1. Exponent Rules

2. Parts
When a number, variable, or expression is raised to a power, the number, variable, or expression is called the base and the power is called the exponent.

3. What is an Exponent? x4 = x ● x ● x ● x 26 = 2 ● 2 ● 2 ● 2 ● 2 ● 2
An exponent means that you multiply the base by itself that many times.
For example
x4 = x
● x
● x
● x
26 = 2
● 2
● 2
● 2
● 2
● 2
= 64

4. The Invisible Exponent
When an expression does not have a visible exponent its exponent is understood to be 1.

5. Exponent Rule #1
When multiplying two expressions with the same base you add their exponents.
For example

6. Exponent Rule #1
Try it on your own:

7. Exponent Rule #2
When dividing two expressions with the same base you subtract their exponents.
For example

8. Exponent Rule #2
Try it on your own:

9. Exponent Rule #3
When raising a power to a power you multiply the exponents
For example

10. Exponent Rule #3
Try it on your own

11. Note
When using this rule the exponent can not be brought in the parenthesis if there is addition or subtraction
You would have to use FOIL in these cases

12. Exponent Rule #4
When a product is raised to a power, each piece is raised to the power
For example

13. Exponent Rule #4
Try it on your own

14. Note
This rule is for products only. When using this rule the exponent can not be brought in the parenthesis if there is addition or subtraction
You would have to use FOIL in these cases

15. Exponent Rule #5
When a quotient is raised to a power, both the numerator and denominator are raised to the power
For example

16. Exponent Rule #5
Try it on your own

17. Zero Exponent
When anything, except 0, is raised to the zero power it is 1.
For example
( if a ≠ 0)
( if x ≠ 0)

18. Zero Exponent
Try it on your own
( if a ≠ 0)
( if h ≠ 0)

19. Negative Exponents
If b ≠ 0, then
For example

20. Negative Exponents
If b ≠ 0, then
Try it on your own:

21. Negative Exponents
The negative exponent basically flips the part with the negative exponent to the other half of the fraction.

22. Math Manners
For a problem to be completely simplified there should not be any negative exponents

23. Mixed Practice

24. Mixed Practice

25. Mixed Practice

26. Mixed Practice

27. Mixed Practice F O I L

28. Mixed Practice

29. Definition of Exponential Function
The exponential function f with base a is defined by
f(x) = ax
where a > 0, a  1, and x is any real number.
For instance,
f(x) = 3x and g(x) = 0.5x
are exponential functions.
Definition of Exponential Function

30. Example: Exponential Function
The value of f(x) = 3x when x = 2 is
f(2) = 32 = 9
The value of f(x) = 3x when x = –2 is
f(–2) = 3–2 =
The value of g(x) = 0.5x when x = 4 is
g(4) = 0.54 =
0.0625
Example: Exponential Function

31. Graph of Exponential Function (a > 1)
The graph of f(x) = ax, a > 1 y
Exponential Growth Function 4
Range: (0, )
(0, 1) x 4
Horizontal Asymptote y = 0
Domain: (–, )
Graph of Exponential Function (a > 1)

32. Graph of Exponential Function (0 < a < 1)
The graph of f(x) = ax, 0 < a < 1 y
Exponential Decay Function 4
Range: (0, )
(0, 1) x 4
Horizontal Asymptote y = 0
Domain: (–, )
Graph of Exponential Function (0 < a < 1)

33. Exponential Function 3 Key Parts 1. Pivot Point (Common Point)
2. Horizontal Asymptote
3. Growth or Decay

34. Manual Graphing Lets graph the following together: f(x) = 2x
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35. Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x))y x
f(x)
(x, f(x)) -2
(-2, ¼) -1
(-1, ½) 1
(0, 1) 2
(1, 2) 4
(2, 4) 4 2 x –2 2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph f(x) = 2x

36. Definition of the Exponential Function
The exponential function f with base b is defined by
f (x) = bx or y = bx
Where b is a positive constant other than and x is any real number.
Here are some examples of exponential functions.
f (x) = 2x g(x) = 10x h(x) = 3x
Base is 2.
Base is 10.
Base is 3.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

37. Calculator Comparison
Graph the following on your calculator at the same time and note the trend
y1 = 2x
y2= 5x
y3 = 10x

38. When base is a fraction
Graph the following on your calculator at the same time and note the trend
y1 = (1/2)x
y2= (3/4)x
y3 = (7/8)x

39. Transformations Involving Exponential Functions
Shifts the graph of f (x) = bx upward c units if c > 0.
Shifts the graph of f (x) = bx downward c units if c < 0.
g(x) = bx+ c
Vertical translation
Reflects the graph of f (x) = bx about the x-axis.
Reflects the graph of f (x) = bx about the y-axis.
g(x) = -bx
g(x) = b-x
Reflecting
Multiplying y-coordintates of f (x) = bx by c,
Stretches the graph of f (x) = bx if c > 1.
Shrinks the graph of f (x) = bx if 0 < c < 1.
g(x) = cbx
Vertical stretching or shrinking
Shifts the graph of f (x) = bx to the left c units if c > 0.
Shifts the graph of f (x) = bx to the right c units if c < 0.
g(x) = bx+c
Horizontal translation
Description
Equation
Transformation

40. Example: Translation of Graph
Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y
f(x) = 2x
The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2
Domain: (–, ) x
y = –1
Range: (–1, )
Example: Translation of Graph

41. Example: Reflection of Graph
Example: Sketch the graph of g(x) = 2-x. State the domain and range. y
f(x) = 2x
The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4
Domain: (–, ) x –2 2
Range: (0, )
Example: Reflection of Graph

42. Discuss these transformations
y = 2(x+1)
Left 1 unit
y = 2x + 2
Up 2 units
y = 2-x – 2
Ry, then down 2 units