Advanced Algebra II: Conceptual Explanations

2010·0.99 MB·English

by  Kenny M. Felder.

Advanced Algebra II: Conceptual
Explanations
By:
Kenny M. Felder

Advanced Algebra II: Conceptual
Explanations
By:
Kenny M. Felder
Online:
< http://cnx.org/content/col10624/1.15/ >
C O N N E X I O N S
RiceUniversity,Houston,Texas
ThisselectionandarrangementofcontentasacollectioniscopyrightedbyKennyM.Felder. Itislicensedunderthe
CreativeCommonsAttribution2.0license(http://creativecommons.org/licenses/by/2.0/).
Collectionstructurerevised:May4,2010
PDFgenerated:November5,2011
Forcopyrightandattributioninformationforthemodulescontainedinthiscollection,seep.237.
Table of Contents
HowtoUseAdvancedAlgebraII .................................................. ...................1
1 Functions
1.1 FunctionConcepts ..........................................................................23
1.2 WhatisaVariable? ..........................................................................23
1.3 WhatisaFunction? .........................................................................24
1.4 TheRuleofConsistency .....................................................................26
1.5 FourWaystoRepresentaFunction ..........................................................27
1.6 DomainandRange ..........................................................................27
1.7 FunctionsintheRealWorld .................................................................30
1.8 FunctionNotation ...........................................................................31
1.9 AlgebraicGeneralizations ...................................................................32
1.10 Graphing ..................................................................................35
1.11 Lines ......................................................................... .............50
1.12 CompositeFunctions ......................................................... .............52
1.13 InverseFunctions ..........................................................................55
2 InequalitiesandAbsoluteValues
2.1 Inequalities .................................................................................61
2.2 AbsoluteValueEquations ...................................................................65
2.3 AbsoluteValueInequalities .................................................................67
2.4 GraphingAbsoluteValues ..................................................... .............69
2.5 GraphingInequalities .......................................................................72
2.6 "PiecewiseFunctions"andAbsoluteValue ...................................................72
3 SimultaneousEquations
3.1 Distance,RateandTime .....................................................................77
3.2 SimultaneousEquationsbyGraphing .......................................................77
3.3 Substitution .................................................................................80
3.4 Elimination .................................................................................80
3.5 SpecialCases ................................................................................81
3.6 WordProblems ................................................................ .............83
3.7 UsingLettersasNumbers ...................................................................84
Solutions ........................................................................................86
4 QuadraticFunctions
4.1 MultiplyingBinomials ......................................................................87
4.2 Factoring ...................................................................... .............89
4.3 SolvingQuadraticEquationsbyFactoring ...................................................93
4.4 SolvingQuadraticEquationsbyCompletingtheSquare .....................................94
4.5 TheQuadraticFormula .....................................................................97
4.6 DifferentTypesofSolutionstoQuadraticEquations .........................................98
4.7 GraphingQuadraticEquations ..............................................................99
4.8 SolvingProblemsbyGraphingQuadraticFunctions ........................................102
4.9 QuadraticInequalities ......................................................................104
Solutions .......................................................................................107
5 Exponents
5.1 ExponentConcepts ........................................................................109
5.2 LawsofExponents .........................................................................109
5.3 Zero,NegativeNumbers,andFractionsasExponents ......................................111
iv
5.4 ExponentialCurves ........................................................................115
6 Logarithms
6.1 LogarithmConcepts .......................................................................123
6.2 TheLogarithmExplainedbyAnalogytoRoots .............................................124
6.3 RewritingLogarithmEquationsasExponentEquations .....................................124
6.4 TheLogarithmDefinedasanInverseFunction .............................................125
6.5 PropertiesofLogarithms ...................................................................126
6.6 CommonLogarithms ......................................................................127
6.7 GraphingLogarithmicFunctions ...........................................................128
7 RationalExpressions
7.1 RationalExpressionConcepts ..............................................................131
7.2 SimplifyingRationalExpressions ...........................................................131
7.3 MultiplyingRationalExpressions ..........................................................133
7.4 AddingandSubtractingRationalExpressions ..............................................135
7.5 RationalEquations .........................................................................137
7.6 DividingPolynomials ......................................................................138
8 Radicals
8.1 RadicalConcepts ...........................................................................143
8.2 PropertiesofRadicals ......................................................................143
8.3 SimplifyingRadicals .......................................................................144
8.4 RadicalEquations ..........................................................................146
9 ImaginaryNumbers
9.1 ImaginaryNumbersConcepts ..............................................................151
9.2 Playingwithi ..............................................................................152
9.3 ComplexNumbers .........................................................................153
9.4 EqualityandInequalityinComplexNumbers ..............................................155
9.5 QuadraticEquationsandComplexNumbers ...............................................155
9.6 AFew“ExtraforExperts”ThoughtsonImaginaryNumbers ...............................156
10 Matrices
10.1 Matrices ..................................................................................159
10.2 MultiplyingMatrices ......................................................................161
10.3 TheIdentityMatrix .......................................................................165
10.4 TheInverseMatrix ........................................................................166
10.5 MatricesonaTI-83orTI-84Calculator ....................................................169
10.6 Determinants .............................................................................175
10.7 SolvingLinearEquations ..................................................................177
11 ModelingDatawithFunctions
11.1 DataModelingConcepts ..................................................................183
11.2 DirectandInverseVariations ..............................................................183
11.3 FindingaLinearFunctionforanyTwoPoints .............................................186
11.4 FindingaParabolicFunctionforanyThreePoints .........................................187
11.5 RegressionontheCalculator ..............................................................190
12 Conics
12.1 ConicConcepts ...........................................................................195
12.2 AMathematicalLookatDistance .........................................................197
12.3 Circles ....................................................................................199
12.4 Parabolas .................................................................................203
12.5 Ellipses ...................................................................................206
v
12.6 Hyperbolas ...............................................................................209
12.7 ABriefRecap: HowDoYouTellWhatShapeItIs? ........................................216
13 SequencesandSeries
13.1 Sequences ................................................................................219
13.2 Series .....................................................................................220
13.3 ProofbyInduction ........................................................................221
14 Probability
14.1 Probability ................................................................................225
14.2 Permutations .............................................................................229
14.3 Combinations .............................................................................230
15 (Untitled)
Glossary .............................................................................................234
Index ................................................................................................235
Attributions .........................................................................................237
vi
How to Use Advanced Algebra II1
Overaperiodoftime, Ihavedevelopedasetofin-classassignments, homeworks, andlessonplans, that
workformeandforotherpeoplewhohavetriedthem. Thecompletesetcomprisesthreeseparatebooks
thatworktogether:
• TheHomeworkandActivitiesBook2 containsin-classandhomeworkassignmentsthataregivento
thestudentsday-by-day.
• TheConceptsBook3providesconceptualexplanations,andisintendedasareferenceorreviewguide
forstudents;itisnotusedwhenteachingtheclass.
• TheTeacher’sGuide4provideslessonplans;itisyourguidetohowIenvisionedthesematerialsbeing
usedwhenIcreatedthem(andhowIusethemmyself).
Instructorsshouldnotethatthisbookprobablycontainsmoreinformationthanyouwillbeabletocoverin
asingleschoolyear.Imyselfdonotteachfromeverychapterinmyownclasses,buthavechosentoinclude
these additional materials to assist you in meeting your own needs. As you will likely need to cut some
sectionsfromthebook, Istronglyrecommendthatyouspendtimeearlyontodeterminewhichmodules
aremostimportantforyourstaterequirementsandpersonalteachingstyle.
One more warning is important: these materials were designed for an Advanced Algebra II course. For
suchacourse,Ihopethiswillprovideyouwithready-to-usetextbookandlessonplans. Ifyouareteaching
aStandardorRemedial-levelcourse,thesematerialswillstillbeuseful,butyouwillprobablyhavetocut
or reduce some of the most conceptual material, and supplement it with more drill-and-practice than I
provide.
The following table of contents provides a list of topics covered in this course with links to each module.
Youcanusetheselinkstomovebetweenthebooksortojumpaheadtoanytopic.
Conceptual Activities& Teacher’sGuide
Explanations Homework [col10687]
[col10624]643 [col10686]644
Functions
Introduction [m19335]645
FunctionConcepts [m18192]646
continuedonnextpage
1Thiscontentisavailableonlineat<http://cnx.org/content/m19435/1.6/>.
2AdvancedAlgebraII:ActivitiesandHomework<http://cnx.org/content/col10686/latest/>
3AdvancedAlgebraII:ConceptualExplanations<http://cnx.org/content/col10624/latest/>
4AdvancedAlgebraII:Teacher’sGuide<http://cnx.org/content/col10687/latest/>
1
2
WhatisaVariable? [m18194]647
WhatisaFunction? [m18189]648
The Rule of Consis- [m19190]649
tency
FourWaystoRepresent [m18195]650
aFunction
DomainandRange [m18191]651
TheFunctionGame [m19121]652 [m19342]653
TheFunctionGame:In- [m19125]654
troduction
The Function Game: [m19126]655
Leader’sSheet
The Function Game: [m19124]656
AnswerSheet
Functions in the Real [m18193]657 [m19331]658
World
Homework: Functions [m19115]659
intheRealWorld
FunctionNotation [m18188]660
Algebraic Generaliza- [m18186]661 [m19114]662 [m19332]663
tions
Homework: Algebraic [m19108]664
Generalizations
Graphing [m18196]665 [m19334]666
Homework: Graphing [m19116]667
Horizontal and Vertical [m19110]668 [m19339]669
Permutations
Homework: Horizon- [m19119]670
tal and Vertical Permu-
tationsI
Homework: Horizon- [m31952]671
tal and Vertical Permu-
tationsII
continuedonnextpage