Exponential Functions and Graphs

1
Section 4.2

• Exponential Functions and Graphs

2
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.

3
Exponent

• Remember

4
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

5
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.

6
Example

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

7
Example

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

8
Example

• Graph the exponential function
  

9
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.

10
Observing Relationships
11
Connecting the Concepts
12
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

13
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

14
Natural Exponential Function

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

15
Graphs of Exponential Functions, Base e

• Graph f(x) ex.

16
Example

• Graph f(x) ex2.

17
Example

• Graph f(x) 2 ? e?3x.

18
Compound Interest Formula
19
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.)