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# Exponential Growth and Decay

## Differentiate between growth and decay by examining functions

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Exponential Growth and Decay

Have you ever been to a condominium? Take a look at this dilemma.

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 \cdot 1.1^t\end{align*} where \begin{align*}f\end{align*} is the fee after \begin{align*}t\end{align*} years.

Understanding exponential growth and decay will help you with this dilemma. Pay attention and you will see it again at the end of the Concept.

### Guidance

Do you know how to tell if a function is an exponential function?

If your function can be written in the form \begin{align*}y=ab^x\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are constants, \begin{align*}a \ne 0, b>0,\end{align*} and \begin{align*}b \ne 1\end{align*}, then it must be exponential. In quadratic equations, your functions were always to the \begin{align*}2^{nd}\end{align*} power. In exponential functions, the exponent is a variable. Their graphs will have a characteristic curve either upward or downward.

Exponential Functions

1. \begin{align*}y=2^x\end{align*}
2. \begin{align*}c=4 \cdot 10^d\end{align*}
3. \begin{align*}y=2 \cdot \left(\frac{2}{3}\right)^x\end{align*}
4. \begin{align*}t=4 \cdot 10^u\end{align*}

Not Exponential Functions

In some cases with exponential functions, as the \begin{align*}x\end{align*} value increased, the \begin{align*}y\end{align*} value increased, too. This was a direct relationship known as exponential growth. As the \begin{align*}x\end{align*} value increases, the \begin{align*}y\end{align*} value grows at a very fast rate! The other graph you saw showed the opposite—as the \begin{align*}x\end{align*} value increased, the \begin{align*}y\end{align*} value decreased. This relationship is an inverse relationship known as decay. The graphs of these functions are opposites, reflected on the \begin{align*}y\end{align*}-axis.

We can also analyze growth and decay functions in real – life situations.

A famous story tells about a courtier who presented a Persian king with a beautiful handmade chessboard. The king asked him what he would like in return for his gift and the courtier surprised the king by asking him for one grain of rice on the first square of the chessboard, two grains of rice on the second, four grains on the third, etc. The king agreed and ordered for the rice to be brought. By the \begin{align*}21^{st}\end{align*} square, over a million grains of rice were required and, by the \begin{align*}41^{st}\end{align*} square, over a quadrillion grains of rice were needed. There was simply not enough rice in all the world for the final squares.

This story reminds us of the drastic increases that we can see in exponential functions. Although this story is a fable, there are many instances in the real-world where exponential growth can be seen.

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

#### Example A

\begin{align*}y=\frac{1}{2}^x\end{align*}

Solution: Exponential decay

#### Example B

\begin{align*}y=4^x\end{align*}

Solution: Exponential growth

#### Example C

\begin{align*}y=5^x\end{align*}

Solution: Exponential growth

Now let's go back to the dilemma from the beginning of the Concept.

First, make a table of values.

\begin{align*}t\end{align*} \begin{align*}f\end{align*}
0 185
1 203.5
2 223.85
3 246.24
4 270.86
5 297.94
6 327.74
7 360.51
8 396.56
9 436.22
10 479.84

Now we graph the function.

### Vocabulary

Exponential Functions
results that expand exponentially. The graph curves upward or downward.
Exponential Growth Graph
a direct relationship graph each variable increases.
Decay Graph
an indirect relationship graph, one variable increases as the other one decreases.

### Guided Practice

Here is one for you to try on your own.

Does the following function represent an exponential function?

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

Solution

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

### Practice

Directions:Graph each function. Then say whether it represents economic growth or decay. There will be two answers for each problem.

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

### Vocabulary Language: English

Exponential decay

Exponential decay

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

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

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