Exponential functions are of the form


Notice:
The variable x
is an exponent. As such, the
graphs of these functions are not straight
lines. In a straight line, the "rate of
change" is the same across the graph.
In these graphs, the "rate of change"
increases or decreases across the graphs. 


Observe how the graphs of
exponential functions change based upon the values of a
and b:
Example: 

when a > 0
and the b
is between 0 and 1, the graph
will be decreasing (decaying).
For this example,
each time x is increased by 1, y decreases to
one half of its previous value.
Such a situation is
called
Exponential
Decay.

Example: 

when a > 0
and the b
is greater than 1, the graph
will be increasing (growing).
For this example,
each time x is increased by 1, y increases by
a factor of 2.
Such a situation is
called
Exponential
Growth. 
Many real world phenomena can
be modeled by functions that describe how things grow or decay
as time passes. Examples of such phenomena include the
studies of populations, bacteria, the AIDS virus, radioactive
substances, electricity, temperatures and credit payments, to
mention a few.
Any quantity that grows or
decays by a fixed percent at regular intervals is said to
possess exponential growth or exponential decay.
At the Algebra level, there are
two functions that can be easily used to illustrate the concepts of
growth or decay in applied situations.
When a quantity grows by a fixed
percent at regular intervals,
the pattern can be represented
by the functions,
Growth:


Decay:



a =
initial amount before
measuring growth/decay
r = growth/decay
rate (often a percent)
x = number of
time intervals that have passed

Example:
A bank account balance, b, for
an account starting with s dollars, earning an annual
interest rate, r, and left untouched for n
years can be calculated as
(an
exponential growth formula).
Find a bank account balance to the nearest dollar, if
the account starts with $100, has an annual rate of 4%, and
the money left in the account for 12 years.
We will now examine rate
of growth and decay in a three step process. We will
(1) build a chart to examine the data and "see" the growth or
decay, (2) write an equation
for the function, and (3) prepare a scatter plot of the data
along with the graph of the function.
Consider these examples of growth
and decay:
Growth:
Cell Phone Users 
In 1985, there were 285
cell phone subscribers in the small town of
Centerville. The number of subscribers
increased by 75% per year after 1985.
How many cell phone subscribers were in
Centerville in 1994? (Don't consider
a fractional part of a person.) 
Years 
x = 1
1986 
2
1987 
3
1988 
4
1989 
5
1990 
6
1991 
7
1992 
8
1993 
9
1994 
Number of
Cell Phone users 
498 
872 
1527 
2672 
4677 
8186 
14325 
25069 
43871 
There are 43871 subscribers in 1994.
Function: 
a = the
initial amount before the growth
begins
r = growth rate
x = the number of
intervals 

as x
ranges from 1 to 9 for this
problem 
The scatter plot of the data
table can be prepared by hand or
with the use of a graphing
calculator. For graphing
the function, employ your
graphing calculator.

See how to
prepare a scatter plot
of your data table using
your TI 83+/84+ graphing
calculator.
Click here.
After the data points
are plotted, set Y1 = to
the function, and graph.
The function and the
scatter plot will overlap
as they did at the
right. 


horizontal axis = year (1986 = 1)
vertical axis = number of cell phone
users 

Growth
by doubling:
Bacteria

One of the most common
examples of exponential growth deals with
bacteria. Bacteria can multiply at an
alarming rate when each bacteria splits into
two new cells, thus doubling. For example, if we
start with only one bacteria which can
double every hour, by the end of one day we
will have over 16 million bacteria. 
End
of Hour 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
... 
24 
Bacteria 
starting with one 
2 
4 
8 
16 
32 
64 
128 
256 
512 
1024 
2048 
4096 
8192 
16384 
... 
16777216 
Pattern: 
2^{1} 
2^{2} 
2^{3} 
2^{4} 
2^{5} 
2^{6} 
2^{7} 
2^{8} 
2^{9} 
2^{10} 
2^{11} 
2^{12} 
2^{13} 
2^{14} 

2^{24} 

At the end of 24 hours,
there are 16,777,216 bacteria.
By looking at
the pattern, we see that the growth in this
situation
can be represented as a function:
Will our formula show this same function?
If an amount doubles, the rate of increase
is 100%.
Function: 
a = the
initial amount before the growth
begins
r = growth rate
x = the number of
intervals 

as x
ranges from 1 to 24 for this
problem 
horizontal axis = end of hour
vertical axis = number of bacteria




Let's examine the graph of our scatter plot
and function. To the left
of the origin we see that the function graph tends to
flatten, but stays slightly above the xaxis.
To the right of the origin the function graph grows
so quickly that it is soon off the graph.
The rate at which the graph changes
increases as time increases. 

When we can see larger yvalues, we
see that the growth still continues at a
rapid rate. This is what is meant by
the expression "increases exponentially". 
Note: In reality, exponential growth does not
continue indefinitely. There would,
eventually, come a time when there would no
longer be any room for the bacteria, or
nutrients to sustain them. Exponential
growth actually refers to only the early
stages of the process and to the manner and
speed of the growth.


Decay:
Tennis Tournament 
Each year the local
country club sponsors a tennis
tournament. Play starts with
128 participants. During each
round, half of the players are
eliminated.
How many players
remain after 5 rounds? 
Rounds 
1 
2 
3 
4 
5 
Number of
Players left 
64 
32 
16 
8 
4 
There are 4 players remaining after 5
rounds.
Function: 
a = the
initial amount before the decay
begins
r = decay rate
x = the number of
intervals 

as x
ranges from 1 to 5 for this
problem 
Notice the shape of this graph
compared to the graphs of the
growth functions.

horizontal axis = rounds
vertical axis = number of players
left 

Let's examine the scatter plot and the
function.
At 0 the yintercept is 100. To the
right of the origin we see that the graph
declines rapidly and then tends to
flatten, staying slightly above the xaxis.
The rate of change
decreases as time increases. 

When we zoom in on the flattened area of the
graph, we
see that the graph does stay above the
xaxis. This makes sense because we
could not have a "negative" number of grams
of DDT leftover. 

