7.1 –Exponential Functions An exponential function has the form y = ab x where a does not equal zero and the base b is a positive number other than 1

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.




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.



Example 1: Graph y = 2x


Example 2: Graph y = 9(3)x


Example 3: Graph y = 22x


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.


Exponential Growth and Decay Models

A(t) = a(1+r)t Rate of growth (r > 0) or decay (r < 0)

Number of Periods

Initial amount

Amount after t time

periods


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


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


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?


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?


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.


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?


Compound Interest


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


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.

Documents

7.1 –Exponential Functions An exponential function has the form y = ab x where a does not equal zero and the base b is a positive number other than 1