Consider these two companies:

• Company A has 100 stores, and expands by opening 50 new stores a year
• Company B has 100 stores, and expands by increasing the number of stores by 50% of their total each year.

Company A is exhibiting linear growth. In linear growth, we have a constant rate of change—a constant number that the output increased for each increase in input. For company A, the number of new stores per year is the same each year. Company B is different—we have a percent rate of change rather than a constant number of stores/year as our rate of change. To see the significance of this difference compare a 50% increase when there are 100 stores to a 50% increase when there are 1000 stores:

• 100 stores, a 50% increase is 50 stores in that year.
• 1000 stores, a 50% increase is 500 stores in that year.

Calculating the number of stores after several years, we can clearly see the difference in results.

Years	Company A	Company B
2	200	225
4	300	506
6	400	1139
8	500	2563
10	600	5767

Graphs of data from A and B, with B fit to a curve. This percent growth can be modeled with an exponential function.

Exponential Function

An exponential growth or decay function is a function that grows or shrinks at a constant percent growth rate. The equation can be written in the form f(x) = a(1 + r)^x or f(x) = ab^x where b = 1 + r. Where

• a is the initial or starting value of the function,
• r is the percent growth or decay rate, written as a decimal,
• b is the growth factor or growth multiplier. Since powers of negative numbers behave strangely, we limit b to positive values.

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Shana Calaway, Dale Hoffman, and David Lippman, Business Calculus, " 1.7: Exponential Functions," licensed under a CC-BY license.