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 (x₁, y₁) and (x₂, y₂) which satisfy the inverse variation, x₁y₁ = k and x₂y₂ = k. Consequently, x₁y₁ = x₂y₂ 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?

x₁y₁ = x₂y₂
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:

xy = 1

Note that the lines never cross the axes -- they get closer and closer to x = 0 and y = 0, but x and y never equal zero.

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.