Algebra I: Variation
Inverse Variation
Inverse Variation
The statement "
y
varies inversely as
x
means that when
x
increases,
y
decreases by the same factor. In other words, the expression
xy
is constant:
| xy = k |
where k is the constant of variation.
We can also express the relationship between x and y as:
y = 
|
where k is the constant of variation.
Since
k
is constant, we can find
k
given any point by multiplying the x-coordinate by the y-coordinate. For example, if
y
varies inversely as
x
, and
x = 5
when
y = 2
, then the constant of variation is
k = xy = 5(2) = 10
. Thus, the equation describing this inverse variation is
xy = 10
or
y = 
.
Example 1: If
y
varies inversely as
x
, and
y = 6
when
x = 
, write an equation describing this inverse variation.
k = (6) = 8
xy = 8
or
y = 
Example 2: If
y
varies inversely as
x
, and the constant of variation is
k = 
, what is
y
when
x = 10
?
xy =
10y =
y = ×
= 
× = 
k is constant. Thus, given any two points ( x 1, y 1 ) and ( x 2, y 2 ) which satisfy the inverse variation, x 1 y 1 = k and x 2 y 2 = k . Consequently, x 1 y 1 = x 2 y 2 for any two points that satisfy the inverse variation.
Example 3: If
y
varies inversely as
x
, and
y = 10
when
x = 6
, then what is
y
when
x = 15
?
x
1
y
1 = x
2
y
2
6(10) = 15y
60 = 15y
y = 4
Thus, when
x = 6
,
y = 4
.
Graphing Inverse Variation
Unlike the graph of direct variation, the graph of inverse variation is not linear. Rather, it is a hyperbola:
To graph an inverse variation, make a data table and plot points. Then connect the points with a smooth (not straight) curve. There should be two curves -- one in the first quadrant (where both x and y are positive) and one in the third quadrant (where both x and y are negative). The result should be qualitatively similar to the graph of xy = 1 above.
To calculate the constant of variation from a graph of inverse variation, simply pick a point and multiply its two coordinates.
