1. Exponential Functions

2. Exponential Functions and Their Graphs

3. Irrational Exponents
If b is a positive number and x is a real number, the expression bx always represents a positive number. It is also true that the familiar properties of exponents hold for irrational exponents.

4. Example 1:
Use properties of exponents to simplify

5. Example 1:
Use properties of exponents to simplify

6. Example 1:
Use properties of exponents to simplify

7. Example 1:
Use properties of exponents to simplify

8. Exponential Functions
An exponential function with base b is defined by the equation
x is a real number.
The domain of any exponential function is the interval
The range is the interval

9. Graphing Exponential Functions

10. Graphing Exponential Functions

11. Example 2:
Let’s make a table and plot points to graph.

12. Example 2:

13. Example 2:

14. Properties:
Exponential Functions

15. Example 3:
Given a graph, find the value of b:

16. Example 3:
Given a graph, find the value of b:

17. Increasing and Decreasing Functions

18. One-to-One Exponential Functions

19. Compound Interest

20. Example 4:
The parents of a newborn child invest $8,000 in a plan that earns 9% interest, compounded quarterly. If the money is left untouched, how much will the child have in the account in 55 years?

21. Future value of account
Example 4 Solution:
Using the compound interest formula:
Future value of account
in 55 years

22. Base e Exponential Functions
Sometimes called the natural base,
often appears as the base of an exponential functions.
It is the base of the continuous compound interest formula:

23. Example 5:
If the parents of the newborn child in Example 4 had invested $8,000 at an annual rate of 9%, compounded continuously, how much would the child have in the account in 55 years?

24. Future value of account
Example 5 Solution:
Future value of account
in 55 years

25. Graphing
Make a table and plot points:

26. Exponential Functions
Horizontal asymptote
Function increases
y-intercept (0,1)
Domain all real numbers
Range: y > 0

27. Translations Up k units Down k units Right k units Left k units
For k>0
y = f(x) + k
y = f(x) – k
y = f(x - k)
y = f(x + k)
Up k units
Down k units
Right k units
Left k units

28. Example 6:
On one set of axes, graph

29. Example 6:
On one set of axes, graph
Up 3

30. Example 7:
On one set of axes, graph
Right 3

31. Non-Rigid Transformations
Exponential Functions with the form f(x)=kbx and f(x)=bkx are vertical and horizontal stretchings of the graph f(x)=bx. Use a graphing calculator to graph these functions.