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When something increases or decreases by the same percentage over a fixed period, you have an exponential (percentage) growth. The exponential growth may be negative, meaning the graph will decrease to the right rather than ascending upwards as it otherwise would.
Theory
Exponential Functions
An exponential function is expressed in this way:
Notice that the variable is now the exponent. Both and are numbers. We call the start value and the growth factor.
When the value of in the function is positive, the graph looks like one of the two graphs below.
Rule
The Graph of an Exponential Function
is the -value when , is the growth factor,
blue graph, red graph.
In general, gives you a fixed percentage increase, gives you a fixed percentage reduction and gives no change. The number acts as a growth factor. The value of affects the sign of the function values.
Example 1
Suppose you deposit into a savings account. You get interest per year on this deposit. How much will you have in your account after 7 years?
This is an example of exponential growth. First you need to find the growth factor associated with a % increase:
You must calculate the following terms to find the amount of money you have after seven years:
Thus, you have in your account after seven years.
Example 2
You have the function . This function intersects the -axis at and grows exponentially. This form of growth is very powerful. References to this type of graph are also often used in everyday speech, when someone says something is experiencing “exponential” growth.
Example 3
You have the function . This intersects the -axis at and decreases exponentially. This form of reduction is very powerful. It’s like the downward version of the previous example.
