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 \begin{align*}f = 185 \times 1.1^t\end{align*} 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 \begin{align*}y = ab^x\end{align*}, 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 \begin{align*}y=-6 \times 0^x\end{align*} is not because .

is an exponential function, butIn some cases with exponential functions, as the value increased, the value increased, too. This was a **direct relationship** known as **exponential growth**. As the \begin{align*}x\end{align*} value increases, the value grows at a very fast rate!

In other cases, as the \begin{align*}y\end{align*} value decreased. This relationship is an **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 \begin{align*}f\end{align*} is the fee after years.

First, make a table of values for the function

.\begin{align*}t\end{align*} | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

\begin{align*}f\end{align*} | 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 \begin{align*}b\end{align*} value is 1.

Next, graph the function.

#### Example 3

Graph the function \begin{align*}y = \left(\frac{1}{2} \right)^x\end{align*} 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 \begin{align*}y=5^x \end{align*} and tell whether it represents exponential growth or decay.

First, graph the function.

Next, answer the question.

From the graph, the function \begin{align*}y=5^x\end{align*} represents exponential growth.

### Review

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

- \begin{align*}y= 12^x\end{align*}