PRECALCULUS I

EXPONENTIAL FUNCTIONS

Dr. Claude S. Moore Danville Community College

DEFINITION The exponential function is

f(x) = a x

where a > 0, a



1, and x is any real number.

VALUES OF a INFLUENCE GRAPHS

The following are true for f(x) = a

x

: 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 = 2 x

This graph of

y = f(x) = 2 x

was generated with the TI-82.

a = 2 > 1, graph goes up to the right.

Graph goes through (0,1).

GRAPHING f(x) = a

-x

Before graphing f(x) = a

-x

, rewrite the function as : f(x) = 1/a

x

= (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) = a x vs. f(x) = a -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.

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

)

 100

e

0

.

2197

t

Find P(0), P(5), and P(10).

Answers:

P(0) = 100 P(5) = 299.97 P(10) = 899.8

COMPOUND INTEREST

•

Compounded n

A



P

   

r n

  

nt

•

times per year.

A = amount in balance P = principal invested r = annual interest rate t = number of years

A



Pe rt

•

Compounded continuously.

EXAMPLE: COMPOUND INTEREST

A



P

   

r n

  

nt

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

A



Pe rt

Find the balance of a $3500 investment compounded continuously at 8% for 5 years.

The answer is:

A = 3500e 0.08(5) = $5221.39.

( $5214.46 compounded monthly)