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MATH 120
The Logistic Function
Elementary Functions
In the past weeks, we have considered the use of linear, exponential, power
and polynomial functions as mathematical models in many different contexts.
Another type of function, called the
logistic function
, occurs often in describing
certain kinds of growth.
These functions, like exponential functions, grow quickly at
first, but because of restrictions that place limits on the size of the underlying
population, eventually grow more slowly and then level off.
y
(0)
C
y
x
As is clear from the graph above, the characteristic Sshape in the graph of a
logistic function shows that initial exponential growth is followed by a period in
which growth slows and then levels off, approaching (but never attaining) a
maximum upper limit.
Logistic functions are good models of biological population growth in species
which have grown so large that they are near to saturating their ecosystems, or of
the spread of information within societies.
They are also common in marketing,
where they chart the sales of new products over time; in a different context, they can
also describe demand curves: the
decline
of demand for a product as a function of
increasing price can be modeled by a logistic function, as in the figure below.
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(0)
C
y
x
The formula for the logistic function,
y
=
C
1
+
Ae

Bx
involves three parameters
A
,
B
,
C
.
(Compare with the case of a quadratic function
y
=
ax
2
+
bx
+
c
which also has three parameters.) We will now investigate the
meaning of these parameters.
First we will assume that the parameters represent positive constants.
As
the input
x
grows in size, the term
–
Bx
that appears in the exponent in the
denominator of the formula becomes a larger and larger negative value.
As a result,
the term
e
–Bx
becomes smaller and smaller (since raising any number bigger than 1,
like
e
, to a negative power gives a small positive answer).
Hence the term
Ae
–Bx
also becomes smaller and smaller.
Therefore, the entire denominator
1 +
Ae
–Bx
is
always a number larger than 1 and decreases to 1 as
x
gets larger.
Finally then, the
value of
y
, which equals
C
divided by this denominator quantity, will always be a
number smaller than
C
and increasing to
C
.
It follows therefore that the parameter
C
represents the limiting value of the output
past which the output cannot grow (see the figures above).
On the other hand, when the input
x
is near 0, the exponential term
Ae
–Bx
in
the denominator is a value close to
A
so that the denominator 1 +
Ae
–Bx
is a value
near 1 +
A
.
Again, since
y
is computed by dividing
C
by this denominator, the value
of
y
will be a quantity much smaller than
C
.
Looking at the graph of the logistic
curve in Figure 1, you see that this analysis explains why
y
is small near
x
= 0 and
approaches
C
as
x
increases.
To identify the exact meaning of the parameter
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This note was uploaded on 05/17/2011 for the course ALGEBRA 098 taught by Professor Johnson during the Spring '09 term at Grand Valley State University.
 Spring '09
 Johnson
 Algebra

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