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Lesson 9-1 (pp. 478–483)
Inverse Variation
Lesson Objectives
1 Using inverse variation
2 Using combined variation
NAEP 2005 Strand: Algebra
Topic: Patterns, Relations, and Functions
Local Standards: ____________________________________
Vocabulary.
k
An inverse variation is a function of the form y x or xy k, where k 0.
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All rights reserved.
A combined variation combines direct and inverse variations in more complicated
relationships.
Examples of Combined Variations
Combined Variation
Equation Form
with the square of x.
y kx2
with the cube of x.
y 5 k3
jointly
with x and y.
z kxy
jointly
with x and y and
y varies
directly
y varies
inversely
z varies
z varies
with w.
directly
z varies
with x and
the product of w and y.
x
inversely
inversely
with
kxy
z5 w
kx
z 5 wy
Examples.
1 Modeling Inverse Variation Suppose that x and y vary inversely, and x 7
when y 4. Write the function that models the inverse variation.
y5k
x
4
28
x and y vary inversely.
k
7
k
y Substitute the given values of x and y.
Find k.
28
x
Use the value of k to write the function.
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Lesson 9-1
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2 Identifying Direct and Inverse Variations Is the relationship between the
variables in each table a direct variation, an inverse variation, or neither? Write
functions to model the direct and inverse variations.
2
4
14
y
0.7
0.35
0.1
decreases
. The product of each pair of x- and y-values is
inversely
with x and the constant of variation is
As x increases, y
1.4 . y varies
So xy b.
and the function is y 1.4
x
2
y
6
1.3
4
decreases
As x increases, y
the products of x and y are the
c.
direct
x
2
y
5
.
7
5
neither a
1.4
x
1.4 .
4
, but this is not an inverse variation. Not all
same
(2 6 1.3 5). This is
variation nor an inverse variation.
6
10 15
decreases
As x increases, y
corresponding x-value, y varies
2.5
variation is
. Since each y-value is 2.5 times the
directly
with x and the constant of
, and the function is y 2.5x
.
3 Finding a Formula The area of an equilateral triangle varies directly as the
square of the radius r of its circumscribed circle. The area of an equilateral
triangle for which r 2 is 3!3. Find the formula for the area A of an equilateral
triangle in terms of r.
A kr2
(
3!3 k
3!3
A varies
2
directly
as the square of r.
2
)
Substitute the values for A and r.
k
Solve for k.
4
A
3!3
r2
Substitute the value for k.
4
174
Lesson 9-1
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x
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a.
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Check Understanding.
1. Suppose that x and y vary inversely, and x 0.3 when y 1.4. Write the function
that models the inverse variation.
y 0.42
x
2. Is the relationship between the values in each table a direct variation, an inverse
variation, or neither? Write functions to model the direct and inverse variations.
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a.
b.
c.
x
0.8
0.6
0.4
y
0.9
1.2
1.8
x
2
4
6
y
3.2
1.6
1.1
x
1.2
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
y
18
1.4
21
1.6
inverse; y 0.72
x
neither
direct; y 15x
24
3. The volume of a square pyramid with congruent edges varies directly as the cube
of the length of an edge. The volume of a square pyramid with edge length 4 is
32 !2
3 . Find the formula for the volume V of a square pyramid with congruent
edges of length e.
3
V !2
6 e
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Lesson 9-1
175