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Inverse Variation
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
  1. 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).


  2. 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.
  3. 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.
  4. 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.

Examples
Example
#1. If y varies inversely with x, and y = 5 when x = 8, what is k, the constant of proportionality?
What is your answer?
 
Example
#2. If y varies inversely with x and k = 0.32, what is x when y = 10?
What is your answer?
 
Example
#3. If y varies inversely with x and k = 1.07, what is y when x = 18?
What is your answer?
 



M Ransom

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