MATH 112 Guided Workbook
Department of Mathematics
The University of Arizona
2015-16 Academic Year
Table of Contents
Table of Contents .......................................................................................................... 1
Syllabus Highlights ........................................................................................................ 3
Quick Reference Guide ................................................................................................. 4
Strategies for Checking Answers .................................................................................. 5
Quadratic Formula Program for Graphing Calculators ................................................. 6
Quadratic Formula - TI 82 ......................................................................................... 6
Quadratic Formula - TI 83, TI 83 Plus, and TI 84 Plus ................................................ 7
Quadratic Formula - TI 85, TI 86................................................................................ 8
Quadratic Formula – Casio models ........................................................................... 9
CHAPTER 1 – Equations, Inequalities, and Applications ............................................. 11
Section 1.1 – Linear Equations ................................................................................ 11
Section 1.2 – Applications of Linear Equations ....................................................... 15
Section 1.4 – Quadratic Equations .......................................................................... 25
Section 1.5 – Applications of Quadratic Equations ................................................. 31
CHAPTER 2 – The Rectangular Coordinate System, Lines, and Circles ....................... 41
Section 2.3 – Lines ................................................................................................... 41
Section 2.4 – Parallel and Perpendicular Lines ....................................................... 46
CHAPTER 3 – Functions ............................................................................................... 51
Section 3.1 – Relations and Functions .................................................................... 51
Section 3.2 – Properties of a Function’s Graph....................................................... 64
Section 3.3 – Graphs of Basic Functions; Piecewise Functions ............................... 80
Section 3.4 – Transformations of Functions ........................................................... 90
Section 3.5 – The Algebra of Functions; Composite Functions............................. 105
Section 3.6 – One-to-One Functions; Inverse Functions....................................... 115
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CHAPTER 4 – Polynomial and Rational Functions ..................................................... 127
Section 4.1 – Quadratic Functions ........................................................................ 127
Section 4.2 – Applications and Modeling of Quadratic Functions ........................ 135
Section 4.3 – The Graphs of Polynomial Functions ............................................... 144
Section 4.4 – Synthetic Division; The Remainder and Factor Theorems .............. 162
Section 4.6 – Rational Functions and Their Graphs .............................................. 170
CHAPTER 5 – Exponential and Logarithmic Functions and Equations ...................... 189
Section 5.1 – Exponential Functions ..................................................................... 189
Section 5.2 – The Natural Exponential Function ................................................... 204
Section 5.3 – Logarithmic Functions ..................................................................... 212
Section 5.4 – Properties of Logarithms ................................................................. 222
Section 5.5 – Exponential and Logarithmic Equations .......................................... 231
Section 5.6 – Applications of Exponential and Logarithmic Functions ................. 243
Answers to End-of-Section Self-Assessments ........................................................... 253
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Syllabus Highlights
Welcome to Math 112! The following highlights some of the most important parts of the full
syllabus. Students are still expected to know the contents of the full syllabus, which is posted in
D2L.
Required Materials



MyMathLab access
Math 112 Guided Workbook
Graphing Calculator
In-Person Class Meetings
An important and mandatory part of the course, the in-person class meetings allow students
time to interact with their instructor and classmates. Written assignments will be collected at
class and constitute a significant portion of the course grade.
Online Work
The book, homework, and online quizzes are all hosted on http://mymathlab.com (MML).
Enrollment in the mymathlab.com course must be completed by the end of the first day of
classes. Failure to enroll by the third day of classes will result in administrative drop.
Exams
There will be two midterm exams and a cumulative final exam.
Course Grade Break Down




Homework:
MML Tests:
Midterms:
Final Exam
100 points
100 points (25 points each)
400 points (200 points each)
200 points
Getting Help


Office hours and appointments by email
Tutoring at ThinkTank
Actions That May Result in an Administrative Drop




Failing to sign up for MML
Failing to come to class the first day
Missing more than 3 classes
Missing more than 5 assignments
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Quick Reference Guide
Math Department Information
Math Dept Home
Math Dept Tutoring
Math 112 Homepage
Math Common Final Exam Schedule
http://math.arizona.edu
http://math.arizona.edu/academics/tutoring
http://math.arizona.edu/~algebra
http://math.arizona.edu/academics/courseinfo/common
University Information
U of A Home
Final Exam schedule
Important Dates and Deadlines
U of A Computing Homepage
24/7 Computing Support
UAccess
D2L
http://www.arizona.edu
http://registrar.arizona.edu/schedules/finals.htm
http://www.em.arizona.edu/datesdeadlines/DatesDeadlines.aspx
http://uits.arizona.edu/
http://uits.arizona.edu/departments/the247
http://uaccess.arizona.edu/
http://d2l.arizona.edu
Important University Policies
Code of Academic Integrity
http://deanofstudents.arizona.edu/policies-and-codes/code-academic-integrity
Student Code of Conduct
http://azregents.asu.edu/rrc/Policy Manual/5-308-Student Code of Conduct.pdf
Services for Students
University Tutoring Services
Disability Resource Center
Campus Health
http://thinktank.arizona.edu/
http://drc.arizona.edu
http://www.health.arizona.edu/main.htm
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Strategies for Checking Answers
One of the components of the written work in the course involves checking your answers.
There are many ways to check your answer to a problem. These are some strategies you can
try:
1. Use the answer you found and substitute it back into the original equation. When you
are asked to solve an equation, this is the simplest and most straight-forward way to
check your results.
2. When you are solving a word problem, check to see whether your solution yields the
desired outcome. For example, if you are solving for the dimensions of a rectangle that
give a certain area, use the length and width you found in your solution to calculate the
area, and see if you get what you expect.
3. Solve the problem using a different technique or algorithm. For example, if you solved a
quadratic equation by factoring, you could check it by using the quadratic formula.
4. Use a different approach to solve the problem. For example, if you solved a problem
strictly algebraically, you can try graphing on your calculator to check your solution.
(This can work the other way around as well!)
5. Check to see whether your answer makes sense. For example, if you are solving a
problem to find the speed of a car driving on the highway, 350 mph is not a reasonable
solution.
6. To take the previous strategy one step farther, you can refine what kind of answer is
reasonable. For example, if the problem states that a person is given one 100 mg
dosage of a drug, and you need to calculate the amount of the drug left in the person’s
system after a certain amount of time, then your answer must be greater than or equal
to 0 mg, and less than or equal to 100 mg. So if you get an answer of 120 mg or -5 mg,
you know you’ve done something incorrectly.
7. Build a concrete example of the more abstract question. For example, if you are asked
how the graph of 𝑦 = 2𝑓(𝑥) is transformed from the graph of 𝑦 = 𝑓(𝑥), you can pick
a specific function, say 𝑦 = √ 𝑥, and take a look at the graph of 𝑦 = √ 𝑥 compared to
the graph of 𝑦 = 2√ 𝑥 .
Note: Sometimes students try to check by re-doing the problem in the same way they
approached it the first time. If you made a mistake the first time, it is very likely to make the
same mistake the second time. This is not recommended as a way to check your work (though it
may help in a pinch on an exam).
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Quadratic Formula Program for Graphing Calculators
Quadratic Formula - TI 82
Introduction
This program solves equations of the form 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 = 0 by using the quadratic
formula.
The Program
:Prompt A,B,C
:(B2-4*A*C) D
:If D<0
:Then
:Disp "NO REAL SOLUTIONS"
:Goto 1
:Else
:((-B+ (D))/(2A)) E
:((-B- (D))/(2A)) F
:Disp "SOLUTIONS",E,F
:Lbl 1
{Prompt is in PRGM under I/O}
{The arrow is STO}
{If is in PRGM under CTL} {< is in TEST }
{Then is in PRGM under CTL}
{Disp is in PRGM under I/O} {Words within "" are
typed using ALPHA}
{Goto is in PRGM under CTL}
{Else is in PRGM under CTL}
{The is square root} {The - is the negative sign}
{The second - is the subtraction sign}
{Disp is in PRGM under I/O} {Words within "" are
typed using ALPHA}
{Lbl is in PRGM under CTL}
Running the program
You will be asked to enter values for A, B, and C according to the quadratic formula. To
test your program, try the following:
A=2, B=3, C=4. Your answer should be NO REAL SOLUTIONS.
A=5, B=4, C=-2. Your answer should be .348331477355, -1.14833147735.
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Quadratic Formula - TI 83, TI 83 Plus, and TI 84 Plus
Introduction
This program solves equations of the form 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 = 0 by using the quadratic
formula.
The Program
:Prompt A,B,C
:(B2-4*A*C) D
:If D<0
:Then
:Disp "NO REAL SOLUTIONS"
:Goto 1
:Else
:((-B+ (D))/(2A)) E
:((-B- (D))/(2A)) F
:Disp "SOLUTIONS",E,F
:Lbl 1
{Prompt is in PRGM under I/O}
{The arrow is STO}
{If is in PRGM under CTL} {< is in TEST }
{Then is in PRGM under CTL}
{Disp is in PRGM under I/O} {Words within "" are
typed using ALPHA}
{Goto is in PRGM under CTL}
{Else is in PRGM under CTL}
{The is square root} {The - is the negative sign}
{The second - is the subtraction sign}
{Disp is in PRGM under I/O} {Words within "" are
typed using ALPHA}
{Lbl is in PRGM under CTL}
Running the program
You will be asked to enter values for A, B, and C according to the quadratic formula. To
test your program, try the following:
A=2, B=3, C=4. Your answer should be NO REAL SOLUTIONS.
A=5, B=4, C=-2. Your answer should be .348331477355, -1.14833147735.
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Quadratic Formula - TI 85, TI 86
Introduction
This program solves equations of the form 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 = 0 by using the quadratic
formula.
The Program
:Prompt A,B,C
:(B2-4*A*C) D
:If D<0
:Then
:Disp "NO REAL SOLUTIONS"
:Goto P
:Else
:((-B+ (D))/(2A)) E
:((-B- (D))/(2A)) F
:Disp "SOLUTIONS",E,F
:Lbl P
{Prompt is in PRGM under I/O}
{The arrow is STO}
{If is in PRGM under CTL} {< is in TEST}
{Then is in PRGM under CTL}
{Disp is in PRGM under I/O} {Words within "" are
typed using ALPHA}
{Goto is in PRGM under CTL}
{Else is in PRGM under CTL}
{The is square root} {The - is the negative sign}
{The second - is the subtraction sign}
{Disp is in PRGM under I/O} {Words within "" are
typed using ALPHA}
{Lbl is in PRGM under CTL}
Running the program
You will be asked to enter values for A, B, and C according to the quadratic formula. To
test your program, try the following:
A=2, B=3, C=4. Your answer should be NO REAL SOLUTIONS.
A=5, B=4, C=-2. Your answer should be .348331477355, -1.14833147735.
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Quadratic Formula – Casio models
Introduction
This program solves equations of the form 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 = 0 by using the quadratic
formula.
The Program
'QUADRATIC'
"A"? A
"B"? B
"C"? C
(B^2-4 A C) D
D<0
Goto 1
((-B+
D)
(2A))
{This will be the name of the program}
{" is in ALPHA} {? is in PRGM} { is on the , button}
E
((-B- D) (2A)) F
"SOLUTIONS"
E
F
Goto 2
Lbl 1
"NO REAL SOLUTIONS"
Lbl 2
{ is the times sign}
{ , Goto are in PRGM under JMP} {< is in PRGM under
REL} {0 is a zero}
{The is the square root symbol} {The - is the negative
sign}
{The second - is a subtraction sign}
{
is in PRGM, do not hit EXE}
{Lbl is in PRGM under JMP}
Running the program
You will be asked to enter values for A, B, and C according to the quadratic formula. To
test your program, try the following:
A=2, B=3, C=4. Your answer should be NO REAL SOLUTIONS.
A=5, B=4, C=-2. Your answer should be .348331477355, -1.14833147735.
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