Table Of ContentAlgebra 2
Contributing Author
Dinah Zike
Consultant
Douglas Fisher, PhD
Director of Professional Development
San Diego State University
San Diego, CA
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the
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ISBN: 0-07-869018-8 Algebra 2(Virginia Student Edition)
Noteables™: Interactive Study Notebook with Foldables™
1 2 3 4 5 6 7 8 9 10 047 09 08 07 06 05 04
Contents
Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 1 4-6 Cramer’s Rule . . . . . . . . . . . . . . . . . 85
Vocabulary Builder . . . . . . . . . . . . . . . . . . 2 4-7 Identity and Inverse Matrices . . . . . 87
1-1 Expressions and Formulas . . . . . . . . 4 4-8 Using Matrices to Solve Systems
1-2 Properties of Real Numbers. . . . . . . 6 Of Equations . . . . . . . . . . . . . . . . . 89
1-3 Solving Equations . . . . . . . . . . . . . . 8 Study Guide . . . . . . . . . . . . . . . . . . . . . . 91
1-4 Solving Absolute Value Equations. . 11
1-5 Solving Inequalities . . . . . . . . . . . . 13
1-6 Solving Compound and Absolute
Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 95
Value Inequalities . . . . . . . . . . . . . 16
Vocabulary Builder . . . . . . . . . . . . . . . . . 96
Study Guide . . . . . . . . . . . . . . . . . . . . . . 19
5-1 Monomials. . . . . . . . . . . . . . . . . . . 98
5-2 Polynomials . . . . . . . . . . . . . . . . . 101
5-3 Dividing Polynomials. . . . . . . . . . 103
Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 23 5-4 Factoring Polynomials . . . . . . . . . 105
Vocabulary Builder . . . . . . . . . . . . . . . . . 24 5-5 Roots of Real Numbers . . . . . . . . 107
2-1 Relations and Functions. . . . . . . . . 26 5-6 Radical Expressions . . . . . . . . . . . 109
2-2 Linear Equations . . . . . . . . . . . . . . 30 5-7 Rational Exponents . . . . . . . . . . . 111
2-3 Slope . . . . . . . . . . . . . . . . . . . . . . . 33 5-8 Radical Equations and Inequalities 114
2-4 Writing Linear Equations. . . . . . . . 36 5-9 Complex Numbers . . . . . . . . . . . . 116
2-5 Model Data: Using Scatter Plots . . 38 Study Guide . . . . . . . . . . . . . . . . . . . . . 119
2-6 Special Functions. . . . . . . . . . . . . . 41
2-7 Graphing Inequalities . . . . . . . . . . 44
Study Guide . . . . . . . . . . . . . . . . . . . . . . 45
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 123
Vocabulary Builder . . . . . . . . . . . . . . . . 124
6-1 Graphing Quadratic Functions. . . 126
Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 49 6-2 Graph Quadratic Equations . . . . . 129
Vocabulary Builder . . . . . . . . . . . . . . . . . 50 6-3 Factor Quadratic Equations. . . . . 132
3-1 Graph Systems of Equations . . . . . 52 6-4 Completing the Square . . . . . . . . 134
3-2 Solving Systems of Equations 6-5 The Quadratic Formula and the
Algebraically . . . . . . . . . . . . . . . . . 55 Discriminant. . . . . . . . . . . . . . . . . 137
3-3 Graph Systems of Inequalities. . . . 58 6-6 Graphs of Quadratic Functions . . 140
3-4 Linear Programming . . . . . . . . . . . 60 6-7 Graphing and Solving Quadratic
3-5 Solving Systems of Equations in Inequalities . . . . . . . . . . . . . . . . . 142
Three Varaiables. . . . . . . . . . . . . . . 64 Study Guide . . . . . . . . . . . . . . . . . . . . . 145
Study Guide . . . . . . . . . . . . . . . . . . . . . . 66
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 149
Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 69 Vocabulary Builder . . . . . . . . . . . . . . . . 150
Vocabulary Builder . . . . . . . . . . . . . . . . . 70 7-1 Polynomial Functions. . . . . . . . . . 152
4-1 Introduction to Matrices . . . . . . . . . 72 7-2 Graphing Polynomial Functions. . 155
4-2 Operations with Matrices . . . . . . . . 75 7-3 Solving Equations Using
4-3 Multiplying Matrices . . . . . . . . . . . . 77 Quadratic Techniques . . . . . . . . . 158
4-4 Transformations with Matrices . . . . 80 7-4 Remainder and Factor Theorems . 161
4-5 Determinants . . . . . . . . . . . . . . . . . . 83 7-5 Roots and Zeros. . . . . . . . . . . . . . 163
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7-6 Rational Zero Theorem . . . . . . . . 165 11-2 Arithmetic Series . . . . . . . . . . . . . 250
7-7 Operations on Functions . . . . . . . 167 11-3 Geometric Sequences. . . . . . . . . . 253
7-8 Inverse Functions and Relations . . 169 11-4 Geometric Series . . . . . . . . . . . . . 255
7-9 Square Root Functions and 11-5 Infinite Geometric Series. . . . . . . 257
Inequalities . . . . . . . . . . . . . . . . . 171 11-6 Recursion and Special Sequences . 259
Study Guide . . . . . . . . . . . . . . . . . . . . . 173 11-7 The Binomial Theorem . . . . . . . . 261
11-8 Proof/Mathematical Induction. . . 263
Study Guide . . . . . . . . . . . . . . . . . . . . . 265
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 177
Vocabulary Builder . . . . . . . . . . . . . . . . 178
8-1 Midpoint and Distance Formulas . 180
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 269
8-2 Parabolas. . . . . . . . . . . . . . . . . . . 181
Vocabulary Builder . . . . . . . . . . . . . . . . 270
8-3 Circles. . . . . . . . . . . . . . . . . . . . . . 184
12-1 The Counting Principle . . . . . . . . 272
8-4 Ellipses. . . . . . . . . . . . . . . . . . . . . 187
12-2 Permutations and Combinations. . 274
8-5 Hyperbolas. . . . . . . . . . . . . . . . . . 190
12-3 Probability . . . . . . . . . . . . . . . . . . 276
8-6 Conic Sections . . . . . . . . . . . . . . . 192
12-4 Multiplying Probabilities. . . . . . . 280
8-7 Solving Quadratic Systems. . . . . . 194
12-5 Adding Probabilities . . . . . . . . . . 282
Study Guide . . . . . . . . . . . . . . . . . . . . . 197
12-6 Statistical Measures . . . . . . . . . . 284
12-7 The Normal Distribution . . . . . . . 286
12-8 Binomial Experiments . . . . . . . . . 288
12-9 Sampling and Error . . . . . . . . . . . 290
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 201
Study Guide . . . . . . . . . . . . . . . . . . . . . 292
Vocabulary Builder . . . . . . . . . . . . . . . . 202
9-1 ((cid:2)) and ((cid:3)) Rational Expressions. . 203
9-2 ((cid:4)) and ((cid:5)) Rational Expressions. . 206
9-3 Graphing Rational Functions. . . . 209
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 297
9-4 Direct, Joint, Inverse Variation . . 211
Vocabulary Builder . . . . . . . . . . . . . . . . 298
9-5 Classes of Functions. . . . . . . . . . . 213
13-1 Right Triangle Trigonometry . . . . 300
9-6 Solving Rational Equations
13-2 Angles and Angle Measure. . . . . 303
and Inequalities. . . . . . . . . . . . . . 215
13-3 Trigonometric Functions of
Study Guide . . . . . . . . . . . . . . . . . . . . . 219
General Angles . . . . . . . . . . . . . . 305
13-4 Law of Sines. . . . . . . . . . . . . . . . . 308
13-5 Law of Cosines. . . . . . . . . . . . . . . 311
13-6 Circular Functions . . . . . . . . . . . . 313
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 223
13-7 Inverse Trigonometric Functions . . 315
Vocabulary Builder . . . . . . . . . . . . . . . . 224
Study Guide . . . . . . . . . . . . . . . . . . . . . 317
10-1 Exponential Functions. . . . . . . . . 226
10-2 Logarithms and Logarithmic
Functions . . . . . . . . . . . . . . . . . . . 229
10-3 Properties of Logarithms . . . . . . . 231
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 321
10-4 Common Logarithms . . . . . . . . . . 233
Vocabulary Builder . . . . . . . . . . . . . . . . 322
10-5 Base e and Natural Logarithms. . 235
14-1 Graph Trigonometric Functions. . 323
10-6 Exponential Growth and Decay. . 238
14-2 Translations of Trig. Graphs. . . . . 325
Study Guide . . . . . . . . . . . . . . . . . . . . . 241
14-3 Trigonometric Identities . . . . . . . 328
14-4 Verifying Trigonometric Identities. 330
14-5 ((cid:4)) and ((cid:5)) of Angles Formulas . . 332
Foldables. . . . . . . . . . . . . . . . . . . . . . . . 245 14-6 Double/Half-Angle Formulas. . . . 334
Vocabulary Builder . . . . . . . . . . . . . . . . 246 14-7 Solving Trigonometric Equations. 337
11-1 Arithmetic Sequences . . . . . . . . . 248 Study Guide . . . . . . . . . . . . . . . . . . . . . 341
iv Glencoe Algebra 2
Organizing Your Foldables
Make this Foldable to help you organize and
store your chapter Foldables. Begin with one
sheet of 11"(cid:2)17" paper.
Fold
Fold the paper in half lengthwise. Then unfold.
Fold and Glue
Fold the paper in half widthwise and glue all of the edges.
Glue and Label
Glue the left, right, and bottom edges of the Foldable
to the inside back cover of your Noteables notebook.
Foldables Organizer
Reading and Taking Notes As you read and study each chapter, record
notes in your chapter Foldable. Then store your chapter Foldables inside
this Foldable organizer.
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This note-taking guide is designed to help you succeed in Algebra 2.
Each chapter includes:
CHAPTER
6
Quadratic Functions and Inequalities
Use the instructions below to make a Foldable to help you
organize your notes as you study the chapter. You will see
Foldable reminders in the margin of this Interactive Study
Notebook to help you in taking notes.
The Chapter Opener
Begin with one sheet of 11" (cid:2)17" paper.
contains instructions and
FFTcrohooleldsdns w i anfoin shledda. liCCfn uul eftto nauglortthnhwgs ise. hapter 6iall uFostlrdaatbiolen st hoant hwoiwll hteol pm yaokue
tthhee medidgde lteo f othlde flarostm Cto organize your notes.
crease as shown.
Refold and Label
Refold along lengthwise
fold and staple uncut
stehcet isoenct aiot nt owpi.t hLa ab el 6-16-26-63-56-66-7Vocab.
lesson number and
close to form a booklet.
CHAPTER
©Glencoe/McGraw-Hill NapbrOeoTsueEtn- TttAhedeK .Io NWrGdr eiTtreI Pi nw: hwWyh hyiceohnu t ythoheui n ctkoa nkthceee np cotost neacsr,ee pt bhtsein iwnkgere BUTAhIsL iysD ois uY a 6cOno UmaRlpp lhVeatObe eCtthiAceaB sl tUluisLdtAy o Rnf oYnteews fvoorc tahbeu clahrayp tteerrm, yso yuo wu iwll islle lee aBrunil din Y Cohuarpter 6.
presented in this sequence. Vocabulary reminders to complete each term’s definition or description on
these pages. Remember to add the textbook page number in the second
column for reference when you study.
A Note-Taking Tip
provides a helpful Glencoe Algebra 2 123 Vocabulary Term oFno Puangde Definition DesEcxraipmtipolne or
axis of symmetry
hint you can use
when taking notes. completing the square
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constant term
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[dihs-KRIH-muh-nuhnt] c
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linear term Within each chapter, e/M
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The Build Your Vocabulary maximum value Build Your Vocabulary G
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table allows you to write minimum value boxes will remind you w
doteefr fimimnsip ttoioorgntaes ntahtn evdro iecnax baomunleparleys p[q[pkuauwrahaad-bhrRoa-AlDtai-Rcb Aeuqh-tu-ilahutkhi]o]n to fill in this table. ©Glencoe/McGraw-Hill -Hill
convenient place.
124 Glencoe Algebra 2
vi Glencoe Algebra 2
Lessons cover the content of
6–6 Analyzing Graphs of Quadratic Functions the lessons in your textbook.
SanOdL l oAgIaI.r1it5hmTihce f usntucdtieonnts .wTilhl ere gcroagpnhiizneg tchael cguelnaetorra lw silhl abpee u osfe pdo alysn ao mtoiaoll, teox ipnovensetnigtiaatle, As your teacher discusses each
the shape and behavior of these functions.
BUILD YOUR VOCABULARY(pages 124–125) example, follow along and
Each leWsHsAToYOnU’L LiLsEARN A function written in the form, y(cid:6)(x(cid:5)h)2(cid:4)k, where complete the fill-in boxes.
correla•tAfyune(cid:6)naclatyid(zoxen(cid:5) sq ouhfat) 2dth(cid:4)roaet kifc.o rmthe (h, k) is the of the parabola and x(cid:6)his its Take notes as appropriate.
Virgini•aWfur nictSeti oan Oq iuna tdhLrea tfisocrm. , is referred to as the vertex form.
y(cid:6)a(x(cid:5)h)2(cid:4)k.
Graph a Quadratic Function in Vertex Form
Analyze y(cid:3)(x(cid:4)3)2(cid:5)2. Then draw its graph.
The vertex is at (h, k) or and the axis of symmetry is
x(cid:6) . The graph has the same shape as the graph of Examples parallel the
y (cid:2)x2, but is translated 3 units right and 2 units up. examples in your textbook.
Now use this information to draw the graph.
STEP 1Plot the vertex, . f(x)
((11,, 66))
STEP 2Draw the axis of symmetry, ((55,, 66))
.
STEP 3Find and plot two points ((22,, ((3333)),, 22))f(x)(( (cid:3)44,, 33(x)) (cid:4) 3)2 (cid:5) 2 6–7
on one side of the axis of O x Solve a Quadratic Inequality
symmetry, such as (2, 3)
and (1, 6). Solve x2(cid:5)x(cid:6)2 algebraically.
OOn RtheG pAagNe fIoZr EIT STEP 4Use symmetry to complete the graph. © First, solve the related equationx2(cid:4)x(cid:6)2.
LgTovaeeh rfha sret6stophn-oi1rche niasz 6 k olp-o6 2eftan- rt6rata6ca,-a hn 63pb-sl5 s kato6tlha-errl6aaatetic nb oahg6sonf-l r7ltaaa Vaeoatcp.arinboh .adn. Your Turn Analyze y(cid:6)(x(cid:4)2)2(cid:5)4. Then dryawO its graxph. Glencoe/McGraw-Hill (cid:6)0 xo2r (cid:4) x(cid:6)(cid:6)(cid:6)020 (cid:6)0 RSFauecblattoZtreare.dcr otq 2uP afrrododrmaut ciect aePcqrhou paseitdioretny.
x(cid:6) x(cid:6) Solve each equation.
Plot the values on a number line. Use closed circles since these
Foldables feature solutions are included. Notice that the number line is
separated into 3 intervals.
140 Glencoe Arlgeebram 2 inds you to take x ≤ (cid:4)2 (cid:4)2 ≤ x ≤ 1 x ≥ 1
notes in your Foldable.
–5 –4 –3 –2 –1 0 1 2 3 4 5
Test a value in each interval to see if it satisfies the original
inequality.
x(cid:6)(cid:4)2 (cid:4)2 (cid:6)x(cid:6)1 x(cid:7)1
Test x(cid:3)(cid:4)3. Test x(cid:3)0. Test x(cid:3)2.
x2(cid:4)x(cid:8)2 x2(cid:4)x(cid:8)2 x2(cid:4)x(cid:8)2
CHAPTER ((cid:5)3)2(cid:5)3 (cid:8)?2 02(cid:4)0 (cid:8)?2 22(cid:4)2 (cid:8)?2
6 BRINGING IT ALL TOGETHER 6 (cid:8)2 (cid:2) 0 (cid:8)2 ✔ 6 (cid:8)2 (cid:2)
The solution set is .
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Exercises:
www.glencoe.com/sec/math/
t_resources/free/index.php
© 6-1
G Graphing Quadratic Functions 144 Glencoe Algebra 2
le Rseenfteern tcoe tsh.e graph at the right as you complete the following Your Turn Exercises allow
n
co 1.The curve is called a . (–2, 4) f(x) you to solve similar
e/M 2.The line x(cid:6)(cid:5)2 is called the . exercises on your own.
cG 3.The point ((cid:5)2, 4) is called the . O(0, –1) x
ra
w Determine whether each function has a maximum or
-H mofi enaimchu mfu nvcatliuoen. .Then find the maximum or minimum value
ill 4.ƒ(x) (cid:6)(cid:5)x2(cid:4)2x(cid:4)5 5.ƒ(x) (cid:6)3x2(cid:5)4x(cid:5)2
© Bringing It All Together
Glencoe/McGraw-Hill Sco6oSn.lovslxeev62 ci-en(cid:5)2uagtc2i hvQx eue(cid:6) qianu8dtareatgiteoircns .E bIqfe uetwaxtaeiceotnn r swo 7boh.ytis xcG 2hcr(cid:4)a tanhp5nehxo ri(cid:5)tno bgo7et (cid:6)s f oa0uren dlo, csatateted .the Smcotuanidcney ipGdteus aifdsr oea mnred ev kiaeecwhy sl ethsseon.
Glencoe Algebra 2 145
Glencoe Algebra 2 vii
NOTE-TAKING TIPS
Your notes are a reminder of what you learned in class. Taking good notes
can help you succeed in mathematics. The following tips will help you take
better classroom notes.
• Before class, ask what your teacher will be discussing in class. Review
mentally what you already know about the concept.
• Be an active listener. Focus on what your teacher is saying. Listen for
important concepts. Pay attention to words, examples, and/or diagrams
your teacher emphasizes.
• Write your notes as clear and concise as possible. The following symbols
and abbreviations may be helpful in your note-taking.
Symbol or Symbol or
Word or Phrase Word or Phrase
Abbreviation Abbreviation
for example e.g. not equal (cid:9)
such as i.e. approximately (cid:2)
with w/ therefore (cid:2)
without w/o versus vs
and (cid:4) angle (cid:3)
• Use a symbol such as a star ((cid:3) ) or an asterisk (*) to emphasis important
concepts. Place a question mark (?) next to anything that you do not
understand.
• Ask questions and participate in class discussion.
• Draw and label pictures or diagrams to help clarify a concept.
• When working out an example, write what you are doing to solve the
problem next to each step. Be sure to use your own words.
• Review your notes as soon as possible after class. During this time, organize
and summarize new concepts and clarify misunderstandings. ©
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• Don’t write every word. Concentrate on the main ideas and concepts. G
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• Don’t use someone else’s notes as they may not make sense.
-H
• Don’t doodle. It distracts you from listening actively. ill
• Don’t lose focus or you will become lost in your note-taking.
viii Glencoe Algebra 2
H A PTE
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Solving Equations and Inequalities
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Use the instructions below to make a Foldable to help you
organize your notes as you study the chapter. You will see
Foldable reminders in the margin of this Interactive Study
Notebook to help you in taking notes.
Begin with one sheet of notebook paper.
Fold
Fold lengthwise to the holes.
Cut and Fold
Label the columns as shown.
EquationsInequalities
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often a good idea to use symbols to emphasize
important concepts.
Glencoe Algebra 2 1
H A PTE
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BUILD YOUR VOCABULARY
This is an alphabetical list of new vocabulary terms you will learn in Chapter 1.
As you complete the study notes for the chapter, you will see Build Your
Vocabulary reminders to complete each term’s definition or description on
these pages. Remember to add the textbook page number in the second
column for reference when you study.
Found Description or
Vocabulary Term Definition
on Page Example
absolute value
algebraic expression
Associative Property
[uh-SOH-shee-uh-tihv]
Commutative Property
[kuh-MYOO-tuh-tihv]
compound inequality
Distributive Property
[dih-STRIH-byuh-tihv]
empty set
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Inverse Property
2 Glencoe Algebra 2
