Blog Post 4--Determining Linear and Exponential Equations

Lesson: Determining whether a function is Linear or Exponential 
(using an example that is neither linear nor exponential) 

Keeping with my normal theme on this blog, I am going to explain a mathematical concept using comic books. In particular, I will be using comic book sales figures for Marvel’s The Avengers from issues #1 to #500 dating from 1965 to 2005. I have calculated the average sales figures of the comic series by decade and have listed the results in the below table.

Decade	1965	1975	1985	1995
Average Total Paid Circulation (Averaged by decade)	204,303	206,010	201,481	106,167

Additionally, I have graphed the results:

With this information, I can figure out whether or not this equation is linear, exponential or neither.
I can predict that since these are sales figures, the relationship is unlikely to be linear or exponential. Likes and dislikes of consumers change at varying rates and. therefore, it is hard to predict sales figures from one year to a next with a consistent rate or change or ratio. This is especially true when dealing with elastic goods such as comic books.
To verify my prediction I will take the following steps:

Step 1: Can this function be linear?

To figure out if a function is linear, the function must have a constant rate of change.
The equation for rate of change is y2-y1/x2-x1. In this case:

Average Paid circulation=y
Decade=x

So:

206,010 (Y2)-204,303 (Y1)/1975 (x2)-1965(x1)

(206,010-204,303)=1707/(10)=170.7

Do this again for decades 3 verses 2 and 4 verses 3: 

(201,481-206,010)=-4529
(1985-1975)=10

-4529/10=-452.9

106,167-201,481= -95314

1995-1985=10

-95314/10= -9531.4

All Rates of Change=

170.7
-452.9
-9,531.4

Since these ROC are NOT all consistent, THIS FUNCTION IS NOT LINEAR.

Step 2: Can this function be exponential?

The formula for finding exponential functions is Y2/Y1. 

This is called the Ratio.

Below I have found each ratio for decades 2 v. 1, 3 v. 2, and 4 v. 3.

206,010/204,303=1.008

201,481/206,010=.978

106,167/201,481=.526

Since none of these ratios are consistent, THIS FUNCTION IS NOT EXPONENTIAL

Conclusion:

As predicted, this function is neither linear nor is it exponential. This means that it cannot be measured by a constant value that shows repeated sums (linear) or repeated products (exponents).

Source for figures used in this example: http://www.comichron.com/titlespotlights/avengers.html