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CAmodule7discussionpdf.docx
CAmodule7lecutrepdf.docx
CAmodule7readingpdf.docx

CAmodule7discussionpdf.docx

Module 7: Mathematical Thinking about Binomials

Discussion Instructions:

Please review the class discussion directions . They tell you how to prepare and reflect before responding to the prompt below. They also include detailed guidelines how you must respond to classmates and then reply to the responses you receive. Please pay close attention to the response directions. To earn your discussion points, you must post, respond, and reply as directed.

Discussion writing prompts

Please create a new forum thread for each question:

· How do you raise an expression with a fraction as the exponent? Explain and detail your answer and provide examples to support your answer.

Response focus

1. In your response, in addition to the standard requirement of noting what you have in common and where you differ with classmates, offer a suggestion as to how to improve your classmates' comfort level with the topics we are studying this week.

Remember to review the academic expectations for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday. Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Sunday.

· Contribute a minimum of 250 words to the initial post.

· Follow established netiquette (Links to an external site.) guidelines when participating in forums.

Reply

CAmodule7readingpdf.docx

Module 6: Exponential Expressions, Equations, and Functions

Exponents

Product Rule for Exponents:

If m and n are natural numbers and a is any real number, then:

am∗an=am+n

Examples:

1) x2∗x4=x2+4=x6

2) 23∗24=27=2.2.2.2.2.2.2=128

Quotient Rule for Exponents:

If a is any nonzero number and m and n are integers, then:

aman=am−n

Examples:

1) x7x4=x7−4=x3

2) 6462=64−2=62=36

Zero Exponent Rule:

If a is any nonzero number and a ≠ 0, then: a∘ = 1

Negative Exponent Rule:

For any nonzero real number a and any whole number m,

a−m=1am

Example:

1) 7−2=172=149

The Power Rule:

If a is a real number and m and n are integers, then:

(am)n=am.n

Raising a Product to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=am∗bm

Raising a Quotient to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=ambmb≠0

Exponential Function

For any real number, a > 0 and a ≠ 1

In an exponential function, the variable x is the exponent.

f(x) = ax or y=ax is an exponential function.

In a polynomial function, the variable x is in the base position (example: g(x) = x2)

Examples of Exponential Functions:

f(x) = 2x

y = 5x

g(x) = (34)x

The exponential functions can be graphed by selecting values for x, finding the corresponding values of y [or f(x)], and plotting the points.

Graphs of Exponential Functions:

For all exponential functions of the form f(x) = ax or y=ax

where: a > 0 and a ≠ 1,

1. The domain of the function is (-∞, ∞).

2. The range of the function is (0, ∞).

The graph of the function passes through the points: (−1,1a),(0,1), and (1, a)

Example 1: Sketch the graphs of the function f(x)=2x and g(x)=(12)x by making a table of values.

Solution:

Step 1: Choose some values of x and find the corresponding values of f(x) and g(x).

x	f(x)=2x	g(x)=(12)x
-3	18	8
-2	14	4
-1	12	2
0	1	1
1	2	12
2	4	14
3	8	18

Example 1 (continued):

Step 2 : Plot the points found in the previous step for f(x) and g(x), and draw a smooth curve connecting them.

Step 3: Notice that

g(x)=(12)x=12x=2−x=f(−x)

and so we could have obtained the graph of g from the graph of f by reflecting in the y-axis.

Source: Chapter 6 Section 1 (alamo.edu)Links to an external site.

Solve Applications of Exponentials Functions

Exponential functions are often used to describe the growth and decay of specific quantities.

Example: Jenny told her daughter that if she did her chores, she would give her 2 cents the first week and double the amount each week for the next 10 weeks. The number of cents her daughter would receive in any given week, w, can be determined by the function n(w) =2w . Determine the number of cents Jenny would give to her daughter in week 8.

Solution: By evaluating 28, in week 8 Jenny would give to her daughter 28=256 cents or $2.56

References

Angel, A., & Rundle, D. (2018). Intermediate algebra for college students (10th edition). Pearson.

CAmodule7readingpdf.docx

Module 6: Exponential Expressions, Equations, and Functions

Exponents

Product Rule for Exponents:

If m and n are natural numbers and a is any real number, then:

am∗an=am+n

Examples:

1) x2∗x4=x2+4=x6

2) 23∗24=27=2.2.2.2.2.2.2=128

Quotient Rule for Exponents:

If a is any nonzero number and m and n are integers, then:

aman=am−n

Examples:

1) x7x4=x7−4=x3

2) 6462=64−2=62=36

Zero Exponent Rule:

If a is any nonzero number and a ≠ 0, then: a∘ = 1

Negative Exponent Rule:

For any nonzero real number a and any whole number m,

a−m=1am

Example:

1) 7−2=172=149

The Power Rule:

If a is a real number and m and n are integers, then:

(am)n=am.n

Raising a Product to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=am∗bm

Raising a Quotient to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=ambmb≠0

Exponential Function

For any real number, a > 0 and a ≠ 1

In an exponential function, the variable x is the exponent.

f(x) = ax or y=ax is an exponential function.

In a polynomial function, the variable x is in the base position (example: g(x) = x2)

Examples of Exponential Functions:

f(x) = 2x

y = 5x

g(x) = (34)x

The exponential functions can be graphed by selecting values for x, finding the corresponding values of y [or f(x)], and plotting the points.

Graphs of Exponential Functions:

For all exponential functions of the form f(x) = ax or y=ax

where: a > 0 and a ≠ 1,

1. The domain of the function is (-∞, ∞).

2. The range of the function is (0, ∞).

The graph of the function passes through the points: (−1,1a),(0,1), and (1, a)

Example 1: Sketch the graphs of the function f(x)=2x and g(x)=(12)x by making a table of values.

Solution:

Step 1: Choose some values of x and find the corresponding values of f(x) and g(x).

x	f(x)=2x	g(x)=(12)x
-3	18	8
-2	14	4
-1	12	2
0	1	1
1	2	12
2	4	14
3	8	18

Example 1 (continued):

Step 2 : Plot the points found in the previous step for f(x) and g(x), and draw a smooth curve connecting them.

Step 3: Notice that

g(x)=(12)x=12x=2−x=f(−x)

and so we could have obtained the graph of g from the graph of f by reflecting in the y-axis.

Source: Chapter 6 Section 1 (alamo.edu)Links to an external site.

Solve Applications of Exponentials Functions

Exponential functions are often used to describe the growth and decay of specific quantities.

Solution: By evaluating 28, in week 8 Jenny would give to her daughter 28=256 cents or $2.56

References

Angel, A., & Rundle, D. (2018). Intermediate algebra for college students (10th edition). Pearson.

CAmodule7discussionpdf.docx

Module 7: Mathematical Thinking about Binomials

Discussion Instructions:

Discussion writing prompts

Please create a new forum thread for each question:

· How do you raise an expression with a fraction as the exponent? Explain and detail your answer and provide examples to support your answer.

Response focus

Remember to review the academic expectations for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday. Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Sunday.

· Contribute a minimum of 250 words to the initial post.

· Follow established netiquette (Links to an external site.) guidelines when participating in forums.

Reply

CAmodule7readingpdf.docx

Module 6: Exponential Expressions, Equations, and Functions

Exponents

Product Rule for Exponents:

If m and n are natural numbers and a is any real number, then:

am∗an=am+n

Examples:

1) x2∗x4=x2+4=x6

2) 23∗24=27=2.2.2.2.2.2.2=128

Quotient Rule for Exponents:

If a is any nonzero number and m and n are integers, then:

aman=am−n

Examples:

1) x7x4=x7−4=x3

2) 6462=64−2=62=36

Zero Exponent Rule:

If a is any nonzero number and a ≠ 0, then: a∘ = 1

Negative Exponent Rule:

For any nonzero real number a and any whole number m,

a−m=1am

Example:

1) 7−2=172=149

The Power Rule:

If a is a real number and m and n are integers, then:

(am)n=am.n

Raising a Product to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=am∗bm

Raising a Quotient to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=ambmb≠0

Exponential Function

For any real number, a > 0 and a ≠ 1

In an exponential function, the variable x is the exponent.

f(x) = ax or y=ax is an exponential function.

In a polynomial function, the variable x is in the base position (example: g(x) = x2)

Examples of Exponential Functions:

f(x) = 2x

y = 5x

g(x) = (34)x

The exponential functions can be graphed by selecting values for x, finding the corresponding values of y [or f(x)], and plotting the points.

Graphs of Exponential Functions:

For all exponential functions of the form f(x) = ax or y=ax

where: a > 0 and a ≠ 1,

1. The domain of the function is (-∞, ∞).

2. The range of the function is (0, ∞).

The graph of the function passes through the points: (−1,1a),(0,1), and (1, a)

Example 1: Sketch the graphs of the function f(x)=2x and g(x)=(12)x by making a table of values.

Solution:

Step 1: Choose some values of x and find the corresponding values of f(x) and g(x).

x	f(x)=2x	g(x)=(12)x
-3	18	8
-2	14	4
-1	12	2
0	1	1
1	2	12
2	4	14
3	8	18

Example 1 (continued):

Step 2 : Plot the points found in the previous step for f(x) and g(x), and draw a smooth curve connecting them.

Step 3: Notice that

g(x)=(12)x=12x=2−x=f(−x)

and so we could have obtained the graph of g from the graph of f by reflecting in the y-axis.

Source: Chapter 6 Section 1 (alamo.edu)Links to an external site.

Solve Applications of Exponentials Functions

Exponential functions are often used to describe the growth and decay of specific quantities.

Solution: By evaluating 28, in week 8 Jenny would give to her daughter 28=256 cents or $2.56

References

Angel, A., & Rundle, D. (2018). Intermediate algebra for college students (10th edition). Pearson.

CAmodule7discussionpdf.docx

Module 7: Mathematical Thinking about Binomials

Discussion Instructions:

Discussion writing prompts

Please create a new forum thread for each question:

· How do you raise an expression with a fraction as the exponent? Explain and detail your answer and provide examples to support your answer.

Response focus

Remember to review the academic expectations for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday. Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Sunday.

· Contribute a minimum of 250 words to the initial post.

· Follow established netiquette (Links to an external site.) guidelines when participating in forums.

Reply

CAmodule7readingpdf.docx

Module 6: Exponential Expressions, Equations, and Functions

Exponents

Product Rule for Exponents:

If m and n are natural numbers and a is any real number, then:

am∗an=am+n

Examples:

1) x2∗x4=x2+4=x6

2) 23∗24=27=2.2.2.2.2.2.2=128

Quotient Rule for Exponents:

If a is any nonzero number and m and n are integers, then:

aman=am−n

Examples:

1) x7x4=x7−4=x3

2) 6462=64−2=62=36

Zero Exponent Rule:

If a is any nonzero number and a ≠ 0, then: a∘ = 1

Negative Exponent Rule:

For any nonzero real number a and any whole number m,

a−m=1am

Example:

1) 7−2=172=149

The Power Rule:

If a is a real number and m and n are integers, then:

(am)n=am.n

Raising a Product to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=am∗bm

Raising a Quotient to a Power:

If a and b are real numbers and m is an integer, then:

(ab)m=ambmb≠0

Exponential Function

For any real number, a > 0 and a ≠ 1

In an exponential function, the variable x is the exponent.

f(x) = ax or y=ax is an exponential function.

In a polynomial function, the variable x is in the base position (example: g(x) = x2)

Examples of Exponential Functions:

f(x) = 2x

y = 5x

g(x) = (34)x

The exponential functions can be graphed by selecting values for x, finding the corresponding values of y [or f(x)], and plotting the points.

Graphs of Exponential Functions:

For all exponential functions of the form f(x) = ax or y=ax

where: a > 0 and a ≠ 1,

1. The domain of the function is (-∞, ∞).

2. The range of the function is (0, ∞).

The graph of the function passes through the points: (−1,1a),(0,1), and (1, a)

Example 1: Sketch the graphs of the function f(x)=2x and g(x)=(12)x by making a table of values.

Solution:

Step 1: Choose some values of x and find the corresponding values of f(x) and g(x).

x	f(x)=2x	g(x)=(12)x
-3	18	8
-2	14	4
-1	12	2
0	1	1
1	2	12
2	4	14
3	8	18

Example 1 (continued):

Step 2 : Plot the points found in the previous step for f(x) and g(x), and draw a smooth curve connecting them.

Step 3: Notice that

g(x)=(12)x=12x=2−x=f(−x)

and so we could have obtained the graph of g from the graph of f by reflecting in the y-axis.

Source: Chapter 6 Section 1 (alamo.edu)Links to an external site.

Solve Applications of Exponentials Functions

Exponential functions are often used to describe the growth and decay of specific quantities.

Solution: By evaluating 28, in week 8 Jenny would give to her daughter 28=256 cents or $2.56

References

Angel, A., & Rundle, D. (2018). Intermediate algebra for college students (10th edition). Pearson.

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CAmodule7discussionpdf.docx

CAmodule7lecutrepdf.docx

CAmodule7readingpdf.docx

CAmodule7discussionpdf.docx

CAmodule7lecutrepdf.docx

CAmodule7readingpdf.docx

CAmodule7discussionpdf.docx