This document discusses power functions and exponents. It defines power functions as operations where a base is taken to an exponent. It provides examples of squaring (x^2), cubing (x^3), and fourth power (x^4) functions. Key properties discussed include that power functions go through the origin, have all real numbers as their domain, and their range depends on whether the exponent is odd or even. Symmetry properties also depend on the exponent. Examples are provided to illustrate calculating probabilities using exponents.

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Chapter 7
Powers

2. Section 7-1
Power Functions
How do powers apply to the real world?

3. Powering/
Exponentiation

4. Powering/
Exponentiation
An operation where a base is taken to an exponent

5. Powering/
Exponentiation
An operation where a base is taken to an exponent
xn

6. Powering/
Exponentiation
An operation where a base is taken to an exponent
xn
base

7. Powering/
Exponentiation
An operation where a base is taken to an exponent
exponent
xn
base

8. Powering/
Exponentiation
An operation where a base is taken to an exponent
exponent
xn
power
base

9. Powering/
Exponentiation
An operation where a base is taken to an exponent
exponent
xn
power
base
xn means “x to the nth power”

10. Base:

11. Base: A number that is multiplied over and over

12. Base: A number that is multiplied over and over
Exponent:

13. Base: A number that is multiplied over and over
Exponent: Number of factors of the base

14. Base: A number that is multiplied over and over
Exponent: Number of factors of the base
7
x

15. Base: A number that is multiplied over and over
Exponent: Number of factors of the base
7
x = x•x•x•x•x•x•x

16. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?

17. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?
y = D•D•D•D

18. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?
4
y = D•D•D•D = D

19. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?
4
y = D•D•D•D = D
b. Make a table for D = {.1, .2, .3, ..., .9, 1}

20. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?
4
y = D•D•D•D = D
b. Make a table for D = {.1, .2, .3, ..., .9, 1}

21. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?
4
y = D•D•D•D = D
b. Make a table for D = {.1, .2, .3, ..., .9, 1}
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
D
y 0.0001 0.0016 0.0081 0.0256 0.0625 0.1296 0.2401 0.4096 0.6561 1

22. Example 1
Matt Mitarnowski drives to school. Suppose there is a
probability D that he will run into a delay on the way.
a. What is the probability y that he will be delayed four days
in a row?
4
y = D•D•D•D = D
b. Make a table for D = {.1, .2, .3, ..., .9, 1}
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
D
y 0.0001 0.0016 0.0081 0.0256 0.0625 0.1296 0.2401 0.4096 0.6561 1

23. Example 1
c. What value of D will give a probability of .5 that Matt will
be delayed four days in a row?

24. Example 1
c. What value of D will give a probability of .5 that Matt will
be delayed four days in a row?
4
.5 = D

25. Example 1
c. What value of D will give a probability of .5 that Matt will
be delayed four days in a row?
4
.5 = D
This answer is not in our table!

26. Example 1
c. What value of D will give a probability of .5 that Matt will
be delayed four days in a row?
4
.5 = D
This answer is not in our table!
We need to take a 4th root of D.

27. Example 1
c. What value of D will give a probability of .5 that Matt will
be delayed four days in a row?
4
.5 = D
This answer is not in our table!
We need to take a 4th root of D.
4
4 4
.5 = D

28. Example 1
c. What value of D will give a probability of .5 that Matt will
be delayed four days in a row?
4
.5 = D
This answer is not in our table!
We need to take a 4th root of D.
4
4 4
.5 = D
D ≈ .8408964153

29. Power Function

30. Power Function
n
f (x ) = x , n > 0

31. Identity Function

32. Identity Function
1
f (x ) = x

33. Squaring Function

34. Squaring Function
2
f (x ) = x

35. Cubing Function

36. Cubing Function
3
f (x ) = x

37. Fourth Power Function

38. Fourth Power Function
4
f (x ) = x

39. Fifth Power Function

40. Fifth Power Function
5
f (x ) = x

41. Properties of Power
Functions

42. Properties of Power
Functions
1. The graph goes through the origin

43. Properties of Power
Functions
1. The graph goes through the origin
n
0 = 0 for all n > 0

44. Properties of Power
Functions
1. The graph goes through the origin
n
0 = 0 for all n > 0
2. The domain is all real numbers

45. Properties of Power
Functions
1. The graph goes through the origin
n
0 = 0 for all n > 0
2. The domain is all real numbers
Any number can be taken to an exponent

46. Properties of Power
Functions
1. The graph goes through the origin
n
0 = 0 for all n > 0
2. The domain is all real numbers
Any number can be taken to an exponent
3. The range has two possibilities:

47. Properties of Power
Functions
1. The graph goes through the origin
n
0 = 0 for all n > 0
2. The domain is all real numbers
Any number can be taken to an exponent
3. The range has two possibilities:
a. If n is odd, R = {y: y is all real numbers}

48. Properties of Power
Functions
1. The graph goes through the origin
n
0 = 0 for all n > 0
2. The domain is all real numbers
Any number can be taken to an exponent
3. The range has two possibilities:
a. If n is odd, R = {y: y is all real numbers}
b. If n is even, R = {y : y ≥ 0}

49. Properties of Power
Functions

50. Properties of Power
Functions
4. Symmetry exists in 2 cases:

51. Properties of Power
Functions
4. Symmetry exists in 2 cases:
a. If n is odd, there is rotational symmetry about
the origin

52. Properties of Power
Functions
4. Symmetry exists in 2 cases:
a. If n is odd, there is rotational symmetry about
the origin
b. If n is even, there is reflection symmetry over
the y-axis

53. Homework

54. Homework
p. 423 #1-23
“If we all did the things we are capable of doing, we would
literally astound ourselves.” - Thomas A. Edison