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 |
y = |
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
.
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.