Chapter 6
Exponential and Logarithmic
Functions
Section 6.1
EXPONENTIAL GROWTH AND
DECAY
Modeling Bacterial Growth
You can use a calculator to model the growth of 25
bacteria, assuming that the entire population
doubles every hour. Copy and complete the table
below:
Time (hr)
0
1
Population
25
50
2
3
4
5
6
Modeling Bacterial Growth
You can represent the growth of an initial population of
100 bacteria that doubles every hour by completing the
table below:
Time (hr)
0
Population
100
1
2
3
4
5
6
The population after n hours can be represented by the
following exponential expression:
n times
100 ´2 ´ 2 ´ 2 ´ 2 ´... ´ 2 =100 ´ 2n
Vocabulary
• The expression, 100  2n, is called an
exponential expression because the
exponent, n, is a variable and the base, 2, is a
fixed number.
• The base of an exponential expression is
commonly referred to as the multiplier.
Modeling Human Population Growth
• Human populations grow much more slowly
than bacterial populations.
• Bacterial populations that double each hour
have a growth rate of 100% per hour.
• The population of the U.S. in 1990 was
growing at a rate of about 8% per decade.
Modeling Human Population Growth
The population of the U.S. was 248,718,301 in 1990 and was projected to grow at
a rate of about 8% per decade. Predict the population, to the nearest hundred
thousand, for the years 2010 and 2025.
1.
To obtain the multiplier for exponential growth, add the growth rate to
100%.
1.
Write the expression for the population n decades after 1990.
1.
How many years past 1990 is 2010? How many decades is this?
1.
How many years past 1990 is 2010? How many decades is this?
2.
Deeper Thinking…
• Why would we ADD to 100% for an exponential
growth?
• What does growth mean?
• Don’t forget to always change your percentages to
decimals and then use this as the multiplier.
• What would it mean for a population to have a
growth rate of 100%?
– Ex: Population = 300  100% growth = ?
– What would a 200% growth rate look like for this
population?
– Would 100% change the population?
• Why do we ADD the growth rate to 100%?
Modeling Biological Decay
• Caffeine is eliminated from the bloodstream
of a child at a rate of about 25% per hour.
• A rate of decay can be though of as a negative
growth rate.
Modeling Biological Decay
The rate at which caffeine is eliminated from the bloodstream of an adult is about
15% per hour. An adult drinks caffeinated soda, and the caffeine in his or her
bloodstream reaches a peak level of 30 milligrams. Predict the amount, to the
nearest tenth of a milligram, of caffeine remaining 1 hour after the peak level and
4 hours after the peak level.
1.
To obtain the multiplier for exponential decay, subtract the growth rate to
100%.
1.
Write the expression for the caffeine level x hours after the peak level.
1.
Find the remaining caffeine amount after 1 hour.
1.
Find the remaining caffeine amount after 4 hours.
2.
Deeper Thinking…
• Why would we SUBTRACT from 100% for an
exponential decay?
• What does the word decay mean?
• Don’t forget to always change your percentages to
decimals and then use this as the multiplier.
• What would it mean for a population to have a
decay rate of 100%?
– Ex: Population = 300  100% decay = ?
– What would a 25% decay rate look like for this
population?
• Why do we SUBTRACT the decay rate from 100%?
Practice
Find the multiplier ofr each rate of exponential growth or decay.
1. 7% growth
2.
2% decay
3.
0.05% decay
4.
9% growth
5.
8.2% decay
6.
0.075% growth
Given x = 5, y = 3/5, and z = 3.3, evaluate each expression.
1. 2x
2.
50(2)3x
3.
3y
4.
25(2)z
5.
10(2)z+2
Practice
Predict the population of bacteria for each situation and time period (be
sure to use the formula and show all work!!).
1.
2.
55 bacteria that double every hour
a.
after 3 hours
b.
after 5 hours
75 E.coli bacteria that double every 30 minutes
a.
after 2 hours
b.
after 3 hours
c.
Homework
Finish worksheet 6.1 