3.1 Exponential Functions and Their Graphs

In this chapter, we will be studying two types of nonalgebraic functions: exponential functions and logarithmic functions.

In this section we'll focus on exponential functions.

Exponential functions are different than other functions that we've studied so far because  is an exponent.

For example:

Note that  is not an exponential function but a polynomial function.

Let's recall some properties of exponents:

• When multiplying terms with the same base but different exponents, add the exponents.

      Example:

• When dividing terms with the same base but different exponents, subtract the exponent in the denominator from the exponent in the numerator.

      Example:

          In this step, two 3s in the numerator and denominator divide out so we're left with:

• When raising an exponent to a power, multiply the two exponents.

      Example:

• Any number to the zero power equals 1

• When a number is raised to a negative power,

• When a number is raised to a fractional exponent,

      Note:

 which also equals  

        and  which also equals

All these expressions are equal

• Zero to the zero power is indeterminate

Unlike other functions that we’ve been studying, exponential functions are actually useful. Things like population, interest, and bacteria grow exponentially. We also use exponential decay for things like radioactive decay and carbon dating.

Definition:

The exponential function  with base  is denoted by  where  and   is any real number

The base  is excluded because it yields  and that is a constant function not an exponential function.

The domain of the function is the set of all real numbers.

The function can also be written as  

where  and  are real numbers

where   is a nonzero real number and  is positive

Graph: 

Domain: (,)

Range: (,)

x-intercepts: none

y-intercepts: (0,1)

Vertical asymptotes: none

Horizontal asymptotes:

(the horizontal asymptote is only relevant on one end of the graph)

     

Transformation of the graph:

     

  Vertical stretch/compression. 

  Horizontal stretch/compression. When , the graph is reflected over the y axis. 

   Horizontal shifts

         Vertical Shifts

The Natural Base :

Since  is called the natural base,  is called the natural exponential function.

Domain: (,)

Range: (,)

x-intercepts: none

y-intercepts: (0,1)

Vertical asymptotes: none

Horizontal asymptotes:

Formulas for Compound Interest:

For n compounding per year:      

For continuous compounding:

 is the balance in the account

 is the principal or initial amount in the account

 is the rate expressed as a decimal

 is the number of compoundings per year (ex. annually would be 1 and monthly would be 12, daily would be 365 and so on)

 is the number of years