7.1 –Exponential Functions
An exponential function has the form y = abx
where a does not equal zero and the base b is a
positive number other than 1.
In an exponential function, the base b is a
constant. The exponent is the independent
variable with domain the set of real numbers.
7.1 –Exponential Functions
7.1 –Exponential Functions
7.1 –Exponential Functions
For exponential growth, as the value of x
increases, the value of y increases.
For exponential decay, as the value of x
increases, the value of y decreases, approaching
zero.
The exponential functions shown are asymptotic
to the x-axis. An asymptote is a line that a graph
approaches as x or y increases in absolute value.
7.1 –Exponential Functions
7.1 –Exponential Functions
Example 1:
Graph y = 2x
7.1 –Exponential Functions
Example 2:
Graph y = 9(3)x
7.1 –Exponential Functions
Example 3:
Graph y = 22x
7.1 –Exponential Functions
Example 4: Identify each function or situation
as an example of exponential growth or
decay.
1. f(x) = 12(0.95)x
2. f(x) = .25(2)x
3. You put $1000 into a college savings account
for four years. The account pays 5% interest
annually.
7.1 –Exponential Functions
Exponential Growth and Decay Models
Amount
after t
time
periods
Number of Periods
A(t) =
t
a(1+r)
Initial amount
Rate of growth (r >
0) or decay (r < 0)
7.1 –Exponential Functions
For exponential growth y = abx, with b > 1, the
value b is the growth factor. A quantity that
exhibits exponential growth increases by a
constant percentage each time period. The
percentage increase r, written as a decimal, is
the rate of increase or growth rate.
For exponential growth b = 1 +r
7.1 –Exponential Functions
For exponential growth y = abx, with 0 < b < 1, the
value b is the decay factor. A quantity that
exhibits exponential decay decreases by a
constant percentage each time period. The
percentage increase r, written as a decimal, is
the rate of decay or decay rate.
Usually a rate of decay is expressed as a
negative quantity, so b = 1 + r
7.1 –Exponential Functions
Example 5:
In 1996, there were 2573 computer viruses and
other computer security incidents. During the
next 7 years, the number of incidents
increased by about 92% each year.
Write an exponential growth model giving the
number n of incidents t years after 1996.
About how many incidents were there in 2003?
When was there 125,000 computer incidents?
7.1 –Exponential Functions
Example 6:
If the rabbit population is growing at a rate of 20%
every year and starts out at 150 rabbits
currently.
How many rabbits are there in 12 years?
How long does it take for the population to reach
5000 rabbits?
7.1 –Exponential Functions
Example 7:
The population of a certain animal species
decreases at a rate of 3.5% per year. You
have counted 80 of the animals in the habitat
you are studying.
a. Write a function that models the change in the
animal population.
b. Graph the function. Estimate the number of
years until the population first drops below 15
animals.
7.1 –Exponential Functions
Example 8:
In the year 2003 there was a world population of
150 Iberian Lynx and in 2004 there were only
120. If this trend continues and the population
is decreasing exponentially, how many lynx
will there be in 2014?
7.1 –Exponential Functions
Compound Interest
7.1 –Exponential Functions
Example 9:
You deposit $4000 in an account that pays 2.92%
annual interest. Find the balance after 3
years if the interest is compounded with the
given frequency.
a. Quarterly
b. Daily
7.1 –Exponential Functions
Example 10:
You want $2000 in an account after 4 years. Find
the amount you should deposit for each of the
situations described below.
a. The account pays 2.5% annual interest
compounded quarterly.
b. The account pays 3.25% annual interest
compounded monthly.