and y that can be written in the form
y = k , where k ≠ 0. For the equation y = k ,
x
x
y varies inversely as x.
Holt Algebra 2
8-1
Variation Functions
Example 4: Writing and Graphing Inverse Variation
Given: y varies inversely as x, and y = 4
when x = 5. Write and graph the inverse
variation function.
y= k
x
4= k
5
k = 20
y = 20
x
Holt Algebra 2
y varies inversely as x.
Substitute 4 for y and
5 for x.
Solve for k.
Write the variation
formula.
Variation Functions
8-1
Example 4 Continued
To graph, make a table of values for both positive
and negative values of x. Plot the points, and
connect them with two smooth curves. Because
division by 0 is undefined, the function is
undefined when x = 0.
x
–2
–4
–6
–8
Holt Algebra 2
y
–10
–5
– 10
3
– 25
x
2
4
6
8
y
10
5
10
3
5
2
8-1
Variation Functions
Example 5
The time t that it takes for a group of volunteers to construct
a house varies inversely as the number of volunteers v. If 20
volunteers can build a house in 62.5 working hours, how
many working hours would it take 15 volunteers to build a
house?
Use t1v1 = k = t2v2.
(62.5)(20) = 1250
1250 = 15t
Substitute and find
constant product.
Find t.
So the number of working hours it would take 15 volunteers
to build a house is approximately 83.3 hours.
Holt Algebra 2
Variation Functions
8-1
Example 7: Identifying Direct and Inverse Variation
Determine whether each data set represents a
direct variation, an inverse variation, or neither.
A.
x
y
C.
Holt Algebra 2
6.5
8
x
y
13 104
4 0.5
3
5
6
14
8
21
B.
x
y
5
30
8
48
12
72
8-1
Variation Functions
Notes: Part I
1. The volume V of a pyramid varies jointly as the
area of the base B and the height h, and
V = 24 ft3 when B = 12 ft2 and h = 6 ft. Find B
when V = 54 ft3 and h = 9 ft.
18 ft2
2. The cost per person c of chartering a tour bus
varies inversely as the number of passengers
n. If it costs $22.50 per person to charter a
bus for 20 passengers, how much will it cost
per person to charter a bus for 36 passengers?
$12.50
Holt Algebra 2
8-1
Variation Functions
Notes 3
Determine whether each data set represents a
direct variation, an inverse variation, or neither.
3a.
3b.
x
y
x
y
Holt Algebra 2
3.75 15
12
3
1
0.2
40
8
5
9
26
5.2
In each case xy = 45.
The ratio is constant, so
this represents an inverse
variation.
In each case y = 0.2.
x
The ratio is constant, so
this represents a direct
variation.