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Exponential Growth and Decay

Differentiate between growth and decay by examining functions

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Exponential Growth and Decay
License: CC BY-NC 3.0
 

A condominium complex charges $185 per month for the homeowners’ association fee. The rates can rise every year because of inflation but they promise not to raise the rates more than 10% each year. Keep in mind, though, that if they raise the rate by 10% the first year, the second year is now more expensive. If they raise the maximum again, they are increasing the original $185 plus the first year’s adjustment by 10%. Graph the situation for 10 years.

How much could the homeowners’ fee be in ten years? Use the function  where  is the fee after  years.

In this concept, you will learn to distinguish between exponential growth and exponential decay.

Exponential Growth and Decay

Sometimes you will need to identify whether a function is an exponential function. If your function can be written in the form , where  and  are constants,  , and  , then it must be exponential. In quadratic equations, your functions were always to the 2nd power. In exponential functions, the exponent is a variable. Their graphs will have a characteristic curve either upward or downward.

Therefore the function  is an exponential function, but   is not because .

In some cases with exponential functions, as the  value increased, the  value increased, too. This was a direct relationship known as exponential growth. As the  value increases, the  value grows at a very fast rate!

In other cases, as the  value increased, the  value decreased. This relationship is an inverse relationship known as exponential decay. The graphs of these functions are opposites, reflected on the -axis.

License: CC BY-NC 3.0
 

License: CC BY-NC 3.0

Examples

Example 1

Earlier, you were given a problem about the rising condominium fees. You need to determine how much the homeowners’ fee will be in ten years using the function  where  is the fee after  years.

First, make a table of values for the function  .

  0 1 2 3 4 5 6 7 8
  185 203.50 223.85 246.24 270.86 297.94 327.74 360.51 396.56

Next, graph the function.

License: CC BY-NC 3.0

Example 2

Does the following function represent an exponential function? Graph the function.

First, answer the question.

No, this function does not represent an exponential function because the  value is 1.

Next, graph the function.

License: CC BY-NC 3.0

Example 3

Graph the function  and tell whether it will represent exponential growth or decay.

First, graph the function.

License: CC BY-NC 3.0

Next, answer the question.

From the graph, the function  represents exponential decay.

Example 4

Graph the function  and tell whether it represents exponential growth or decay.

First, graph the function.

License: CC BY-NC 3.0

Next, answer the question.

From the graph, the function  represents exponential growth.

Example 5

Graph the function  and tell whether it represents exponential growth or decay.

First, graph the function.

License: CC BY-NC 3.0

Next, answer the question.

From the graph, the function  represents exponential growth.

Review

Graph each function. Then say whether it represents economic growth or decay.

  1.  

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.17.  

 

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Vocabulary

Exponential decay

Exponential decay occurs when a quantity decreases by the same proportion in each given time period.

Exponential Function

An exponential function is a function whose variable is in the exponent. The general form is y=a \cdot b^{x-h}+k.

Exponential growth

Exponential growth occurs when a quantity increases by the same proportion in each given time period.

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