1. 6.3 Exponential Functions
MAT SPRING 2009
6.3 Exponential Functions
In this section, we will study the following topics:
Evaluating exponential functions with base a
Graphing exponential functions with base a
Evaluating exponential functions with base e
Graphing exponential functions with base e

2. Transcendental Functions
In this chapter, we continue our study of functions with two very important ones— exponential and logarithmic functions. These two functions are types of nonalgebraic functions, known as transcendental functions.

3. So, in an exponential function, the variable is in the exponent.

4. Exponential Functions
Which of the following are exponential functions?

5. Graphs of Exponential Functions
Just as the graphs of all quadratic functions have the same basic shape, the graphs of exponential functions have the same basic characteristics.
They can be broken into two categories—
exponential growth
exponential decay (decline)

6. The Graph of an Exponential Growth Function
We will look at the graph of an exponential function that increases as x increases, known as the exponential growth function. It has the form
Example: f(x) = 2x x
f(x) -5 -4 -3 -2 -1 1 2 3
f(x)=2x
Notice the rapid increase in the graph as x increases
The graph increases slowly for x < 0.
y-intercept is (0, 1)
Horizontal asymptote is y = 0.

8. The Graph of an Exponential Decay (Decline) Function
We will look at the graph of an exponential function that decreases as x increases, known as the exponential decay function. It has the form
Example: g(x) = 2-x
g(x)=2-x x
f(x) -3 -2 -1 1 2 3 4 5
Notice the rapid decline in the graph for x < 0.
The graph decreases more slowly as x increases.
y-intercept is (0, 1)
Horizontal asymptote is y = 0.

9. Graphs of Exponential Functions
Notice that f(x) = 2x and g(x) = 2-x are reflections of one another about the y-axis.
Both graphs have: y-intercept (___,___) and horizontal asymptote y =
The domain of f(x) and g(x) is _________; the range is _______.

10. Graphs of Exponential Functions
MAT SPRING 2009
Graphs of Exponential Functions
Also, note that , using the properties of exponents.
So an exponential function is a DECAY function if
The base a is greater than one and the function is written as f(x) = a-x
-OR-
The base a is between 0 and 1 and the function is written as f(x) = ax

11. Graphs of Exponential Functions
MAT SPRING 2009
Graphs of Exponential Functions
Examples:
In this case, a = 0.25 (0 < a < 1).
In this case, a = 5.6 (a > 1).

12. Transformations of Graphs of Exponential Functions
Look at the following shifts and reflections of the graph of f(x) = 2x.
The new horizontal asymptote is y = 3

13. Transformations of Graphs of Exponential Functions

14. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each. 1.

15. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each. 2.

16. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each. 3.

17. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each. 4.

18. Graph using transformations and determine the domain, range and horizontal asymptote.B) A) D) C)

19. Natural base e
It may seem hard to believe, but when working with exponents and logarithms, it is often convenient to use the irrational number e as a base.
The number e is defined as
This value approaches as x approaches infinity.
Check this out using the TABLE on your calculator:
Enter and look at the value of y as x gets larger and larger.

20. Evaluating the Natural Exponential Function
To evaluate the function f(x) = ex, we will use our calculators to find an approximation. You should see the ex button on your graphing calculator (Use ).
Example: Evaluate to three decimal places.
e-0.5 ≈ ____________
e ≈ _______________

21. Graphing the Natural Exponential Function
Domain:___________________
Range: ___________________
Asymptote: _______________
x-intercept: _______________
y-intercept: _______________
Increasing/decreasing over ___________
List four points that are on the graph of f(x) = ex.

22. Solving Exponential Equations

23. MAT SPRING 2009

25. MAT SPRING 2009
End of Section 6.3