PRECALCULUS I

EXPONENTIAL FUNCTIONS

Dr. Claude S. MooreDanville Community

College

DEFINITION

The exponential function isf(x) = ax

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) = a-x

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).

1. The graph goesthrough (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 goesthrough (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.

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

EXAMPLE:BACTERIA GROWTH

A certain bacteria increases by the model with t in hours.

tetP 21970100 .)( Find P(0), P(5), and P(10). Answers:

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

COMPOUND INTEREST• Compounded n

times per year.• A = amount in balance

P = principal investedr = annual interest ratet = number of years

• Compounded continuously.

nt

n

rPA

1

rtPeA

EXAMPLE:COMPOUND INTEREST

Find the balance of a $3500 investment compounded monthly at

8% for 5 years.

nt

n

rPA

1

The answer is:

A = 3500(1+.08/12)12(5) = $5214.46

EXAMPLE:COMPOUND INTEREST

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)

rtPeA