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# Exponential Decay

## Rational functions with x as an exponent in the denominator

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Exponential Decay Function

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?

### Guidance

In the last concept, we only addressed functions where $|b|>1$ . So, what happens when $b$ is less than 1? Let’s analyze $y=\left(\frac{1}{2}\right)^x$ .

#### Example A

Graph $y=\left(\frac{1}{2}\right)^x$ and compare it to $y=2^x$ .

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

$x$ $\left(\frac{1}{2}\right)^x$ $2^x$
3 $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$ $2^3=8$
2 $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ $2^2=4$
1 $\left(\frac{1}{2}\right)^1 = \frac{1}{2}$ $2^1=2$
0 $\left(\frac{1}{2}\right)^0 = 1$ $2^0=1$
-1 $\left(\frac{1}{2}\right)^{-1} = 2$ $2^{-1}=\frac{1}{2}$
-2 $\left(\frac{1}{2}\right)^{-2} = 4$ $2^{-2}=\frac{1}{4}$
-3 $\left(\frac{1}{2}\right)^3 = 8$ $2^{-3}=\frac{1}{8}$

Notice that $y=\left(\frac{1}{2}\right)^x$ is a reflection over the $y$ -axis of $y=2^x$ . Therefore, instead of exponential growth, the function $y=\left(\frac{1}{2}\right)^x$ decreases exponentially, or exponentially decays . Anytime $b$ is a fraction or decimal between zero and one, the exponential function will decay. And, just like an exponential growth function, and exponential decay function has the form $y=ab^x$ and $a>0$ . However, to be a decay function, $0 . The exponential decay function also has an asymptote at $y=0$ .

#### Example B

Determine which of the following functions are exponential decay functions, exponential growth functions, or neither. Briefly explain your answer.

a) $y=4(1.3)^x$

b) $f(x)=3 \left(\frac{6}{5}\right)^x$

c) $y = \left(\frac{3}{10}\right)^x$

d) $g(x)= -2(0.65)^x$

Solution: a) and b) are exponential growth functions because $b>1$ . c) is an exponential decay function because $b$ is between zero and one. d) is neither growth or decay because $a$ is negative.

#### Example C

Graph $g(x)=-2 \left(\frac{2}{3}\right)^{x-1}+1$ . Find the $y$ -intercept, asymptote, domain, and range.

Solution: To graph this function, you can either plug it into your calculator (entered Y= -2(2/3)^(X-1)+1) or graph $y=-2 \left(\frac{2}{3}\right)^x$ and shift it to the right one unit and up one unit. We will use the second method; final answer is the blue function below.

The $y$ -intercept is:

$y=-2 \left(\frac{2}{3}\right)^{0-1}+1=-2 \cdot \frac{3}{2}+1=-3+1=-2$

The horizontal asymptote is $y=1$ , the domain is all real numbers and the range is $y < 1$ .

Intro Problem Revisit This is an example of exponential decay, so we can once again use the exponential form $f(x)=a \cdot b^{x-h}+k$ , but we have to be careful. In this case, 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 .

$P = 10,000 \cdot 0.95^5\\= 10,000 \cdot 0.7738 = 7738$

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

### Guided Practice

Graph the following exponential functions. Find the $y$ -intercept, asymptote, domain, and range.

1. $f(x)=4 \left(\frac{1}{3}\right)^x$

2. $y=-2 \left(\frac{2}{3}\right)^{x+3}$

3. $g(x)= \left(\frac{3}{5}\right)^x-6$

4. Determine if the following functions are exponential growth, exponential decay, or neither.

a) $y=2.3^x$

b) $y=2 \left(\frac{4}{3}\right)^{-x}$

c) $y=3\cdot 0.9^x$

d) $y=\frac{1}{2} \left(\frac{4}{5}\right)^{x}$

1. $y$ -intercept: $(4, 0)$ , asymptote: $y=0$ , domain: all reals, range: $y < 0$

2. $y$ -intercept: $\left(0, -\frac{16}{27}\right)$ , asymptote: $y=0$ , domain: all reals, range: $y<0$

3. $y$ -intercept: $(-5, 0)$ , asymptote: $y=-6$ , domain: all reals, range: $y>-6$

4. a) exponential growth

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

c) exponential decay

d) neither; $a < 0$

### Vocabulary

Exponential Decay Function
An exponential function that has the form $y=ab^x$ where $a>0$ and $0 .

### Practice

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

1. $y= -\left(\frac{2}{3}\right)^x$
2. $y= \left(\frac{4}{3}\right)^x$
3. $y=5^x$
4. $y= \left(\frac{1}{4}\right)^x$
5. $y= 1.6^x$
6. $y= -\left(\frac{6}{5}\right)^x$
7. $y= 0.99^x$

Graph the following exponential functions. Find the $y$ -intercept, the equation of the asymptote and the domain and range for each function.

1. $y= \left(\frac{1}{2}\right)^x$
2. $y=(0.8)^{x+2}$
3. $y=4 \left(\frac{2}{3}\right)^{x-1}-5$
4. $y= -\left(\frac{5}{7}\right)^x +3$
5. $y= \left(\frac{8}{9}\right)^{x+5} -2$
6. $y=(0.75)^{x-2}+4$
7. Is the domain of an exponential function always all real numbers? Why or why not?
8. 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.
1. Write an exponential decay function to model the price of the item $x$ weeks after it is first put on the rack.
2. What will the price be after the item has been on display for 5 weeks?
3. After how many weeks will the item be half its original price?

### Vocabulary Language: English

Exponential Decay Function

Exponential Decay Function

An exponential decay function is a specific type of exponential function that has the form $y=ab^x$, where $a>0$ and $0.
Exponential Function

Exponential Function

An exponential function is a function whose variable is in the exponent. The general form is $y=a \cdot b^{x-h}+k$.
Model

Model

A model is a mathematical expression or function used to describe a physical item or situation.