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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 f=185×1.1t where f is the fee after t 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 y=abx, where a and b are constants, a0,b>0 , and b1 , 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 c=4×10a is an exponential function, but  y=6×0x is not because b1.

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

In other cases, as the x value increased, the y value decreased. This relationship is an inverse relationship known as exponential decay. The graphs of these functions are opposites, reflected on the y-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 f=185×1.1t where f is the fee after t years.

First, make a table of values for the function f=185×1.1x .

 t 0 1 2 3 4 5 6 7 8
 f 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.

y=3×1x

First, answer the question.

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

Next, graph the function.

License: CC BY-NC 3.0

Example 3

Graph the function y=(12)x 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 y=(12)x represents exponential decay.

Example 4

Graph the function y=4x 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 y=4x represents exponential growth.

Example 5

Graph the function y=5x 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 y=5x represents exponential growth.

Review

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

  1. y=4x
  2. y=(12)x
  3. y=(13)x
  4. y=7x
  5. y=5x
  6. y=2x
  7. y=(14)x
  8. y=(34)x
  9. y=6x
  10. y=11x
  11. y=9x
  12. y=(18)x
  13.  y=12x
  14. y=(25)x
  15. y=13x

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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