Algebra Textbook

409 Pages·2016·35.42 MB·English

by  Eric Bright

8 G A :
TH RADE LGEBRA
at CMS 2016-2017
A Common Core State Standards Textbook

By Mr. Bright and Ms. Hittinger
Brain Image courtesy of Master Isolated Images at http://www.freedigitalphotos.net
©2016 Any part of this document may be freely modified, copied or distributed in any way as long as you
claim ownership of your modifications and continue to provide the resources without cost.
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Table of Contents
Unit 1: Functions Unit 5: Polynomials and Quadratics
1.1 Defining Functions 5.1 Polynomial Functions
1.2 Graphing Functions 5.2 Polynomial Operations
1.3 Rate of Change and Initial Values 5.3 Graphing Quadratics
1.4 Other Function Qualities 5.4 Factoring for Zeros
1.5 Contextualizing Function Qualities 5.5 Completing the Square for the Vertex
1.6 Sketching a Piecewise Function 5.6 Transforming Quadratics
Unit 2: Linear Functions Unit 6: Solving with Quadratics
2.1 Transforming Linear Functions 6.1 Solving Quadratic Equations
2.2 Contextualizing Linear Functions 6.2 The Quadratic Formula
2.3 Comparing Linear Functions 6.3 Creating and Comparing Quadratics
2.4 Solving Linear Equations 6.4 Solving Systems with Quadratics
2.5 Infinite and No Solutions 6.5 One-Variable Quadratic Inequalities
2.6 Inverse Linear Functions 6.6 Two-Variable Quadratic Inequalities
2.7 Linear Inequalities
Unit 7: Pythagorean Theorem
Unit 3: Systems of Linear Equations
7.1 Pythagorean Theorem and Converse
3.1 Solving Systems via Graphing 7.2 2D Applications
3.2 Solving Systems via Substitution 7.3 3D Applications
3.3 Solving Systems via Elimination 7.4 The Distance Formula
3.4 Solving Systems via Inspection
3.5 Solving Systems of Linear Inequalities
Unit 8: Piecewise Functions
8.1 Absolute Value Functions
Unit 4: Exponential Functions
8.2 Step Functions
4.0 Exponent Property Review 8.3 Other Piecewise Functions
4.1 Rational Exponents
4.2 Exponential Functions
Unit 9: Descriptive Statistics
4.3 Transforming Exponential Functions
4.4 Solving Exponential Functions 9.1 Representing Univariate Data
4.5 Comparing Exponential Functions 9.2 Analyzing Univariate Data
9.3 Representing Bivariate Data
9.4 Analyzing Bivariate Data
9.5 Functions of Best Fit
9.6 Two-Way Tables

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How to Use the Book
The two primary purposes for this book are to be a resource for understanding and a single place for
homework assignments. As a resource, you should read the sections in the book that you have a hard time
understanding in class. This book won’t replace the instruction that you receive from your teacher, but it should
supplement that instruction. That means it should help you understand better if you actually read through the
examples and think about what is being said.
As a place for homework assignments, this book puts all the homework directly after the explanation of
each section. Since you can write in this book directly, you are welcome to do your homework right in this book if
you have room to show your work. You will probably end up using a separate sheet of paper to do homework on
concepts like solving equations, but most units you’ll have room to do the homework in this book.
Included as a homework assignment are unit pre-tests. These pre-tests should be completed at the start
of each unit so that your teacher can really zero in on what specific skills you still need help with and what skills you
already have mastered. After that you should work on correcting the pre-test which acts like a study guide for the
post-test (or end of the unit test). Use the pre-test to help you study.
Please take care of this book as the construction is basic in nature in order to keep the costs down and allow
you to write in it. This is your book and only yours. It will not be passed on to students next year. However, if you
lose this book, you will be asked to pay for a replacement. Please treat this book gently and with respect.
If along the way, you notice any errors, please let your teacher know so that the error can be corrected for
next year’s students. We need your help to make this book better and better. Thank you in advance and enjoy!

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Unit 1: Functions
1.1 Defining Functions

1.2 Graphing Functions

1.3 Rate of Change and Initial Values

1.4 Other Function Qualities

1.5 Contextualizing Function Qualities

1.6 Sketching a Piecewise Function

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Pre-Test Unit 1: Functions
No calculator necessary. Please do not use a calculator.
Determine if each of the following is a function or not. Explain how you know. (4 pts; 2 pts for answer,
2 pts for explanation)

1. 2. (cid:2)
(cid:1) (cid:3) (cid:4)

(cid:4) (cid:5)2 (cid:5)1 0 1 2

(cid:1) 4 1 0 1 4

3. Input: Type of an individual bird (robin, finch, etc.) 4.
Output: That bird’s exact weight

5. 6. Give an example of a function in words explaining the
1,3,9,27,81…
input, output, domain, range and how you know it is a
function.

Give the domain and the range of the following functions. (4 pts; 2 pts for each)

7. (cid:16)(cid:17)(cid:4)(cid:18)(cid:3) (cid:17)(cid:4)(cid:5)4(cid:18)(cid:2)(cid:5)7 8. Input: Teenager’s identity
Output: Weekly allowance as of today

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Given the functions (cid:19)(cid:17)(cid:20)(cid:18)(cid:3) (cid:5)(cid:21)(cid:20)(cid:21)(cid:22)(cid:23) and (cid:24)(cid:17)(cid:25)(cid:18)(cid:3)(cid:25)(cid:26)(cid:5)(cid:21) evaluate the following. (4 pts; 2 pts for correct input,
2 pts for correct answer)

9. 10.
(cid:16)(cid:17)3(cid:18) (cid:27)(cid:17)(cid:5)2(cid:18)

Graph the following functions by filling out the chart using inputs ( values) that you think are appropriate.
(cid:20)/(cid:29) (cid:20)
(4 pts; 1 pt for appropriate values, 1 pt for correct table, 2 pts for graph following table)
(cid:20)

11. (cid:2) 12. (cid:30)
(cid:1) (cid:3) (cid:4) (cid:22)2 (cid:1) (cid:3) (cid:4)(cid:22)1
(cid:31)

(cid:4) (cid:4)

(cid:1) (cid:1)

Answer the following questions using the given function showing the value of an investment in ten thousands of
dollars ( ) based on the number of years from this year ( ). (4 pts; partial credit at teacher discretion)
(cid:25)

13. What does the point on the graph represent?

14. Based on the graph, what is causing the value of the (cid:25)
investment to decrease and how do you know?

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Calculate/estimate the average rate of change and the initial value over the interval for each function.
!(cid:5)(cid:21),(cid:21)"
(4 pts; 2 pts for each)

15. 16. (cid:2)
(cid:16)(cid:17)(cid:4)(cid:18) (cid:3) (cid:4) (cid:5)3

17. A banker has been gaining per year on an investment and expects to do the same the next few years.
$2,000
The banker’s investment currently has a value of . Think of the function that shows how much money the
$18,000
investment is worth based on how many years it has been since this year.

18.

(cid:4) (cid:5)4 (cid:5)2 0 2 4

(cid:16)(cid:17)(cid:4)(cid:18) 7 4 1 (cid:5)2 (cid:5)5

For the following functions tell whether they are linear or non-linear, give the domain where they are increasing
and where they are decreasing, and then give the max or min of the function.
(4 pts; 1 pt for linear/non-linear, 2 pts for increasing/decreasing domain, 1 pt for max/min)

19. (cid:1) (cid:3) (cid:30)(cid:4)(cid:5)3 20. (cid:1) (cid:3) (cid:5)(cid:17)(cid:4)(cid:5)3(cid:18)(cid:2)(cid:22)5
%

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Use the following graph showing a function modeling the miles per gallon ( ) a car gets in terms of its speed
&
( ) to answer the questions. (4 pts; partial credit at teacher discretion)
'

21. What appears to be the best mileage this car will get and at
what speed does it occur?

g
p
m
22. What are all the possible speeds this car can drive at? e in &(cid:18)
g (cid:17)
a
e
Mil

23. What are all the possible mileages this car can get?

Speed in mph

(cid:17)'(cid:18)
Determine which graph matches the story and explain why. (5 pts; 2 pts for correct answer with no explanation)

24. I started to walk to class, but I realized I had forgotten my notebook, so I went back to my locker and then I
went quickly at a constant rate to class.

Sketch a graph modeling a function for the following situations. (5 pts; partial credit at teacher discretion)

25. A dog is sleeping when he hears the cat “meow” in the next room.
He quickly runs to the next room where he slowly walks around looking
for the cat. When he doesn’t find the cat, he sits down and goes back
to sleep. Sketch a graph of a function of the dog’s speed in terms of
time.

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1.1 Defining Functions
Functions govern many interactions in our society today. Whether buying a cup of coffee at the local coffee
shop or playing a video game, we are using a function in some fashion.
Definition of a Function
A function is a rule or relationship between two quantities, often referred to as the input and output, such
that for every input there is exactly one output. If we input a specific value into the function, we get a specific
output as an answer. We won’t get the possibility of two answers or else it wouldn’t be a function. The most
common example of a function is an equation such as:

(cid:1) (cid:3)
2(cid:4)+3
In this case, is the input and is the output. If we substitute a value for , say , then we will get an
answer for , namely(cid:4) , as the out(cid:1)put. Notice that every time we input (cid:4)you wi(cid:4)ll g=et3 the output of .
Since we alw(cid:1)ays get o(cid:1)nly= a9 single output for any value we input, this is a true(cid:4) f=un3ction. (cid:1) = 9
An example of an equation that is not a function would be . Notice that if we input , then
(cid:2)
could be the output or could be the output. Therefore(cid:1) thi=s i(cid:4)s not a true function unle(cid:4)ss= w4e make (cid:1)th=e
f2unction where w(cid:1)e t=ak−e 2only the principal (or positive) square root.
(cid:1) = √(cid:4)
Function Notation
Functions represented by equations have a different notation. We are used to an equation looking like this:
. However, from this point forward function equations will be written as follows:
(cid:1) = 2(cid:4)+3

(cid:16)(cid:17)(cid:4)(cid:18)= 2(cid:4)+3
We would say that is a function of such that if you input , you will output . The advantage of
this notation is that we clear(cid:16)ly know what our (cid:4)input and output are. Fo(cid:4)r example, we mig2h(cid:4)t u+se3 the function
to represent the height of ball thrown in the air over time. We would say that the height of thℎe(cid:17) )b(cid:18)a=ll,
(cid:2)
−,1 is6 )a fu+nc4t8io)n of the time since you have thrown it in the air, .
ℎ )

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