NAME
DATE
8-5
PERIOD
Study Guide and Intervention
Variation Functions
Direct Variation and Joint Variation
Direct Variation
y varies directly as x if there is some nonzero constant k such that y = kx. k is called the
constant of variation.
Joint Variation
y varies jointly as x and z if there is some number k such that y = kxz, where k ≠ 0.
Example 1
If y varies directly as
x and y = 16 when x = 4, find x when
y = 20.
y
y
2
1
−
x1 = −
x2
20
16
−
=−
x2
4
16x2 = (20)(4)
x2 = 5
Example 2
If y varies jointly as x and
z and y = 10 when x = 2 and z = 4, find y
when x = 4 and z = 3.
y
Simplify.
The value of x is 5 when y is 20.
y
4·3
2
10
−
=−
y1 = 16, x1 = 4, and y2 = 20
Cross multiply.
y
1
2
−
x 1z 1 = −
x2 z2
Direct variation
2·4
Joint variation
y1 = 10, x1 = 2, z1 = 4, x2 = 4,
and z2 = 3
120 = 8y2
Simplify.
15 = y2
Divide each side by 8.
The value of y is 15 when x = 4 and z = 3.
Exercises
2. If y varies directly as x and y = 16 when
x = 36, find y when x = 54.
3. If y varies directly as x and x = 15
when y = 5, find x when y = 9.
4. If y varies directly as x and x = 33 when
y = 22, find x when y = 32.
5. Suppose y varies jointly as x and z.
Find y when x = 5 and z = 3, if y = 18
when x is 3 and z is 2.
6. Suppose y varies jointly as x and z. Find y
when x = 6 and z = 8, if y = 6 when x is 4
and z is 2.
7. Suppose y varies jointly as x and z.
Find y when x = 4 and z = 11, if y = 60
when x is 3 and z is 5.
8. Suppose y varies jointly as x and z. Find y
when x = 5 and z = 2, if y = 84 when
x is 4 and z is 7.
9. If y varies directly as x and y = 39
when x = 52, find y when x = 22.
11. Suppose y varies jointly as x and z.
Find y when x = 7 and z = 18, if
y = 351 when x is 6 and z is 13.
Chapter 8
10. If y varies directly as x and x = 60 when
y = 75, find x when y = 42.
12. Suppose y varies jointly as x and z. Find y
when x = 5 and z = 27, if y = 480 when
x is 9 and z is 20.
32
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. If y varies directly as x and y = 9 when
x = 6, find y when x = 8.
NAME
DATE
8-5
Study Guide and Intervention
PERIOD
(continued)
Variation Functions
Inverse Variation and Combined Variation
Inverse Variation
y varies inversely as x if there is some nonzero constant k such that xy = k or y = −kx .
Combined
Variation
y varies in combination with x and z if there is some nonzero constant k such that yz = kx or
kx
y=−
z.
Example
If a varies directly as b, and a varies inversely as c, find b when a
equals 10 and c equals -5, if b equals 4 when a equals -2 and c equals 3.
kb
kb
1
2
a1 = −
c1 and a2 = −
c2
ac
ac
b1
ac
b2
a2c2
b1
b2
1 1
2 2
k= −
and k = −
1 1
−
= −
(-2)3
4
10(-5)
b2
− = −
(-2)3 · b2 = 10(-5)4
1
b2 = 33 −
3
Joint Variation Proportions
Solve for k.
Set proportions equal to each other.
Substitute values from problem.
Cross multiply.
Simplify.
1. If y varies inversely as x and y = 12 when x = 10, find y when x = 15.
2. If y varies inversely as x and y = 100 when x = 38, find y when x = 76.
3. If y varies inversely as x and y = 32 when x = 42, find y when x = 24.
4. If y varies inversely as x and y = 36 when x = 10, find y when x = 30.
5. If y varies inversely as x and y = 18 when x = 124, find y when x = 93.
6. If y varies inversely as x and y = 90 when x = 35, find y when x = 50.
7. If y varies inversely as x and y = 42 when x = 48, find y when x = 36.
8. If y varies inversely as x and y = 44 when x = 20, find y when x = 55.
9. If y varies inversely as x and y = 80 when x = 14, find y when x = 35.
10. If y varies inversely as x and y = 3 when x = 8, find y when x = 40.
11. If y varies directly as z and inversely as x and y = 16 and z = 2 when x = 42, find y
when x = 14 and z = 8.
12. If y varies directly as z and inversely as x and y = 23 and z = 1 when x = 12, find y
when x = 15 and z = -3.
Chapter 8
33
Glencoe Algebra 2
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises