Inverse Variation

Suppose that The constant of proportionality is 3. A graph of this relationship for x > 0 is shown below.

This is in fact one branch of a hyperbola. The other branch is located in Quadrant III and is found if the values of x are negative. Features of the graph to notice are the characteristic shape sloping negatively and the fact that the graph approaches the x-axis as x gets large (end behavior).

Suppose that y varies inversely with x. When x = 5, y = 9, how do we find k, the constant of proportionality?

Step 1: We know that

because y varies inversely with x.

Substitute 5 for x and 9 for y and we get

Step 2: Solve for k we get k = 45.

Suppose that y varies inversely with x and k = 7. What is y when x = 2?

Step 1: We know that

because y varies inversely with x.

Substitute 2 for x and 7 for k and we get

Step 2: Solve for k we get k = 14.

Suppose that y varies inversely with x and k = 0.76. What is x when y = 4?

Step 1: We know that

because y varies inversely with x.

Substitute 0.76 for k and 4 for y:

Step 2: Solve for x:

We multiply both sides of the equation by x and get

Dividing both sides of the equation by 4 gives x = 0.19.

Step 2b: Note that we could also solve this problem using the fact that

Substitute 0.76 for k and 4 for y and we get 4x = 0.76.
Dividing both sides of this equation by 4 yields the same result that x = 0.19.