.
9.2
Direct and Inverse Variation
1
Direct Variation
The graph of a direct variation is a line that goes
through the origin.
Formal definition: y varies directly as x
if there is some nonzero constant k such that y = kx.
k is called the constant of variation
EX. y = 17x

k = 17
2
Example: y varies directly as x.
When x = 5, y = -15.
Find y when x = 3.
There are many ways to solve this,
let’s use a direct variation.
y = -3(3)
y = kx
-15 = k(5)
k = -3
y = -9
3
Practice Problem
9
y varies directly as x. When x 
, y = -6.
16
4
Find k and x when y  .
5
y  kx
32
y x
9

3
6  k  
4
32
 16 


x
16


5
3
6    k
9
4 3 


x


32
5  32 
k
3
3
 x
40
4
Inverse Variation
Formal Definition: y varies inversely as x
if there is some nonzero constant k such that
k
xy = k or y  .
x
5
Example:
If y varies inversely as x
and y = -6 when x = 2,
find y when x = -7.
Use the equation for an inverse variation.
k
6 
2
12  k
k
y
x
12
y
7
12
y
7
6
Example 2: y varies inversely as x,
7
4 when
x ,
y
18 25
7
find k and x when y  
.
6
k
2
y
4
k
25
x

  9
7
7
6
x
25
2
18

x

4 7 
6
9

k
 
7  18 
2 6 
x   
9  25 
2
k
4
9
x
75 7
If y varies inversely as x,
and y = 5 when x = 10,
1
find y when x  
.
4
k
y
x
k
5
10
50  k
50
y
1

4
y  200
8
The graph of an Inverse Variation
is a Rectangular Hyperbola
k
y
x
k > 0 (k is positive)
-10
k < 0 (k is negative)
8
8
6
6
4
4
2
2
-5
5
10
-10
-5
5
-2
-2
-4
-4
-6
-6
-8
-8
As x gets larger, y
must get smaller.
10
9