Transcript Exponential Functions

PRECALCULUS I
EXPONENTIAL
FUNCTIONS
Dr. Claude S. Moore
Danville Community College
DEFINITION
The exponential function is
x
f(x) = a
where a > 0, a  1,
and x is any real number.
VALUES OF a
INFLUENCE GRAPHS
The following are true for f(x) = ax :
1. The graph goes through (0,1).
2. The x-axis is a horizontal asymptote.
3. As a 0, the graph tends to flatten more.
4. If a > 1, the graph of f(x) goes up to the
right.
5. If 0 < a < 1, the graph of f(x) goes
down to the right.
EXAMPLE: y = 2x
This graph of
y = f(x) = 2x
was generated
with the TI-82.
a = 2 > 1, graph goes up to the right.
Graph goes through (0,1).
GRAPHING f(x) =
-x
a
Before graphing f(x) = a -x, rewrite the
function as : f(x) = 1/ax = (1/a) x
1. The graph goes through (0,1).
2. The x-axis is a horizontal asymptote.
3. If (1/a) > 1, the graph of f(x) goes up to
the right.
4. If 0 < (1/a) < 1, the graph of f(x) goes
down to the right.
EXAMPLE: y = 2-x
This graph of
y = f(x) = 2-x
= (1/2) x
was generated
with the TI-82.
0<1/2<1, graph goes down to the right.
Graph goes through (0,1).
f(x) =
x
a
vs. f(x) =
1. The graph goes
through (0,1).
2. The x-axis is a
horizontal asymptote.
3. If a > 1, the graph of
f(x) goes up to the
right.
4. If 0 < a < 1, the
graph of f(x) goes
down to the right.
-x
a
1. The graph goes
through (0,1).
2. The x-axis is a
horizontal asymptote.
3. If a > 1, the graph of
f(x) goes down to the
right.
4. If 0 < a < 1, the graph
of f(x) goes up to the
right.
EXAMPLE:
BACTERIA GROWTH
A certain bacteria increases by the
model with t in hours.
P(t )  100e
0.2197t
Find P(0), P(5), and P(10).
Answers:
P(0) = 100
P(5) = 299.97 P(10) = 899.8
COMPOUND INTEREST
 r
A  P 1  
 n
A  Pe
rt
nt
• Compounded n
times per year.
• A = amount in balance
P = principal invested
r = annual interest
rate
t = number of years
• Compounded
continuously.
EXAMPLE:
COMPOUND INTEREST
nt
 r
A  P 1  
 n
Find the balance of a $3500
investment compounded monthly at
8% for 5 years.
The answer is:
A = 3500(1+.08/12)12(5) = $5214.46
EXAMPLE:
COMPOUND INTEREST
rt
A Pe
Find the balance of a $3500
investment compounded continuously
at 8% for 5 years.
The answer is: 0.08(5)
A = 3500e
= $5221.39.
($5214.46 compounded monthly)