Chapter 7Powers

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Section 7-1Power Functions

How do powers apply to the real world?

Powering/Exponentiation

An operation where a base is taken to an exponent

Powering/Exponentiation

An operation where a base is taken to an exponent

xn

Powering/Exponentiation

An operation where a base is taken to an exponent

xn

base

Powering/Exponentiation

An operation where a base is taken to an exponent

xn

base

exponent

Powering/Exponentiation

An operation where a base is taken to an exponent

xn

base

exponent

power

Powering/Exponentiation

An operation where a base is taken to an exponent

xn

base

exponent

power

xn means “x to the nth power”

Powering/Exponentiation

Base:

Base: A number that is multiplied over and over

Base: A number that is multiplied over and over

Exponent:

Base: A number that is multiplied over and over

Exponent: Number of factors of the base

Base: A number that is multiplied over and over

Exponent: Number of factors of the base

x7

Base: A number that is multiplied over and over

Exponent: Number of factors of the base

x7 = x • x • x • x • x • x • x

Example 1Matt 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?

Example 1Matt 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

Example 1Matt 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 = D4

Example 1Matt 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 = D4

b. Make a table for D = {.1, .2, .3, ..., .9, 1}

Example 1Matt 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 = D4

b. Make a table for D = {.1, .2, .3, ..., .9, 1}

Example 1Matt 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 = D4

b. Make a table for D = {.1, .2, .3, ..., .9, 1}

D 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 0.0001 0.0016 0.0081 0.0256 0.0625 0.1296 0.2401 0.4096 0.6561 1

Example 1Matt 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 = D4

b. Make a table for D = {.1, .2, .3, ..., .9, 1}

D 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 0.0001 0.0016 0.0081 0.0256 0.0625 0.1296 0.2401 0.4096 0.6561 1

Example 1c. What value of D will give a probability of .5 that Matt will

be delayed four days in a row?

Example 1c. What value of D will give a probability of .5 that Matt will

be delayed four days in a row?

.5 = D4

Example 1c. What value of D will give a probability of .5 that Matt will

be delayed four days in a row?

.5 = D4

This answer is not in our table!

Example 1c. What value of D will give a probability of .5 that Matt will

be delayed four days in a row?

.5 = D4

We need to take a 4th root of D.

This answer is not in our table!

Example 1c. What value of D will give a probability of .5 that Matt will

be delayed four days in a row?

.5 = D4

We need to take a 4th root of D.

.54 = D44

This answer is not in our table!

Example 1c. What value of D will give a probability of .5 that Matt will

be delayed four days in a row?

.5 = D4

We need to take a 4th root of D.

.54 = D44

D ≈ .8408964153

This answer is not in our table!

Power Function

Power Function

f (x ) = xn ,n > 0

Identity Function

Identity Function

f (x ) = x1

Squaring Function

Squaring Function

f (x ) = x 2

Cubing Function

Cubing Function

f (x ) = x3

Fourth Power Function

Fourth Power Function

f (x ) = x 4

Fifth Power Function

Fifth Power Function

f (x ) = x5

Properties of Power Functions

Properties of Power Functions

1. The graph goes through the origin

Properties of Power Functions

1. The graph goes through the origin

0n = 0 for all n > 0

Properties of Power Functions

1. The graph goes through the origin

2. The domain is all real numbers 0

n = 0 for all n > 0

Properties of Power Functions

1. The graph goes through the origin

2. The domain is all real numbers

Any number can be taken to an exponent

0n = 0 for all n > 0

Properties of Power Functions

1. The graph goes through the origin

2. The domain is all real numbers

Any number can be taken to an exponent

3. The range has two possibilities:

0n = 0 for all n > 0

Properties of Power Functions

1. The graph goes through the origin

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}

0n = 0 for all n > 0

Properties of Power Functions

1. The graph goes through the origin

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}

0n = 0 for all n > 0

Properties of Power Functions

Properties of Power Functions

4. Symmetry exists in 2 cases:

Properties of Power Functions

4. Symmetry exists in 2 cases:

a. If n is odd, there is rotational symmetry about the origin

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

Homework

Homework

p. 423 #1-23

“If we all did the things we are capable of doing, we would literally astound ourselves.” - Thomas A. Edison