1. Exploring Exponential Functions
You should be able to recognize all exponential functions from an equation, graph, or a table of values and determine if it is an exponential growth or decay model.

2. Exponential Functions
What do equations of exponential functions look like?
Exponential functions have the following forms:
Exponential Growth:
y = abx, b > 1
Exponential Decay:
y = abx, 0 < b < 1
b: Represents the growth or decay factor
b - 1: Represents the growth or decay rate (percentage)

3. Determine if the exponential function is a growth or decay model and then identify the initial value, growth or decay factor and the growth or decay rate.
Example 1: f(x) = 3.42(1.78)x
Example 2: f(x) = 24.5(0.78)x
Initial Value:
3.42
Initial Value:
24.5
Growth/Decay Factor:
1.78
Growth/Decay Factor:
0.78
Growth/Decay Rate:
Growth
78%
Growth/Decay Rate:
Decay
22%
Example 3: f(x) = 5.98(0.28)x
Example 4: f(x) = (2.12)x
Initial Value:
5.98
Initial Value:
135.97
Growth/Decay Factor:
0.28
Growth/Decay Factor:
2.12
Growth/Decay Rate:
Decay
72%
Growth/Decay Rate:
Growth
112%

4. How can you determine if a function is an exponential growth or exponential decay by examining a table of values?
Example 5:
Example 6: X Y 2
8.6672 3
7.1071 4
5.8278 5
4.7788 6
3.9186 x -2 -1 1 2 y
.25 .5 4
Exponential Growth
Exponential Decay

5. Example 7: Determine if a function is an exponential growth or exponential decay by examining a table of values then write its equation.
Exponential Growth X Y 4
13.429 5
16.652 6
20.648 7
25.604 8
31.748
Growth Factor:
1.24
Growth Rate:
24%
Initial Value:
5.68
Equation:
f(x) = 5.68(1.24)x

6. Example 8: Determine if a function is an exponential growth or exponential decay by examining a table of values then write its equation.
Exponential Decay X Y 2
8.6672 3
7.1071 4
5.8278 5
4.7788 6
3.9186
Decay Factor:
0.82
Decay Rate:
18%
Initial Value:
Equation:
f(x) = (0.82)x

7. Writing and Predicting with Exponential Functions
The population of Johnson City found during the 2000 census was 25,876. Since then the city has been growing at a rate of 3.2% per year.
Write an equation to represent the population of Johnson City since 2000.
Exponential Growth
Initial Value:
25,876
Growth Rate:
3.2%
Equation:
f(x) = 25,876(1.032)x
Growth Factor:
1.032
Use this equation to predict the current population of Johnson City.
f(x) = 25,876(1.032)12
37,761 people
f(x) = 37,

8. Writing and Predicting with Exponential Functions
The Garcias have $12,000 in a savings account. The account is increasing by an average rate of 3.76% each year.
Write an equation to represent the amount in the savings account since the initial deposit.
Exponential Growth
Initial Value:
$12,000
Growth Rate:
3.76%
Equation:
f(x) = 12,000(1.0376)x
Growth Factor:
1.0376
Use this equation to predict the amount in the savings account 8 years after the initial deposit.
f(x) = 12,000(1.0376)8
f(x) = $16,122.08

9. Writing and Predicting with Exponential Functions
A new car costs $32, It is expected to depreciate 12% each year for 4 years and then depreciate 8% each year thereafter.
Find the value of the car 7 years after it was purchased.
First 4 years:
Exponential Decay
Initial Value:
32,000
Decay Rate:
12%
Equation:
f(x) = 32,000(.88)x
Decay Factor:
.88
f(x) = 32,000(.88)4
f(x) = $19,190.25
Next 3 years:
Exponential Decay
Initial Value:
19,190.25
Decay Rate: 8%
Equation:
f(x) = 19,190.25(.92)x
Decay Factor:
.92
f(x) = 19,190.25(.92)3
f(x) = $14,943.22