3.1 Exponential Functions and Their Graphs

Exponential Functions
Section 3.1 discusses exponential functions and their graphs.

Rules for Exponents

Polynomial and Exponential Functions Differ
In polynomial functions, a variable is raised to a term, while in an exponential functions, a term is raised to a variable.

Here a is the base and n is the exponent:


When two of the same bases are multiplied, the exponents are added together:


When two of the same bases are divided, the exponents are subtracted:


When a base and its exponent are raised by another power, the exponents are multiplied:


When the exponent is negative, the result if the reciprocal of the base and positive exponent:


Anything raised to the first power is equal to itself:

Anything raised to the zero power is one:

Exponential Functions and Their Graphs
The basic exponential function:

Here a is the base and x is the exponent. a is multiplied by itself x number of times.

Graphs of Exponential Functions:

The basic exponential function graph for the above exponential function (where a is the base and x is the exponent).

The y-intercept for all unmanipulated exponential functions is y=1.

The horizontal asymptote is at .
The domain is .
The range is .

Manipulating Exponential Functions:

describes the possible ways exponential functions and consequently their graphs can be manipulated.

When a is changed:

-The graph is vertically stretched by a factor of a if a>1.

-The graph is vertically compressed by a factor of a if a<1.

-If a is negative, the graph is reflected in the x-axis.

When b (the base) is changed:

-The y-intercept is unchanged

-The graph increases of decreases by a larger factor depending on whether b is less than or greater than 1, respectively.

When c exists:

-The graph moves right a factor of c when x-c.

-The graph moves left a factor of c when x+c.

When d exists:

-The graph moves up a factor of d when d is added.

-The graph moves down a factor of d when d is subtracted.

Examples:


Vertical compression.

Increased growth factor, shifted up 6 units (note change in y-intercept).

Compound Interest

When finding the compound interest, use:

A=Present value

P= Principal (initial value)

r=Annual interest rate (APR)

n= Number of compounding periods per year

t=Time in years

Example:

A total of $14,000 is invested at an annual interest rate of 8%. Find the balance after 4 years if it is compounded monthly:

Substitute 14,000 for P.

Substitute .08 for r (remember to convert 8% to decimal form).

Subsitute 12 for n (there are 12 months in 1 year).

Sustitute 4 for t.

Solve: $19,259.33

For continuously compounded interest, use:

e is the natural base, and is a button found on your calculator.

Example:

Using the same figures from the first example, the answer is: $19,279.79