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 . So, what happens when is less than 1? Let’s analyze .

#### Example A

Graph and compare it to .

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

3 | ||

2 | ||

1 | ||

0 | ||

-1 | ||

-2 | ||

-3 |

Notice that is a reflection over the -axis of . Therefore, instead of exponential growth, the function *decreases exponentially,* or *exponentially decays*. Anytime 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 and . However, to be a decay function, . The exponential decay function also has an asymptote at .

#### Example B

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

a)

b)

c)

d)

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

#### Example C

Graph . Find the -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 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 -intercept is:

The horizontal asymptote is , the domain is all real numbers and the range is .

**Intro Problem Revisit** This is an example of exponential decay, so we can once again use the exponential form , 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*.

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

### Guided Practice

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

1.

2.

3.

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

a)

b)

c)

d)

#### Answers

1. -intercept: , asymptote: , domain: all reals, range:

2. -intercept: , asymptote: , domain: all reals, range:

3. -intercept: , asymptote: , domain: all reals, range:

4. a) exponential growth

b) exponential decay; recall that a negative exponent flips whatever is in the base. is the same as , which looks like our definition of a decay function.

c) exponential decay

d) neither;

### Explore More

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

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

- 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 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?