NAME _____________________________________________ DATE ____________________________ PERIOD _____________
8-5 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
Example 2: If y varies jointly as x and z and y = 10
when x = 4, find x when y = 20.
when x = 2 and z = 4, find y when x = 4 and z = 3.
𝑦1
𝑥1
16
4
𝑦2
=𝑥
2
20
=𝑥
2
16𝑥2 = (20)(4)
𝑥2 = 5
Direct variation
𝑦1
𝑥1𝑧1
10
𝑦2
=𝑥
2 𝑧2
𝑦2
𝑦1 = 16, 𝑥1 = 4, and 𝑦2 = 20
2·4
Cross multiply.
120 = 8𝑦2
Simplify.
The value of x is 5 when y is 20.
=
4·3
Joint variation
𝑦1 = 10, 𝑥1 = 2, 𝑧1 = 4, 𝑥2 = 4, and 𝑧2 = 3
Simplify.
15 = 𝑦2
Divide each side by 8.
The value of y is 15 when x = 4 and z = 3.
Exercises
1. If y varies directly as x and y = 9 when x = 6,
find y when x = 8.
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.
10. If y varies directly as x and x = 60 when y = 75,
find x when y = 42.
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
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
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
8-5 Study Guide and Intervention (continued)
Variation Functions
Inverse Variation and Combined Variation
𝑘
Inverse Variation
y varies inversely as x if there is some nonzero constant k such that xy = k or y = 𝑥.
Combined Variation
y varies in combination with x and z if there is some nonzero constant k such that
𝑘𝑥
yz = kx or 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.
𝑎1 =
𝑘𝑏1
k=
and 𝑎2 =
𝑐1
𝑎1𝑐1
𝑏1
and k =
𝑎1𝑐1
𝑏1
(−2)3
4
=
=
𝑘𝑏2
𝑐2
𝑎2 𝑐2
𝑏2
𝑎2 𝑐2
𝑏2
10(−5)
𝑏2
(–2)3 · 𝑏2 = 10(–5)4
1
𝑏2 = 333
Joint Variation Proportions
Solve for k.
Set proportions equal to each other.
Substitute values from problem.
Cross multiply.
Simplify.
Exercises
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
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Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
8-5 Skills Practice
Variation Functions
State whether each equation represents a direct, joint, inverse, or combined variation. Then name the constant of
variation.
4
1
1. c = 12m
2. p = 𝑞
3. A = 2bh
4. rw = 15
5. y = 2rgt
6. f = 5280m
7. y = 0.2d
8. vz = –25
9. t = 16rh
8
10. R = 𝑤
1
12. C = 2πr
11. b = 3a
13. If y varies directly as x and y = 35 when x = 7, find y when x = 11.
14. If y varies directly as x and y = 360 when x = 180, find y when x = 270.
15. If y varies directly as x and y = 540 when x = 10, find x when y = 1080.
16. If y varies directly as x and y = 12 when x = 72, find x when y = 9.
17. If y varies jointly as x and z and y = 18 when x = 2 and z = 3, find y when x is 5 and z is 6.
18. If y varies jointly as x and z and y = –16 when x = 4 and z = 2, find y when x is –1 and z is 7.
19. If y varies jointly as x and z and y = 120 when x = 4 and z = 6, find y when x is 3 and z is 2.
20. If y varies inversely as x and y = 2 when x = 2, find y when x = 1.
21. If y varies inversely as x and y = 6 when x = 5, find y when x = 10.
22. If y varies inversely as x and y = 3 when x = 14, find x when y = 6.
23. If y varies directly as z and inversely as x and y = 27 and z = –3 when x = 2, find x when y = 9 and z = 5.
24. If y varies directly as z and inversely as x and y = –15 and z = 5 when x = 5, find x when y = –36 and z = –3.
Chapter 8
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Glencoe Algebra 2