Exploring Exponential Functions
For each equation below, identify the equation as exponential growth or decay, its initial value, the growth (or
decay) factor, and the growth(or decay) rate.
1. y  3.34(1.67) x
2. y  8(.54) x
3. y  4.5(4) x
For each graph below, identify the graph as exponential growth or decay, and estimate its initial value.
4.
5.
6.
For each table below, identify the equation as exponential growth or decay, its initial value, the growth (or
decay) factor, the growth (or decay) rate, and write the equation of the exponential represented in the table.
7.
8.
x
2
3
4
5
6
y
136.89
106.77
82.284
64.961
50.67
9.
x
4
5
6
7
8
y
180.55
241.94
324.2
434.43
582.14
x
1
2
3
4
5
y
17.325
26.681
41.088
63.275
97.444
10. It is estimated that the population of the world is increasing at an average annual rate of 1.3%. The
population was about 6,472,416,997 in the year 2005.
a. Write an equation for the population t years after 2005.
b. Use the equation to predict the population of the world in 2015 and 2025.
11. During the 1990s, the forested area in Guatemala decreased at an average rate of 1.7%. The forested
area on Guatemala was about 34,400 square kilometers in 1990.
a. Write an equation for the amount of forested area t years after 1990.
b. If the trend has continued as described above, use the equation to predict the amount of forested
area in 2010 and 2020.
12. Wilma and Walder’s Weaving Wanders bought a piece of weaving equipment for $60,000. It is
expected to depreciate at an average rate of 10% per year.
a. Write an equation representing the value of the equipment t years after its purchase.
b. Use the equation to predict the value of the equipment 6 years after its purchase.
13. A biologist is studying a newly-discovered species of bacteria. He places 100 bacteria in a petri dish in
order to study its behavior. The bacteria is estimated to be growing at a rate of 17% per hour.
a. Write an equation for the amount of bacteria t hours after its placement in the petri dish.
b. How much bacteria will there be after 12 hours?
14. The population of rabbits in a national forest has been declining by 1/20 each year since 2003 when its
population was measure at 4,578 rabbits.
a. Write an equation for the population t years after 2003.
b. Use the equation to predict the population of the rabbits in the forest in 2015.
15. The table below shows the ending balance of a college savings account for each year listed.
Year
2008
2009
2010
2011
Amount
$10,991
$11,343
$11,706
$12,080
a. If the savings account was opened in 2005, what was the initial amount invested in the college
savings account?
b. Write an exponential equation that models the amount in the savings account n years since 2005.
c. If the student that the account was created for is to enter college in the fall of 2017, how much
money will be in the account?
Answer Key
For each equation below, identify the equation as exponential growth or decay, its initial value, the growth
factor, and the growth rate.
1. y  3.34(1.67) x
2. y  8(.54) x
3. y  4.5(2.34) x
Exponential Growth
Initial Value: 3.34
Growth Factor: 1.67
Growth Rate: 67%
Exponential Decay
Initial Value: 8
Growth Factor: .54
Growth Rate: -46%
Exponential Growth
Initial Value: 4.5
Growth Factor: 2.34
Growth Rate: 134%
For each graph below, identify the graph as exponential growth or decay, and estimate its initial value.
4.
5.
6.
Exponential Growth
Initial Value: 2.3
Exponential Growth
Initial Value: 7
Exponential Decay
Initial Value: 8
For each table below, identify the equation as exponential growth or decay, its initial value, the growth factor,
the growth rate, and write the equation of the exponential represented in the table.
7.
8.
x
2
3
4
5
6
y
136.89
106.77
82.284
64.961
50.67
Exponential Decay
Initial Value: 225
Growth Factor: .78
Growth Rate: -22%
Equation: y  225(.78) x
9.
x
4
5
6
7
8
y
180.55
241.94
324.2
434.43
582.14
Exponential Growth
Initial Value: 56
Growth Factor: 1.34
Growth Rate: 34%
Equation: y  56(1.34) x
x
1
2
3
4
5
y
17.325
26.681
41.088
63.275
97.444
Exponential Growth
Initial Value: 11.25
Growth Factor: 1.54
Growth Rate: 54%
Equation: y  11.25(1.54) x
10. It is estimated that the population of the world is increasing at an average annual rate of 1.3%. The
population was about 6,472,416,997 in the year 2005.
a. Write an equation for the population t years after 2005. P  6, 472, 416,997(1.013)t
b. Use the equation to predict the population of the world in 2015 and 2025.
2015 = 7,364,799,758
2025 = 8,380,219,554
11. During the 1990s, the forested area in Guatemala decreased at an average rate of 1.7%. The forested
area on Guatemala was about 34,400 square kilometers in 1990.
a. Write an equation for the amount of forested area t years after 1990. A  34, 400(.983)t
b. If the trend has continued as described above, use the equation to predict the amount of forested
area in 2010 and 2020.
2010 = 24,413.43
2020 = 20,566.67
12. Wilma and Walder’s Weaving Wanders bought a piece of weaving equipment for $60,000. It is
expected to depreciate at an average rate of 10% per year.
a. Write an equation representing the value of the equipment t years after its purchase.
V  60, 000(0.9)t
b. Use the equation to predict the value of the equipment 6 years after its purchase.
V=$31,886.46
13. A biologist is studying a newly-discovered species of bacteria. He places 100 bacteria in a petri dish in
order to study its behavior. The bacteria is estimated to be growing at a rate of 17% per hour.
a. Write an equation for the amount of bacteria t hours after its placement in the petri dish.
A  100(1.17)t
b. How much bacteria will there be after 12 hours? A = 658
14. The population of rabbits in a national forest has been declining by 1/20 each year since 2003 when its
population was measure at 4,578 rabbits.
a. Write an equation for the population t years after 2003. P  4,578(.95)t
b. Use the equation to predict the population of the rabbits in the forest in 2015? P = 2473.77
15. The table below shows the ending balance of a college savings account for each year listed.
Year
2008
2009
2010
2011
Amount
$10,991
$11,343
$11,706
$12,080
a. If the savings account was opened in 2005, what was the initial amount invested in the college
savings account? $10,000
b. Write an exponential equation that models the amount in the savings account n years since 2005.
A  10, 000(1.032) n
c. If the student that the account was created for is to enter college in the fall of 2017, how much
money will be in the account? A = $14593.40