Exponential Functions and Their Graphs

Base is 2 Base is 10 Base is 3 Base is ½

The base is a positive number excluding 1 and the exponent is a variable.

The parent form of the graph has a y-intercept at (0,1).

The value of b determines the steepness of the curve.

The following functions are not exponential functions:

Variable is the base and not the exponent

The base of an exponential function must be a positive number other than 1

The base of an exponential function must be positive

Variable is both the base and the exponent

Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2x cannot be zero. In this case, the x-axis is an asymptote. An asymptote is a line that a graphed function approaches as the value of x gets very large or very small.

Construct a t-table.

ExampleUse the graph of f(x)=4 to obtain the graph of g(x)=4 3.

What is the domain and range of each function?

x x

How will the graph shift?

It will shift up 3 units.

How will the graph shift?

It will shift to the left 1 unit.

Example

The exponential equation 13.49 .967 1 predicts the number of O-rings

that are expected to fail at the temperature x F on the space shuttles. The

O-rings were used to seal the connections between d

x

o

f x

ifferent sections of the shuttle

engines. Use a calculator to find the number expected to fail at the temperature of

40 degrees.

We expect two O-rings to fail at the temperature of 40 degrees.

The Natural Base e

•The letter e is the initial of the last name of Leonhard Euler (1701-1783) who introduced the notation.

• Since has special calculus properties that simplify many calculations, it is the natural base of exponential functions.

•The value of e is defined as the number that the expression approaches as n approaches infinity.

• The value of e to 16 decimal places is 2.7182818284590452.

• The function is called the Natural Exponential Function

11

n

n

( ) xf x e

( ) xf x e

1The values of 1+ for increasingly

large values of n. As n the

approximate value of e to nine decimal

places is e 2.718281827.

The irrational number e, approximately

2.72, is called the natura

n

n

l base. The

function f(x)=e is called the natural

exponential function.

x

Example