Download Inverse and Joint Variation

9.1
Inverse and Joint Variation
Goals p Write and use inverse variation models.
p Write and use joint variation models.
Your Notes
VOCABULARY
Inverse variation Two variables x and y show inverse
k
variation provided y ! "" where k is a nonzero
x
constant.
Constant of variation The nonzero constant in a direct
variation equation, inverse variation equation, or joint
variation equation
Joint variation A relationship that occurs when a
quantity varies directly as the product of two or
more other quantities
Example 1
Classifying Direct and Inverse Variation
Tell whether x and y show direct variation, inverse variation,
or neither.
Given Equation
Rewritten Equation
Type of Variation
Neither
a. y ! x # 2
b. 2 ! yx
2
y ! ""
x
Inverse
y
c. x ! ""
7
y ! 7x
Direct
Checkpoint Tell whether x and y show direct variation,
inverse variation, or neither.
1. yx ! 1
inverse
variation
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Algebra 2 Notetaking Guide • Chapter 9
y
2. "" ! x
1.3
direct
variation
3. y ! x $ 1
neither
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Your Notes
Writing an Inverse Variation Equation
Example 2
The variables x and y vary inversely, and y ! 2 when x ! 3.
a. Write an equation that relates x and y.
b. Find y when x ! "2.
Solution
a.
k
y ! ##
x
k
2 !
3
6 !k
Write general equation for inverse variation.
Substitute for y and for x.
Solve for k.
6
The inverse variation equation is y ! ## .
x
b. When x ! "2, the value of y is:
6
y ! ## ! "3
"2
Checking Data for Inverse Variation
Example 3
Tell whether the following data show inverse variation. If they
do, find a model for the relationship between a and b.
a
5
10
15
20
25
b
21
10.5
7
5.25
4.2
Solution
Each product ab is equal to 105 . For instance,
(5)(21) ! 105 and (20)(5.25) ! 105 . So, the
data do show inverse variation. A model for the
105
relationship is b ! ## .
a
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Your Notes
Comparing Different Types of Variation
Example 4
Write an equation for the given relationship.
Relationship
Equation
a. v varies directly with u.
v ! ku
b. v varies inversely with u.
k
v ! ""
u
c. w varies jointly with t, u, and v.
w ! ktuv
d. v varies inversely with the cube of u.
k
v ! ""
u3
e. w varies directly with u and inversely with v.
ku
w ! ""
v
Checkpoint Complete the following exercises.
4. The variables x and y vary inversely, and y ! 3.5 when
x ! 4. Find an equation that relates x and y. Find y when
x ! 6.
14 7
y ! ""; ""
x 3
5. Do the data below show inverse variation? If so, find a
model for the relationship between x and y.
x
2
4
6
8
10
y
12
6
4
3
2.4
24
yes; y ! ""
x
Homework
6. Write an equation for the given relationship: y varies
jointly with x and z and varies inversely with the square
of v.
kxz
y ! ""
v2
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Algebra 2 Notetaking Guide • Chapter 9