Roanoke County Public Schools
Pre-Calculus
Curriculum Guide
Revised
2012
Pre-Calculus Curriculum Guide
2012
Mathematics Curriculum Guide
Revised 2012. Available at www.rcs.k12.va.us.
Roanoke County Public Schools does not discriminate with regard to race, color, age, national origin, gender, or handicapping condition in an
educational and/or employment policy or practice. Questions and/or complaints should be addressed to the Deputy Superintendent/Title IX
Coordinator at (540) 562-3900 ext. 10121 or the Director of Pupil Personnel Services/504 Coordinator at (540) 562-3900 ext. 10181.
Acknowledgements
The following people have made tremendous contributions to the completion of this curriculum guide and all are appreciated.
Rick Marciniec
Hidden Valley High
Sherri Mays
William Byrd High
Bob Powers
Cave Spring High
Kelly Shilling
Cave Spring High
Butch Tyree
Northside High
Roanoke County Public Schools Administration
Dr. Lorraine Lange
Superintendent
Cecil Snead
Director of Secondary Instruction
Rebecca Eastwood
Director of Elementary Instruction
Linda Bowden
Mathematics Coordinator
Preface
This curriculum guide is written for the teachers to assist them in using the textbooks/resources in a most effective way. This guide will assist the mathematics
teacher in preparing students for the challenges of the twenty-first century. As established by the National Council of Teachers of Mathematics Principles and
Standards for School Mathematics, educational goals for students are changing. Students should have many and varied experiences in their mathematical
training to help them learn to value mathematics, become confident in their ability to do mathematics, become problem solvers, and learn to communicate and
reason mathematically. This guide, along with the available textbook resources, other professional literature, alternative assessment methods, and varied
instruction in-service activities will assist the mathematics teacher in continuing to integrate these student goals into the curriculum.
Pre-Calculus Curriculum Guide
2012
Table of Contents
Introduction/General Comments ............................................................................................................................................. i
Textbook/Resources Overview ................................................................................................................................................ i
Sequence of Instruction and Pacing Suggestions ................................................................................................................... ii
Sequence of Instruction and Pacing Suggestions .................................................................................................................. iii
Mapping for Instruction - First Nine Weeks ............................................................................................................................ 1
Mapping for Instruction - Second Nine Weeks ........................................................................................................................ 3
Mapping for Instruction - Third Nine Weeks ........................................................................................................................... 4
Mapping for Instruction - Fourth Nine Weeks ......................................................................................................................... 5
Supplemental Resources ......................................................................................................................................................... 7
SOL 2009 Framework .............................................................................................................................................................. 8
Pre-Calculus Curriculum Guide
2012
Introduction/General Comments
The purpose of the curriculum guide is to provide the teacher with a sequence of topics in Precalculus and a suggested time table for topics. There
are no state SOL objectives for this course, but the Standard of Learning objectives listed under Trigonometry and Mathematical Analysis will apply
to the topics being taught.
A graphing calculator should be used to reinforce those graphical techniques that are initially developed through pencil and paper.
An Instructor’s Guide is included with the teacher’s resources which provides a suggested time and emphasis for each section, along with points to
stress, sample questions to ask, examples, group work, and homework problems.
Textbook/Resources Overview
Course Title: Precalculus
Course Text: Precalculus Mathematics for Calculus; 6th edition
Publisher: Brooks/Cole
Supplemental Materials:
Teacher’s Classroom Resources:
•
Textbook
•
Exam View Pro Testmaker
•
Complete Solutions Manual
•
Student Solutions Manual
•
Study Guide
•
Test Bank
•
Instructors Guide
i
Pre-Calculus Curriculum Guide
2012
Sequence of Instruction and Pacing Suggestions
First Nine Weeks
SOL
MA.1
Chapter/Sections/Topic
*Time Frame
Introduction, Syllabus, Lesson 1.1, and Pretest (Given as a take home)
MA.1, MA.2, MA.3, MA.8
MA.1
MA.1, MA.3, MA.8
Chapter 2: Functions
Lessons 2.1 - 2.7
Chapter 1: Fundamentals
Lessons 1.3 and 1.5
Chapter 3: Polynomial and Rational Functions
Lesson 3.1 - 3.6
1.00 blocks
10.00 blocks
1.00 blocks
10.50 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45
days in the middle schools.
First Nine Weeks Total
22.5 blocks
Second Nine Weeks
SOL
MA.1
MA.1, MA.3
MA.9
Chapter/Sections/Topic
*Time Frame
Chapter 1: Fundamentals
Lesson 1.4
Chapter 3: Polynomial and Rational Functions
Lesson 3.7
Chapter 4: Exponential and Logarithmic Functions
Lessons 4.1 - 4.6
1.00 blocks
6.00 blocks
12.00 blocks
Exam Review and 1st Semester Exam
3.50 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45
days in the middle schools.
ii
Second Nine Weeks Total
22.5 blocks
Pre-Calculus Curriculum Guide
2012
Sequence of Instruction and Pacing Suggestions
Third Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
Chapter 1: Fundamentals
Lesson 1.2
T.1, T.2, T.3, T.4, T.5, T.6, T.9, Chapters 5 and 6: Trigonometric Functions
MA.13
Lessons 6.1, 5.1, 6.2, 5.2, 6.3-6.6, 5.3, 5.4
Chapter 7: Analytic Trigonometry
T.5
Lesson 7.1
MA.1
1.00 blocks
18.50 blocks
3.00 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for
45 days in the middle schools.
Third Nine Weeks Total
22.5 blocks
Fourth Nine Weeks
SOL
T.5, T.7, T.8
MA.7
MA.10
MA.4, MA.5
MA.8
Chapter/Sections/Topic
*Time Frame
Chapter 7: Analytic Trigonometry
Lessons 7.2 - 7.5
8.00 blocks
Chapter 13: Limits
Lessons 13.1 - 13.3
Chapter 8: Polar Coordinates
Lessons 8.1 - 8.3
Chapter 12: Sequences and Series
Lessons 12.1 and 12.6
Chapter 11: Conic Sections (If time allows)
Lessons 11.1 - 11.4
4.00 blocks
5.00 blocks
3.50 blocks
0.00 blocks
Exam Review and Exam
2.00 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for
45 days in the middle schools.
iii
Fourth Nine Weeks Total
22.5 blocks
Pre-Calculus Curriculum Guide
2012
Mapping for Instruction - First Nine Weeks
SOL with Essential Knowledge and Skill
MA.1
Textbook Chapters/Sections/Topics
Introduction, Syllabus, and Pre-Test
0.5 block
This pre-test is located in the
supplemental worksheets section in the
back of the curriculum guide.
1-1 Real Numbers
Instructor’s Guide 1-1
0.5 block
Chapter: 2
SOL with Essential Knowledge and Skill
Supporting Materials
Comments
The pre-test should be
assigned as a take home test
on the first day of class and
due at the end of the first
week.The results of this pretest will determine if students
have the previous knowledge
needed for this course and
are prepared for the rigorous
content this course requires.
Functions
Textbook Chapters/Sections/Topics
Supporting Materials
1 block
Instructor’s Guide 2-1
1.5 blocks
Instructor’s Guide 2-2
MA.1
2-1 What is a Function
MA.1, MA.3, MA.8
2-2 Graphs of Functions
MA.1
2-3 Getting Information from the Graph of
a Function
0.5 block
Instructor’s Guide 2-3
MA.1
2-4 Average Rate of Change of a
Function
1 block
Instructor’s Guide 2-4
Review/Assessment
ExamView Pro and Test Bank
1 block
MA.1, MA.3, MA.8
2-5 Transformations of Functions
1.5 blocks
Instructor’s Guide 2-5
MA.2
2-6 Combining Functions
1.5 blocks
Instructor’s Guide 2-6
MA.2
2-7 One-to-One Functions and Their
Inverses
1 block
Instructor’s Guide 2-7
Review/Assessment
ExamView Pro and Test Bank
1 block
1
Comments
optional; can be omitted if
additional time needed for
other lessons.
Pre-Calculus Curriculum Guide
2012
Chapter: 3
SOL with Essential Knowledge and Skill
Polynomial and Rational Functions
Textbook Chapters/Sections/Topics
Supporting Materials
MA.1
1-3 Algebraic Expressions
0.5 block
Instructor’s Guide 1-3
MA.1
1-5 Equations
0.5 block
Instructor’s Guide 1-5
MA.1, MA.3, MA.8
3-1 Quadratic Functions and Models
1.5 blocks
Instructor’s Guide 3-1
MA.1, MA.3
3-2 Polynomial Functions and Their
Graphs
1.5 blocks
Instructor’s Guide 3-2
MA.1
3-3 Dividing Polynomials
2 blocks
Instructor’s Guide 3-3
Review/Assessment
1 block
ExamView Pro and Test Bank
MA.1
3-4 Real Zeros of Polynomials 1 block
Instructor’s Guide 3-4
MA.1
3-5 Complex Numbers
Instructor’s Guide 3-5
MA.1
3-6 Complex Zeros and the Fundamental
Theorem of Algebra
1 block
Instructor’s Guide 3-6
Review/Assessment
ExamView Pro and Test Bank
1 block
1.5 blocks
2
Comments
Review the concept of
complex numbers. Omit the
operations.
Pre-Calculus Curriculum Guide
2012
Mapping for Instruction - Second Nine Weeks
Chapter: 3
SOL with Essential Knowledge and Skill
Polynomial and Rational Functions
Textbook Chapters/Sections/Topics
Supporting Materials
MA.1
1-4 Rational Expressions
1 block
Instructor’s Guide 1-4
MA.1, MA.3
3-7 Rational Functions
4 blocks
Instructor’s Guide 3-7
Review/Assessment
2 block
ExamView Pro and Test Bank
Chapter: 4
SOL with Essential Knowledge and Skill
Comments
Exponential and Logarithmic Functions
Textbook Chapters/Sections/Topics
Supporting Materials
MA.9
4-1 Exponential Functions
1.5 blocks
Instructor’s Guide 4-1
MA.9
4-2 The Natural Exponential Function
1 block
Instructor’s Guide 4-2
MA.9
4-3 Logarithmic Functions
1.5 blocks
Instructor’s Guide 4-3
MA.9
4-4 Laws of Logarithms
1.5 blocks
Instructor’s Guide 4-4
Review/Assessment
1.5 blocks
ExamView Pro and Test Bank
MA.9
4-5 Exponential and Logarithmic
Equations
2 blocks
Instructor’s Guide 4-5
MA.9
4-6 Modeling w/ Exponential and
Logarithmic Functions
1.5 blocks
Instructor’s Guide 4-6
Review/Assessment
ExamView Pro and Test Bank
Exam Review and Exam
1.5 blocks
3.5 blocks
3
ExamView Pro and Test Bank
Comments
Pre-Calculus Curriculum Guide
2012
Mapping for Instruction - Third Nine Weeks
Chapter: 5 & 6
SOL with Essential Knowledge and Skill
Trigonometric Functions
Textbook Chapters/Sections/Topics
1-2 Exponents and Radicals
Supporting Materials
1 block
Instructor’s Guide 1-2
Omit Scientific Notation
Angular Speed and Linear Speed
are optional
T.3
6-1 Angle Measure
1.5 blocks
Instructor’s Guide 6-1
T.2, T.3
5-1 The Unit Circle
1.5 blocks
Instructor’s Guide 5-1
Review/Assessment
Comments
1 block
ExamView Pro and Test Bank
T.1, T.2, T.3, T.4, T.9, MA.13
6-2 Trigonometry of Right Triangles
1.5 blocks
Instructor’s Guide 6-2
Trigonometric Reference Chart
T.1, T.2, T.5
5-2 Trigonometric Functions of Real
Numbers
1 block
Instructor’s Guide 5-2
Trigonometric Reference Chart
T.1, T.2, T.5
6-3 Trigonometric Functions of Angles
1.5 blocks
Instructor’s Guide 6-3
Trigonometric Reference Chart
T.7
6-4 Inverse Trigonometric Functions
1 block
Instructor’s Guide 6-4
Review/Assessment
ExamView Pro and Test Bank
1 block
T.9, MA.13
6-5 The Law of Sines
1.5 blocks
Instructor’s Guide 6-5
T.9, MA.13
6-6 The Law of Cosines
1.5 blocks
Instructor’s Guide 6-6
Review/Test
1 block
ExamView Pro and Test Bank
T.6
5-3 Trigonometric Graphs
2 blocks
Instructor’s Guide 5-3
Use calculator to explore changes
Technology Integrated Lesson Plan in graphs; Graph functions without
calculator.
T.6
5-4 More Trigonometric Graphs 1.5 blocks Instructor’s Guide 5-4
Technology Integrated Lesson Plan
Review/Quiz
1 block
4
ExamView Pro and Test Bank
Pre-Calculus Curriculum Guide
2012
Chapter: 7
SOL with Essential Knowledge and Skill
T.5
Analytic Trigonometry
Textbook Chapters/Sections/Topics
Supporting Materials
7-1 Trigonometric Identities
2 blocks
Instructor’s Guide 7-1
Review/Assessment
1 block
ExamView Pro and Test Bank
Comments
Mapping for Instruction - Fourth Nine Weeks
Chapter: 7
SOL with Essential Knowledge and Skill
Analytic Trigonometry
Textbook Chapters/Sections/Topics
Supporting Materials
T.5
7-2 Addition and Subtraction Formulas
1.5 blocks
Instructor’s Guide 7-2
T.5
7-3 Double Angle Formulas
Instructor’s Guide 7-3
T.8
1.5 blocks
Review/Assessment
1 block
7-4 Trigonometric Equations
1.5 blocks Instructor’s Guide 7-4
ExamView Pro and Test Bank
1.5 blocks ExamView Pro and Test Bank
Chapter: 13
SOL with Essential Knowledge and Skill
Omit Half-Angle and ProductSum Formulas
Instructor’s Guide 7-5
7-5 More Trigonometric Equations
1 block
Review/Assessment
Comments
Limits: A Preview of Calculus
Textbook Chapters/Sections/Topics
Supporting Materials
MA.7
13-1 Finding Limits Numerically and
Graphically
1 block
Instructor’s Guide 13-1
MA.7
13-2 Finding Limits Algebraically
1 block
Instructor’s Guide 13-2
MA.7
13-3 The Definition of the Derivative
1 block
Instructor’s Guide 13-3
Review/Quiz
ExamView Pro and Test Bank
1 block
5
Comments
Pre-Calculus Curriculum Guide
2012
Chapter: 8
SOL with Essential Knowledge and skill
Polar Coordinates
Textbook Chapters/Sections/Topics
Supporting Materials
MA.10
8-1 Polar Coordinates
1 block
Instructor’s Guide 8-1
MA.10
8-2 Graphs of Polar Equations 1 block
Instructor’s Guide 8-2
MA.10
8-3 Polar Form of Complex Numbers;
DeMoivre's Theorem
1 block
Instructor’s Guide 8-3
Review/Assessment
ExamView Pro and Test Bank
1 block
Chapter: 12
SOL with Essential Knowledge and skill
Sequences and Series
Textbook Chapters/Sections/Topics
Supporting Materials
12-1 Sequences and Summation
Notation
1 block
MA.4
12-6 The Binomial Theorem
1.5 blocks Instructor’s Guide 12-6
Review/Assessment
1 block
Chapter: 11
Comments
Instructor’s Guide 12-1
MA.5
SOL with Essential Knowledge and Skill
Comments
ExamView Pro and Test Bank
Conic Sections
Textbook Chapters/Sections/Topics
Supporting Materials
Comments
MA.8
11-1 Parabolas
*1 block
Instructor’s Guide 11-1
*optional; if time allows
MA.8
11-2 Ellipses
*1 block
Instructor’s Guide 11-2
*optional; if time allows
MA.8
11-3 Hyperbolas
*1 block
Instructor’s Guide 11-3
*optional; if time allows
MA.8
11-4 Shifted Conics
*1 block
Instructor’s Guide 11-4
*optional; if time allows
Review/Quiz
*1 block
ExamView Pro and Test Bank
*optional; if time allows
Exam Review and Exam
2 blocks
6
ExamView Pro and Test Bank
Pre-Calculus Curriculum Guide
2012
Supplemental Resources
2.5 Transformations of Functions:
http://www.youtube.com/watch?v=tZZB9P3P5Yc
4.1 Graphing the Exponential Function:
http://www.youtube.com/watch?v=gl1R6vJrrSM
4.3 Logarithmic Functions:
http://video.google.com/videoplay?docid=-9145601263198670406#
http://www.youtube.com/watch?v=VuL890FP6iQ&feature=player_embedded
5.1 The Unit Circle:
http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php [Unit circle practice]
http://www.mathlearning.net/learningtools/Flash/unitCircle/unitCircle.html [Unit circle practice]
http://www.youtube.com/watch?v=ao4EJzNWmK8&feature=related [Trick to remembering unit circle]
5.3 Trignometric Graphs:
http://www.analyzemath.com/unitcircle/unitcircle.html [Unit Circle and the Trig Functions sin(x) cos(x) tan(x)]
http://www.youtube.com/watch?v=CWjVy4iTVAI [Video of Graphing the Sine Function]
6.1 Angle Measure:
http://www.themathpage.com/aTrig/measure-angles.htm [Defintion of an Angle, Degree Measure, Standard Position, The Four Quadrants, Coterminal Angles]
http://themathpage.com/aTrig/radian-measure.htm [Radian to degrees, Degrees to radians, Coterminal angles, The multiples of pi]
http://www.youtube.com/watch?v=ZUrZHF_WBeI [Greek Alphabet Song]
http://www.mathwarehouse.com/trigonometry/reference-angle/finding-reference-angle.php [Finding Reference Angles ( Interactive)]
http://zonalandeducation.com/mmts/trigonometryRealms/radianDemo1/RadianDemo1.html [Radian- Visual of 1 radian and arc length]
6.5 The Ambiguous Case of the Law of Sines:
http://www.youtube.com/watch?v=ksBaHrVqhyo
13.1 & 13.2 Limits:
http://www.mastermathmentor.com/
7
Pre-Calculus Curriculum Guide
2012
SOL 2009 Framework
Trigonometry
The standards below outline the content for a one-semester course in trigonometry. Students enrolled in trigonometry are assumed to have
mastered those concepts outlined in the Algebra II standards. A thorough treatment of trigonometry will be provided through the study of
trigonometric definitions, applications, graphing, and solving trigonometric equations and inequalities. Emphasis should also be placed on using
connections between right triangle ratios, trigonometric functions, and circular functions. In addition, applications and modeling should be included
throughout the course of study. Emphasis should also be placed on oral and written communication concerning the language of mathematics,
logic of procedure, and interpretation of results.
Graphing calculators, computers, and other appropriate technology tools will be used to assist in teaching and learning. Graphing utilities enhance
the understanding of realistic applications through modeling and aid in the investigation of trigonometric functions and their inverses. They also
provide a powerful tool for solving and verifying solutions to trigonometric equations and inequalities.
T.1
The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric
functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions
defined on the unit circle will be related to trigonometric functions defined in right triangles.
T.2
The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions
and properties of the trigonometric functions.
T.3
The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related
angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.
T.4
The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.
T.5
The student will verify basic trigonometric identities and make substitutions, using the basic identities.
T.6
The student, given one of the six trigonometric functions in standard form, will
T.7
a)
state the domain and the range of the function;
b)
determine the amplitude, period, phase shift, vertical shift, and asymptotes;
c)
sketch the graph of the function by using transformations for at least a two-period interval; and
d)
investigate the effect of changing the parameters in a trigonometric function on the graph of the function.
The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions.
Restrictions on the domains of the inverse trigonometric functions will be included.
8
Pre-Calculus Curriculum Guide
2012
T.8
The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic
trigonometric inequalities.
T.9
The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric
functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.
Mathematical Analysis
The standards below outline the content for a one-year course in Mathematical Analysis. Students enrolled in Mathematical Analysis are assumed
to have mastered Algebra II concepts and have some exposure to trigonometry. Mathematical Analysis develops students’ understanding of
algebraic and transcendental functions, parametric and polar equations, sequences and series, and vectors. The content of this course serves as
appropriate preparation for a calculus course.
Graphing calculators, computers, and other appropriate technology tools will be used to assist in teaching and learning. Graphing utilities enhance
the understanding of realistic applications through modeling and aid in the investigation of functions and their inverses. They also provide a
powerful tool for solving and verifying solutions to equations and inequalities.
MA.1
The student will investigate and identify the characteristics of polynomial and rational functions and use these to sketch the graphs of
the functions. This will include determining zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the
function is increasing or decreasing, and maximum or minimum points. Graphing utilities will be used to investigate and verify these
characteristics.
MA.2
The student will apply compositions of functions and inverses of functions to real-world situations. Analytical methods and graphing
utilities will be used to investigate and verify the domain and range of resulting functions.
MA.3
The student will investigate and describe the continuity of functions, using graphs and algebraic methods.
MA.4
The student will expand binomials having positive integral exponents through the use of the Binomial Theorem, the formula for
combinations, and Pascal’s Triangle.
MA.5
The student will find the sum (sigma notation included) of finite and infinite convergent series, which will lead to an intuitive approach
to a limit.
MA.6
The student will use mathematical induction to prove formulas and mathematical statements.
MA.7
The student will find the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity. A
graphing utility will be used to verify intuitive reasoning, algebraic methods, and numerical substitution.
MA.8
The student will investigate and identify the characteristics of conic section equations in (h, k) and standard forms. Transformations in
the coordinate plane will be used to graph conic sections.
9
Pre-Calculus Curriculum Guide
MA.9
2012
The student will investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions
and solve equations and real-world problems. This will include the role of e, natural and common logarithms, laws of exponents and
logarithms, and the solution of logarithmic and exponential equations.
MA.10 The student will investigate and identify the characteristics of the graphs of polar equations, using graphing utilities. This will include
classification of polar equations, the effects of changes in the parameters in polar equations, conversion of complex numbers from
rectangular form to polar form and vice versa, and the intersection of the graphs of polar equations.
MA.11 The student will perform operations with vectors in the coordinate plane and solve real-world problems, using vectors. This will include
the following topics: operations of addition, subtraction, scalar multiplication, and inner (dot) product; norm of a vector; unit vector;
graphing; properties; simple proofs; complex numbers (as vectors); and perpendicular components.
MA.12 The student will use parametric equations to model and solve application problems.
MA.13 The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric
functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.
MA.14 The student will use matrices to organize data and will add and subtract matrices, multiply matrices, multiply matrices by a scalar, and
use matrices to solve systems of equations.
10
Pre-Calculus Curriculum Guide
2012
Precalculus Pre-Test
Section 1.1:
1.
2.
3.
Real Numbers
2
5
In the set {-10,  , 0,
, 2.13 , π}, which numbers are;
3
6
a.
rational
b.
natural
c.
integer
State the property of real numbers show by:
a.
5(y + 3z) = (y + 3z)5
b.
5(y + 3z) = 5y + 15z
c.
5 + (y + 3z) = (5 + y) + 3z
Write the following expressions without parenthesis:
a.
(3x)(4y + 2z)
b.
3(4y)
4.
Compare the two numbers using a <, >, or = symbol.
7
3.5
a.
3
5.
Evaluate:
2  6
a.
b.
1
4

3
5
11
Pre-Calculus Curriculum Guide
Section 1.2:
6.
Exponents are Radicals
Write the following in exponential notation.
x3
a.
b.
7.
8.
2012
1
5
3
Write the following in radical notation.
2
3
a.
6
b.
a 4
Evaluate the following radical and exponential expressions.
a.
-52
4
b.
83
c.
 2 5
   
 3 4
3
d.
e.
9.
3
2
64
 16 
 
 25 

1
2
Simplify the following radical and exponential expressions. Eliminate any negative exponents.
1 
2 s 3t 1  s 6  16t 4
a.
4 




12
Pre-Calculus Curriculum Guide
 2 x   3x 
x 
2
3
b.
2012
3
4
4
2
c.
 34  3
y 
 
1
10 5 5
d.
10.
z


1
2 3 3
z
Rationalize the denominators of the following fractions.
2
a.
5
b.
Section 1.3:
11.
y
y
5
4
x3
Algebraic Expressions
Simplify:
a.
3(x + 1) – 4(2x – 6)
b.
2(2 – 5t) + t2(t – 1) – (t4 – 1)
c.
(2 + 3a)2
d.
(x + 3y + z)(2x – 3y + 5)
13
Pre-Calculus Curriculum Guide
12.
Factor the following polynomials completely.
a.
12x3 + 18x
b.
4x2 – 25
c.
y3 – 3y2 – 4y + 12
d.
x2 – 6xy + 5y2
e.
2x2 + 5x + 3
Section 1.4:
13.
14.
2012
Rational Expressions
Simplify the following rational expressions.
x2  6 x  8
a.
x2  5x  4
b.
4x x  2
x  4 16 x
c.
4 y2  9
2 y2  y  3

2 y 2  9 y  18 y 2  5 y  6
d.
1
1
 2
x3 x 9
2
Rationalize the denominators.
1
a.
2 3
b.
y
3 y
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Pre-Calculus Curriculum Guide
Section 1.5:
15.
16.
17.
2012
Equations
Solve the following equations:
1
2
x 8  3 x
a.
2
3
b.
2x 1 4

x2 5
c.
3
1
1
 
x 1 2 2x  2
d.
F G
mM
for m.
r2
Solve the following quadratics by the indicated method.
a.
x2 – 7x = - 12; by factoring
b.
x2 + 3x + 1 = 0; by quadratic formula (leave in radical form)
c.
x2 – 4x + 2 = 0; by completing the square
Solve the following equations:
1
2
 2 0
a.
x 1 x
b.
x4 – 5x2 + 4 = 0
c.
3x  5  1
15
Pre-Calculus Curriculum Guide
Section 1.7:
18.
Inequalities
Solve the following inequalities. Express the solution using interval notation and graph the solution set on the real number line.
a.
3x  2  11
b.
1  2x  5  3
c.
x 2  4 x  12  0
Section 1.8:
19.
2012
Coordinate Geometry
Find the distance between and midpoint of the following pairs of points.
a.
(-3, 3) and (5, -3)
y
20.
Sketch the region {(x, y) | 1 < x < 3}
x
Section 1.10: Lines
21.
Find the slope of the line that passes through:
a.
(0, 2) and (-4, 7)
b.
22.
(3, 5) and (3, -7)
Find the equation of the line (in slope intercept form) that:
a.
passes through (2, 3) and has a slope of ¼
16
Pre-Calculus Curriculum Guide
23.
2012
b.
passes through the points (3, 6) and (4, -1)
c.
parallel to the line y = - ½x + 5 and passes through (4, 6)
Graph the following lines.
a.
y = -3x + 5
y
b.
x + 3y = 5
y
x
Section 2.1:
24.
25.
c.
y
x
x
What is a function?
Evaluate the following for f(x) = 2x2 + 3x – 4
a.
f(0)
b.
f(-2)
c.
f(x + 1)
Find the domain of the following functions:
x2
f  x  2
a.
x 1
b.
x=2
f  x   2x  5
17
Pre-Calculus Curriculum Guide
2012
Section 2.2: Graphs of Functions
26.
Determine whether or not the following are functions.
a.
b.
c
y
x
27.
Determine whether or not y is a function of x.
a.
x + y2 = 9
b.
x2 + 3y = 8
c.
2 x 3  y  7
Section 2.6: Combining Functions
28.
Use the functions f(x) = x2 + 2x and g(x) = 3x2 – 1 to find:
a.
(f + g)(x)
b.
(f – g)(x)
c.
(fg)(x)
d.
f 
   x
g
e.
f(g(0))
f.
f(g(2))
g.
f(g(x))
18
Pre-Calculus Curriculum Guide
2012
Section 2.7:
29.
30.
One-to-One Functions and Their Inverses
x5
Show that f  x   2 x  5 and f  x  
are inverses of each other.
2
Find the inverse of:
a.
f(x) = 4x+7
b.
Section 3.1:
31.
f  x   2  5x
Quadratic Functions and Models
For the function f(x) = x2 + 4x – 3:
a.
find its vertex
b.
find its x intercept(s)
c.
find its y intercept(s)
d.
sketch the graph
y
x
32.
Find the minimum value for the graph of f(t) = 10t2 + 40t + 1
19