The population of a city was 10,000 in 2012 and is declining at a rate of 5% each year. If this decay rate continues, what will the city's population be in 2017?

### Exponential Delay Function

Previously, we have only addressed functions where

Graph

Let’s make a table of both functions and then graph.

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Notice that *decreases exponentially,* or *exponentially decays*. Anytime **exponential decay function** has the form

Let's determine which of the following functions are exponential decay functions, exponential growth functions, or neither and briefly explain our answers.

y=4(1.3)x f(x)=3(65)x y=(310)x g(x)=−2(0.65)x

a. and b. are exponential growth functions because

c. is an exponential decay function because

d. is neither growth or decay because

Let's graph

To graph this function, you can either plug it into your calculator (entered Y= -2(2/3)^(X-1)+1) or graph

The

The horizontal asymptote is

### Examples

#### Example 1

Earlier, you were asked to find the city's population in 2017 if the population was 10,000 in 2012 and is declining at a rate of 5% each year.

This is an example of exponential decay, so we can once again use the exponential form *a* = 10,000, the starting population, *x-h* = 5 the number of years, and *k* = 0, but *b* is a bit trickier. If the population is decreasing by 5%, each year the population is (1 - 5%) or (1 - 0.05) = 0.95 what it was the previous year. This is our *b*.

Therefore, the city's population in 2017 is 7,738.

**For Examples 2-4, graph the exponential functions. Find the y-intercept, asymptote, domain, and range.**

#### Example 2

#### Example 3

\begin{align*}y=-2 \left(\frac{2}{3}\right)^{x+3}\end{align*}

\begin{align*}y\end{align*}-intercept: \begin{align*}\left(0, -\frac{16}{27}\right)\end{align*}, asymptote: \begin{align*}y=0\end{align*}, domain: all reals, range: \begin{align*}y<0\end{align*}

#### Example 4

\begin{align*}g(x)= \left(\frac{3}{5}\right)^x-6\end{align*}

\begin{align*}y\end{align*}-intercept: \begin{align*}(-5, 0)\end{align*}, asymptote: \begin{align*}y=-6\end{align*}, domain: all reals, range: \begin{align*}y>-6\end{align*}

**For Examples 5-8, determine if the functions are exponential growth, exponential decay, or neither.**

#### Example 5

\begin{align*}y=2.3^x\end{align*}

exponential growth

#### Example 6

\begin{align*}y=2 \left(\frac{4}{3}\right)^{-x}\end{align*}

exponential decay; recall that a negative exponent flips whatever is in the base. \begin{align*}y=2 \left(\frac{4}{3}\right)^{-x}\end{align*} is the same as \begin{align*}y=2 \left(\frac{3}{4} \right)^{x}\end{align*}, which looks like our definition of a decay function.

#### Example 7

\begin{align*}y=3\cdot 0.9^x\end{align*}

exponential decay

#### Example 8

\begin{align*}y=\frac{1}{2} \left(\frac{4}{5}\right)^{x}\end{align*}

neither; \begin{align*}a < 0\end{align*}

### Review

Determine which of the following functions are exponential growth, exponential decay or neither.

- \begin{align*}y= -\left(\frac{2}{3}\right)^x\end{align*}
- \begin{align*}y= \left(\frac{4}{3}\right)^x\end{align*}
- \begin{align*}y=5^x\end{align*}
- \begin{align*}y= \left(\frac{1}{4}\right)^x\end{align*}
- \begin{align*}y= 1.6^x\end{align*}
- \begin{align*}y= -\left(\frac{6}{5}\right)^x\end{align*}
- \begin{align*}y= 0.99^x\end{align*}

Graph the following exponential functions. Find the \begin{align*}y\end{align*}-intercept, the equation of the asymptote and the domain and range for each function.

- \begin{align*}y= \left(\frac{1}{2}\right)^x\end{align*}
- \begin{align*}y=(0.8)^{x+2}\end{align*}
- \begin{align*}y=4 \left(\frac{2}{3}\right)^{x-1}-5\end{align*}
- \begin{align*}y= -\left(\frac{5}{7}\right)^x +3\end{align*}
- \begin{align*}y= \left(\frac{8}{9}\right)^{x+5} -2\end{align*}
- \begin{align*}y=(0.75)^{x-2}+4\end{align*}
- Is the domain of an exponential function always all real numbers? Why or why not?
- A discount retailer advertises that items will be marked down at a rate of 10% per week until sold. The initial price of one item is $50.
- Write an exponential decay function to model the price of the item \begin{align*}x\end{align*} weeks after it is first put on the rack.
- What will the price be after the item has been on display for 5 weeks?
- After how many weeks will the item be half its original price?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.2.