Introduction
- In this lesson you will learn how to write equations of quantities which vary inversely.
- Graphs of inverse relationships will be investigated.
- A constant of proportionality will be defined and its effect on a graph investigated.
Definitions
- Quantities vary inversely if they are related by the relationship
 .
- Another way to express this is
 .
- We also say that y varies inversely with x.
- When quantities vary inversely, the constant k is called the constant of proportionality.
- Quantities which vary inversely are also said to be inversely proportional.
Examples
- 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.
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