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 2^{nd} 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 **inverse relationship** known as **exponential decay**. The graphs of these functions are opposites, reflected on the -axis.

### 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.

#### 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.

#### Example 3

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

First, graph the function.

Next, answer the question.

From the graph, the function exponential decay.

represents#### Example 4

Graph the function

and tell whether it represents exponential growth or decay.First, graph the function.

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.

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.