November 25, 2013
3.1 Exponential and Logistic Functions
Exponential Function:
f(x) = a b
where:
x
a (initial value) is a nonzero real number
*occurs when x = 0
b (base) is positive, and b ≠1
Which are exponential functions?
a. f(x)=5x
b. f(x)=3x2 c. f(x)=4x-3
d. f(x)=7 2-x
November 25, 2013
Determine the formulas for g(x) and h(x).
Values for Two Exponential Functions
x
g(x)
h(x)
-2
4/9
128
-1
4/3
32
0
4
8
1
12
2
2
36
1/2
November 25, 2013
Exponential Growth and Decay
x
For any exponential function f(x) = a b and any real number x,
If a > 0 and b > 1, the function is increasing (exponential growth)
and b is the growth factor.
If a > 0 and b < 1, the function is decreasing (exponential decay)
and b is the decay factor.
November 25, 2013
Exploration #1 (pg. 279)
domain: ________________
range: _________________
continuity: _______________
incr/decr: ________________
symmetry: ________________
boundedness: ______________
extrema: _________________
asymptotes: _______________
end behavior: ______________
November 25, 2013
Describe how each graph is transformed from f(x) = 5x.
a. g(x) = 5x+1
b. h(x) = 5-x
c. k(x) = 3 5x
November 25, 2013
f(x) =
1+
1
x
x
x
10
1 x
(1+ x )
100
1000
10,000
100,000
November 25, 2013
The Natural Base e.
e =
lim
x ∞
1 +
1
x
x
November 25, 2013
Exponential Functions and the Base e.
Any exponential function f(x) = a b
f(x) = a ekx
x
can be rewritten as
for an appropriately chosen real number k.
If a > 0 and k > 0, f(x) is an exp. growth fn.
If a > 0 and k < 0, f(x) is an exp. decay fn.
Ex.
1. Graph f(x) = 2x.
2. Overlay the graphs for
0.7, and 0.8.
g(x) = ekx for k=0.4, 0.5, 0.6,
3. For which value of k does the graph of g most closely
match the graph of f?
4. Using tables, find the 3-decimal-place value of k for
which the values of g most closely approximate the
values of f.
November 25, 2013
The number B of bacteria in a petri dish culture after t
hours is given by
0.693t
B = 100e
a. What was the initial number of bacteria present?
b. How many bacteria are present after 6 hours?
November 25, 2013
Logistic Growth Functions:
f(x) =
where
c
1 + a bx
or f(x) =
c
1 + a e-kx
a,b,c, and k are postive, with b < 1
c is the limit to growth
If a = c = k = 1, then we get the
logistic growth function
f(x) =
1
1 + e-x
November 25, 2013
Graph the function. Find the y-intercept and horizontal asymptotes.
18
f(x) = 1 + 5(0.2)x
November 25, 2013
Write the logistic function that satisfies the given information.
initial value = 12, limit to growth = 60, passing through (1,24)
November 25, 2013
Graph and find the y-int. and horizontal asymptotes.
1.
f(x) =
20
1 + 2e-3x