### Introduction to Algebra

```Variation
Direct and Inverse
Variation

Direct Variation

A variable y varies directly as variable x if
y = kx
for some constant k
 The constant k is called the constant of
variation
 K is also known as the constant of
proportionality
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Variation
2
Variation

Direct Variation Example


State sales tax t varies directly as
the amount of sale s , i.e. t = ks
For tax of \$200 on a \$12.50 sale,
what is the constant of variation ?
t
k = s
t
12.50
= 200.00
= .0625
Question:
Does this look like y = mx + b ?
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Variation
s
3
Direct and Inverse Variation

Direct Variation
 Output varies directly with input

Example: y = kx


OR
x =k
k is the constant of variation
Newton’s Second Law

The resultant force acting on a mass m is directly
proportional to the acceleration a of the mass:
F = ma OR
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y
Variation
F
a =m
4
Variation
 Inverse Variation
Variable y varies inversely as variable x if
y = k
x
for constant of variation k
y
k is also known as the
constant of inverse
proportionality
x
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Variation
5
Direct and Inverse Variation

Inverse Variation Functions
n
 Output varies inversely with x

Example: y = kx–n OR


k is the constant of variation
The Inverse Square Law

The earth’s gravitational force F acting on an
object of mass m is inversely proportional to
the square of the distance r between the mass
and the center of the earth
F =
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yxn = k
GMm
r2
OR Fr2 = GMm
Variation
6
Variation

Inverse Variation Example
At constant temperature
the pressure P of a gas
in a balloon is inversely
proportional to its volume
V so that
k
P =
V
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Variation
P
V
7
Variation Review

Direct Variation




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Output varies directly with input
Example: y = kx OR
y
x =k
Inverse Variation

Output varies inversely with input

Example: y = kx–1 OR yx = k
constant of
variation is k
Inverse Variation Functions

Output varies inversely with xn

Example: y = kx–n OR yxn = k
Variation
8