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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\begin{align*}f = 185 \times 1.1^t\end{align*} where f\begin{align*}f\end{align*} is the fee after t\begin{align*}t\end{align*} 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\begin{align*}y = ab^x\end{align*}, where a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are constants, a0,b>0\begin{align*}a \neq 0, b > 0\end{align*} , and b1\begin{align*}b \neq 1\end{align*} , 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\begin{align*} c = 4 \times 10^a\end{align*} is an exponential function, but  y=6×0x\begin{align*}y=-6 \times 0^x\end{align*} is not because b1\begin{align*} b \le 1 \end{align*}.

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

In other cases, as the x\begin{align*}x\end{align*} value increased, the y\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 y\begin{align*}y\end{align*}-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\begin{align*}f=185 \times 1.1^t\end{align*} where f\begin{align*}f\end{align*} is the fee after t\begin{align*}t\end{align*} years.

First, make a table of values for the function f=185×1.1x\begin{align*}f=185 \times 1.1^x\end{align*} .

 t\begin{align*}t\end{align*} 0 1 2 3 4 5 6 7 8 f\begin{align*}f\end{align*} 185 203.5 223.85 246.24 270.86 297.94 327.74 360.51 396.56

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\begin{align*}y = 3 \times 1^x\end{align*}

First, answer the question.

No, this function does not represent an exponential function because the b\begin{align*}b\end{align*} value is 1.

Next, graph the function.

License: CC BY-NC 3.0

#### Example 3

Graph the function y=(12)x\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.

License: CC BY-NC 3.0

Next, answer the question.

From the graph, the function y=(12)x\begin{align*}y = \left(\frac{1}{2} \right)^x\end{align*} represents exponential decay.

#### Example 4

Graph the function y=4x\begin{align*} y=4^x \end{align*} 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\begin{align*} y=4^x\end{align*} represents exponential growth.

#### Example 5

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

### Review

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

1. y=4x\begin{align*}y = 4^x\end{align*}
2. y=(12)x\begin{align*}y= \left(\frac{1}{2}\right)^x\end{align*}
3. y=(13)x\begin{align*}y= \left(\frac{1}{3}\right)^x\end{align*}
4. y=7x\begin{align*}y=7^x\end{align*}
5. y=5x\begin{align*}y=5^x\end{align*}
6. y=2x\begin{align*}y=2^x\end{align*}
7. y=(14)x\begin{align*}y= \left(\frac{1}{4}\right)^x\end{align*}
8. y=(34)x\begin{align*}y= \left(\frac{3}{4}\right)^x\end{align*}
9. y=6x\begin{align*}y=6^x\end{align*}
10. y=11x\begin{align*}y=11^x\end{align*}
11. y=9x\begin{align*}y=9^x\end{align*}
12. y=(18)x\begin{align*}y= \left(\frac{1}{8}\right)^x\end{align*}
13.  y=12x\begin{align*}y= 12^x\end{align*}
14. y=(25)x\begin{align*}y= \left(\frac{2}{5}\right)^x\end{align*}
15. y=13x\begin{align*}y=13^x\end{align*}

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

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### Vocabulary Language: English

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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