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Rectifying Inverse Relations into Lines
Introduction
  • In this lesson you will learn how to write equations of quantities which vary inversely.
  • Graphs of inverse relationships will be modified to show a linear relationship.

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.

Consider the following example:
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).
A table of values containing x, , and is shown below.

x
0.42.57.5
0.526
0.6671.54.5
113
1.50.6672
20.51.5
2.50.41.2
30.3331
3.5
This initial graph plots points from the table using (x, y) as coordinates.



This second graph also plots the points (w,y) or as coordinates.



Notice that these points are collinear with the slope 3.

Notice in the second graph that the points lie on a straight line. The slope of this line is 3. This is no accident. The equation can be written as . In this form, acts as the independent variable. If we let , we get y = 3w and can clearly see a linear equation with slope 3. The process of rewriting as y = 3w is known as “rectifying into a line.” The equation for the original branch of a hyperbola is rewritten so that the data can be represented linearly.

An Application
According to Boyle’s Law, the product of the pressure, P, and volume, V, of a gas under constant temperature is a constant. Calling this constant c, we get PV = c. This can also be written as where w = . In this case, a graph of would be a hyperbolic curve, whereas a graph of P = cw would be a line of slope c.

For example, if P = 4 x 105 N/m2 when V = 0.5 m3, we have PV = c = 2 x 105 Nm.
Notice that our variables need units of measurement since they represent experimental data. Pressure is the ratio of force per unit area. One atmosphere of pressure equals 1.01 x 105 N/m2 where the units N/m2 are often called a Pascal. The volume of the confined gas is measured in cubic meters, or m3.
The linear representation of this data is given by rectifying this data into a line. . This would be a line with slope 2 x 105 Nm and is easier to visualize graphically.
To see Boyle’s Law in action, visit this site. Or, if you would like to view Boyle’s original data from his experimentation in the 17th century, visit here.

Example
Example Suppose that y varies inversely with x and x = 0.5 when y = 0.9. What is the equation of its rectified line?
What is your answer?
 


The table below shows the values for x, , and .

x
0.42.51.125
0.520.9
0.81.250.56
110.45
1.50.6670.3
20.50.23
2.50.40.18
30.3330.15
3.50.290.13
This first graph shows the hyperbolic representation . The WINDOW for this graph is X: (0, 3, 1) and Y: (-1, 6, 1)



Using the same WINDOW, this second graph plots the points where as shown in the table.



Notice that these points are collinear with the slope 0.45.

Examples
Example
If y varies inversely as x, and y = 0.3 when x = 1.2, find the constant of proportionality.
What is your answer?
 
Example Describe the graphs of both the hyperbolic and linear representations of this inverse variation.
What is your answer?
 



M Ransom

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