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Section 4.2

- Exponential Functions and Graphs

Exponential Function

- The exponential function is very important in

math because it is used to model many real life

situations. - For example population growth and decay,

compound interest, economics, and much more.

Exponent

- Remember

Exponential Function

- The function f(x) ax, where x is a real number,

a gt 0 and a ? 1, is called the exponential

function, base a. - The base needs to be positive in order to avoid

the complex numbers that would occur by taking

even roots of negative numbers. - Examples

Graphing Exponential Functions

- To graph an exponential function, follow the

steps listed - 1. Compute some function values and list
- the results in a table.
- 2. Plot the points and connect them with a
- smooth curve. Be sure to plot enough
- points to determine how steeply the
- curve rises.

Example

- Graph the exponential function y f(x) 3x.

Example

- Graph y 3x 2.
- The graph is the graph of y 3x shifted _____ 2

units.

Example

- Graph the exponential function

Example

- Graph y 4 ? 3?x
- The graph is a reflection of the graph of y 3x

across the _______, followed by a reflection

across the _______ and then a shift _______ of 4

units.

Observing Relationships

Connecting the Concepts

The Number e

- e is known as the natural base
- (Most important base for exponential
- functions.)
- e is an irrational number
- (cant write its exact value)
- We approximate e

The Number e

- Find each value of ex, to four decimal places,

using the ex key on a calculator. - a) e4
- b) e?0.25
- c) e2
- d) e?1

Natural Exponential Function

- Remember
- e is a number
- e lies between 2 and 3

Graphs of Exponential Functions, Base e

- Graph f(x) ex.

Example

- Graph f(x) ex2.

Example

- Graph f(x) 2 ? e?3x.

Compound Interest Formula

Example

- A father sets up a savings account for his

daughter. He puts 1000 in an account that is

compounded quarterly at an annual interest rate

of 8. - How much money will be in the account at the end

of 10 years? (Assume no other deposits were made

after the original one.)