# Exponential Functions & Their Applications

Exponential Functions & Their Applications

An exponential function is any positive ( not equal to one raised to some ( valued exponent.

f(x) = ax where a > 0, but a ( 1

The domain of the function, the values for which x can equal, are all ( (-(,().

Why? Because an exponent can be any real number. And the exponent is the independent variable.

The range however, is all positive ( greater than zero (0, ().

Why? Because any number raised to some power will always be a positive number.

For all exponential functions when the exponent is equal to:

-1 function value will be 1/a Why? That’s the reciprocal!

0 function value will be 1 Why? Anything to zero power is 1

1 function value will be a Why? Anything to 1st power is itself.

The shape of the curve will always be:

Just some notes:

1) The general exponential family graph will always approach the x-axis but will

never touch the axis. This is called a horizontal asymptote. As the graph is

translated along the vertical axis the asymptote will change accordingly.

2) The graph will grow larger and larger if the number (a) is greater than 1. This is

like population. Think of 2 people, they have children, then those have children

and so on, this is an exponential function, and you should be able to see that it

grows to an infinite number.

3) If we have a –x, we are looking at the reciprocal of a to the x, so it will look like

the graph of the function when 0 < a < 1, when a is greater than 1, and when a is

between 0 and 1 the graph will look like the graph when a > 1.

Exponential functions are all over in the world. Some of the most prominent areas are economics, biology, archeology, and some cross these boundaries into our everyday lives. One of the most prevalent cases is that of compounded interest. We would like to believe (based upon all interest problems that we have computed) that interest is simple interest, but it is generally compound, which increases at an exponential rate. The Challenger Space Shuttle disaster, caused by the decay of o-rings can be described by an exponential function!

Let’s first get some practice graphing some very basic exponential functions by hand. In order to graph these let’s refer back to the 3 values listed above, where the exponent is 0, -1 and 1. In this special case, the general exponential function, the points that we graph are (0, 1), (-1, 1/a) & (1, a), where a is the base.

Example: On the graph below, graph both f(x) = 2x and f(x) = 2-x

Now let’s use our TI calculators to graph the functions in the last example.

Example: Using you calculator graph

a) f(x) = 2x b) f(x) = 2-x

In case you have forgotten the process, I have included the instructions for graphing, modifying the viewing window and getting and modifying a table of values for a function.

Recall that we must first put the equations into the equation editor in the following manner:

Y= ( 2 ^ ALPHA STO>

Y2= ( 2 ^ ( ( ALPHA STO> )

To graph the function we then just hit the GRAPH key. I have adjusted my viewing window, by hitting WINDOW and setting the parameters as the text box on the left. The graph is the text box on the right.

To see a table of values used to calculate this graph hit 2nd GRAPH . See the text box on the left. You should notice that the values for x = 0, 1 and -1 are those predicted in our discussion of the general exponential function. If we needed different values given in our table or a different scale we can set those by using the 2nd WINDOW and selecting a different starting value and incremental values leaving the indep & depend as Auto. See the text box on the right.

The following examples will involve the base e, which is a mathematical constant sometimes called Euler’s number. It is an irrational number just as ( is and is therefore approximated just as (. Euler’s number is a naturally occurring number that models growth and decay. It is equal to approximately 2.718. It is defined most often as the base of the natural logarithm, the limit of a certain sequence or as the sum of a certain series.

The model for exponential growth and decay looks like:

f(t) = A0ekt or A = A0ekt A0 represents amount at time zero

k is a constant representing growth/decay rate

t is the time of growth/decay

f(t) or A is the amount at time t

This looks just like our simplistic examples above except the base is now Euler’s number, the exponent is multiplied by a constant and the base is multiplied by a constant. Now let’s take a look at an exponential function that models exponential growth – namely population growth.

Example: a) *The population of a small country in Central America is growing

according to the function P(t) = 2,400,000e0.03t, where t is the

number of years since 1990. Create a table of values using your

calculator to show the population in 1990, 2000 and 2010.

Note: There are 2 ways of doing this using your calculator. One you can make your starting value 0 and increments by 10 or you can ask for independent values and auto for dependent. You will need to use ZOOM to see this graph. Use ZoomFit.

b) Calculate the population for 2050 using the formula. Use the table

you created to see if you got the correct answer.

Note: Make sure that you used the correct independent value: 60. Did you multiply 60 by 0.03 and then raise e to that number and then multiplied by 2.4x106. The use of exponents can simplify your life.

Now let’s look at an example showing exponential decay. Notice the negative in the exponent and thus a decreasing function.

Example: a) *The number of grams of a radioactive substance present at time t,

in years, is given by the formula A(t) = 200e-0.01t. Find the number

of particles at the beginning of the decay (t = 0), 10 years after

decay has begun, 50 years after the decay has begun, and 100 years

after the decay has begun.

b) The half-life of a radio-active particle is defined as the time that it

takes for half of the original particles to decay. Use your

calculator and the values you got from the above to narrow in on

the half-life of this radio-active particle. What value represents the

half-life, A(t) or t?

* Examples modified from examples used in: Dugopolski, 2002, Precalculus Functions and Graphs, Ed. 1, p. 259, Boston, Addison Wesley.

Homework Project

I have put a project on my website on the Intermediate Algebra page in the right column. Follow the instructions given on the referenced page to complete the project. This project requires interaction with at least one classmate (it was designed for an on-line course so students do not need to interact face to face), but can be done for the most part on your own time. It is due one-week from today during the 1st 15-minutes of class. Standard procedure applies for late work. The purpose of this project is to show how technology can make a decay model come alive, and to get you to think about what you are seeing on a graph, how it relates to a function and how both relate to a real world example.

The link to my website:

The direct link to the project which links to the applet:

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(-1,1/a)

(0,1)

(-1,1/a)

(1,a)

(1,a)

(0,1)

Exponential Growth

Where a > 1

Increasing Function

Exponential Decay

Where 0< a < 1 or negative exponent

Decreasing Function

y

x

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( Note: -5

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