college algebra

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

Module 7:  Readings and Multimedia

  Lecture

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

        Chapter 1: Basic Concepts - Section 1.5 Exponents

        Chapter 9: Exponential and Logarithmic Functions - Section 9.2: Exponential Functions

Download and use the Word handout  A Brief Guide to Exponential Expressions, Equations, and Functions Download A Brief Guide to Exponential Expressions, Equations, and Functions

Exponents. (2017). Math Is Fun.  https://www.mathsisfun.com/exponent.htmlLinks to an external site.

Exponential function reference. (2017). Math is Fun.  https://www.mathsisfun.com/sets/function-exponential.htmlLinks to an external site.

Using exponents in algebra. (2017). Math is Fun.  https://www.mathsisfun.com/algebra/exponents-using.htmlLinks to an external site.

  

Multimedia

Introduction to exponents | Pre-Algebra | Khan Academy

Khan Academy. (n.d.).  Introduction to exponents | Pre-Algebra | Khan Academy [Video]. YouTube.   https://www.youtube.com/watch?v=XZRQhkii0h0Links to an external site.

Exponent rules part 1 | Exponents, radicals, and scientific notation | Pre-Algebra | Khan Academy

Khan Academy. (n.d.).  Exponent rules part 1 | Exponents, radicals, and scientific notation | Pre-Algebra | Khan Academy  [Video]. YouTube.  https://www.youtube.com/watch?v=kITJ6qH7jS0Links to an external site.

What are exponential functions

McLogan, B. (n.d.).  What are exponential functions [Video]. YouTube .   https://www.youtube.com/watch?v=Gr4v3I_XBgkLinks to an external site.

Links to an external site.

Exponential growth functions | Exponential and logarithmic functions | Algebra II | Khan Academy

Khan Academy. (n.d.).  Exponential growth functions | Exponential and logarithmic functions | Algebra II | Khan Academy. [Video]. YouTube.   https://www.youtube.com/watch?v=6WMZ7J0wwMILinks to an external site.

Links to an external site.

Graphing exponential functions | Exponential and logarithmic functions | Algebra II | Khan Academy Links to an external site.

Khan Academy. (n.d.).  Graphing exponential functions | Exponential and logarithmic functions | Algebra II | Khan Academy [Video]. YouTube.   https://www.youtube.com/watch?v=9SOSfRNCQZQLinks to an external site.

Solving exponential equation | Exponential and logarithmic functions | Algebra II | Khan Academy

 Khan Academy. (n.d.).  Solving exponential equation | exponential and logarithmic functions | Algebra II | Khan Academy  [Video]. YouTube.    https://www.youtube.com/watch?v=7Ig6kVZaWoULinks to an external site.

Exponential growth and decay word problems | Algebra II | Khan Academy

Khan Academy. (n.d.).  Exponential growth and decay word problems | Algebra II | Khan Academy  [Video]. YouTube.    https://www.youtube.com/watch?v=m5Tf6vgoJtQLinks to an external site.

 

Links to an external site.

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.