This document defines and explains exponential functions, which are functions that include the expression bx where b is a positive number other than 1. It discusses the shape of exponential graphs, asymptotes, graphing the general forms y=abx and y=a(b^x)+h+k, exponential growth models, and compound interest formulas. Examples are provided to illustrate key concepts.

2. What are Exponential Functions?
 Exponential functions – functions that
include the expression bx where b is a
positive # other than 1.
 b is called the base.

3. What’s the Shape?
 Let’s make a table to find the general shape.
 If we use f(x) = 2x as an example:
x f(x) = 2x
-3
-2
-1
0
1
2
3

4. Asymptotes
 An asymptote is a line that a graph
approaches (but does not touch) as you
move away from the origin.
 For example:
 Our graph has a
horizontal asymptote
at y = 0.

5. Graphing y = abx
 If a > 0 and b > 1, y = abx is an
exponential growth function.
 For all y = abx , b > 1:
 Graphs pass through (0, a) (a is the y-int)
 x-axis is an asymptote
 Domain: all real #s
 Range: y > 0 if a > 0
y < 0 if a < 0

6. To graph:
 Plot 2 points: (0, a) and (1, __)
 Plug in 1 for x to fill the blank
 Connect with a smooth curve that:
 Starts left of the origin, close to the x-axis
 Moves up or down quickly to the right

7. Examples
 Graph:

8. Your Turn!
 Graph

9. General Exponential Functions
 General form:
 As usual:
 h is horizontal shift
 k is vertical shift
 To graph:
 Sketch the “parent graph” y = abx
 Shift using h and k

10. Examples
 Graph and state the domain and range:

11. Your Turn!
 Graph and state the domain and range:

12. Exponential Growth Models
 When a real-life quantity increases by a
fixed % each year, the amount of the
quantity after t years can be modeled by:
y = a(1 + r)t
where a is the initial amount and r is the %
increase (as a decimal).
 (1 + r) is the growth factor.

13. Example:
 In January, 1993, there were about 1,313,000
Internet hosts. During the next five years, the
number of hosts increased by about 100% per
year.
 Write a model giving the number h (in millions)
of hosts t years after 1993.
 How many hosts were there in 1996?

14. Compound Interest
 Compound interest is interest paid on the
original principal and on previously earned
interest.
 Modeled by an exponential function.
 If interest is compounded n times per
year, the amount A in the account after t
years is:
where P is the initial principal and r is the
annual interest rate.

15. Example:
 You deposit $1000 in an account that pays
8% annual interest. Find the balance after 1
year if interest is compounded:
 A. annually
 B. quarterly
 C. daily
 Which is the best investment?