CK 12 Algebra I Second Edition

2015·33 MB·English

by  Eve Rawley, Anne Gloag, Andrew Gloag

CK-12 Algebra I - Second
Edition
Eve Rawley
Anne Gloag
Andrew Gloag
Eve Rawley, (EveR)
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iii
Contents www.ck12.org
Contents
1 EquationsandFunctions 1
1.1 VariableExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 OrderofOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 PatternsandEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 EquationsandInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 FunctionsasRulesandTables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6 FunctionsasGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7 Problem-SolvingPlan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.8 Problem-SolvingStrategies: MakeaTableandLookforaPattern . . . . . . . . . . . . . . . . . 61
2 RealNumbers 72
2.1 IntegersandRationalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 AddingandSubtractingRationalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3 MultiplyingandDividingRationalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.4 TheDistributiveProperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.5 SquareRootsandRealNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.6 Problem-SolvingStrategies: GuessandCheck,WorkBackward . . . . . . . . . . . . . . . . . . 110
3 EquationsofLines 116
3.1 One-StepEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2 Two-StepEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.3 Multi-StepEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.4 EquationswithVariablesonBothSides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.5 RatiosandProportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.6 PercentProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4 GraphsofEquationsandFunctions 156
4.1 TheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.2 GraphsofLinearEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.3 GraphingUsingIntercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.4 SlopeandRateofChange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.5 GraphsUsingSlope-InterceptForm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.6 DirectVariationModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.7 LinearFunctionGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.8 Problem-SolvingStrategies-Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5 WritingLinearEquations 223
5.1 FormsofLinearEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5.2 EquationsofParallelandPerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
5.3 FittingaLinetoData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.4 PredictingwithLinearModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
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6 LinearInequalities 266
6.1 SolvingInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.2 UsingInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.3 CompoundInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
6.4 AbsoluteValueEquationsandInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
6.5 LinearInequalitiesinTwoVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
7 SolvingSystemsofEquationsandInequalities 309
7.1 LinearSystemsbyGraphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
7.2 SolvingLinearSystemsbySubstitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
7.3 SolvingLinearSystemsbyElimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
7.4 SpecialTypesofLinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
7.5 SystemsofLinearInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
8 ExponentialFunctions 364
8.1 ExponentPropertiesInvolvingProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
8.2 ExponentPropertiesInvolvingQuotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
8.3 Zero,Negative,andFractionalExponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
8.4 ScientificNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
8.5 GeometricSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
8.6 ExponentialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8.7 ApplicationsofExponentialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
9 Polynomials 415
9.1 AdditionandSubtractionofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
9.2 MultiplicationofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
9.3 SpecialProductsofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
9.4 PolynomialEquationsinFactoredForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
9.5 FactoringQuadraticExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
9.6 FactoringSpecialProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
9.7 FactoringPolynomialsCompletely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
10 QuadraticEquationsandQuadraticFunctions 473
10.1 GraphsofQuadraticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
10.2 QuadraticEquationsbyGraphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
10.3 QuadraticEquationsbySquareRoots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
10.4 SolvingQuadraticEquationsbyCompletingtheSquare . . . . . . . . . . . . . . . . . . . . . . 505
10.5 SolvingQuadraticEquationsbytheQuadraticFormula . . . . . . . . . . . . . . . . . . . . . . 516
10.6 TheDiscriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
10.7 Linear,ExponentialandQuadraticModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
11 AlgebraandGeometryConnections 548
11.1 GraphsofSquareRootFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
11.2 RadicalExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
11.3 RadicalEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
11.4 ThePythagoreanTheoremandItsConverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
11.5 DistanceandMidpointFormulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
12 RationalEquationsandFunctions 595
12.1 InverseVariationModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
12.2 GraphsofRationalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
12.3 DivisionofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
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12.4 RationalExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
12.5 MultiplyingandDividingRationalExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . 624
12.6 AddingandSubtractingRationalExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
12.7 SolutionsofRationalEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
13 ProbabilityandStatistics 643
13.1 TheoreticalandExperimentalProbability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
13.2 ProbabilityandPermutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
13.3 ProbabilityandCombinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
13.4 ProbabilityofCompoundEvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
13.5 MeasuresofCentralTendencyandDispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
13.6 Stem-and-LeafPlotsandHistograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
13.7 Box-and-WhiskerPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
13.8 SurveysandSamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
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www.ck12.org Chapter1. EquationsandFunctions
C 1
HAPTER
Equations and Functions
Chapter Outline
1.1 VARIABLE EXPRESSIONS
1.2 ORDER OF OPERATIONS
1.3 PATTERNS AND EQUATIONS
1.4 EQUATIONS AND INEQUALITIES
1.5 FUNCTIONS AS RULES AND TABLES
1.6 FUNCTIONS AS GRAPHS
1.7 PROBLEM-SOLVING PLAN
1.8 PROBLEM-SOLVING STRATEGIES: MAKE A TABLE AND LOOK FOR A PATTERN
1
1.1. VariableExpressions www.ck12.org
1.1 Variable Expressions
Learning Objectives
• Evaluatealgebraicexpressions.
• Evaluatealgebraicexpressionswithexponents.
Introduction - The Language of Algebra
No one likes doing the same problem over and over again—that’s why mathematicians invented algebra. Algebra
takesthebasicprinciplesofmathandmakesthemmoregeneral, sowecansolveaproblemonceandthenusethat
solutiontosolveagroupofsimilarproblems.
Inarithmetic,you’vedealtwithnumbersandtheirarithmeticaloperations(suchas+, −, ×, ÷). Inalgebra,weuse
symbolscalledvariables(whichareusuallyletters, suchasx, y, a, b, c, ...)torepresentnumbersandsometimes
processes.
For example, we might use the letter x to represent some number we don’t know yet, which we might need to
figure out in the course of a problem. Or we might use two letters, like x and y, to show a relationship between
two numbers without needing to know what the actual numbers are. The same letters can represent a wide range
of possible numbers, and the same letter may represent completely different numbers when used in two different
problems.
Usingvariablesoffersadvantagesoversolvingeachproblem“fromscratch.”Withvariables,wecan:
• Formulatearithmeticallawssuchasa+b=b+aforallrealnumbersaandb.
• Referto“unknown”numbers. Forinstance: findanumberxsuchthat3x+1=10.
• Write more compactly about functional relationships such as, “If you sell x tickets, then your profit will be
3x−10dollars,or“f(x)=3x−10,”where“f”istheprofitfunction,andxistheinput(i.e. howmanytickets
yousell).
Example1
Writeanalgebraicexpressionfortheperimeterandareaoftherectanglebelow.
To find the perimeter, we add the lengths of all 4 sides. We can still do this even if we don’t know the side lengths
innumbers,becausewecanusevariableslikel andwtorepresenttheunknownlengthandwidth. Ifwestartatthe
topleftandworkclockwise,andifweusetheletterPtorepresenttheperimeter,thenwecansay:
2
www.ck12.org Chapter1. EquationsandFunctions
P=l+w+l+w
Weareadding2l’sand2w’s,sowecansaythat:
P=2·l+2·w
It’scustomaryinalgebratoomitmultiplicationsymbolswheneverpossible. Forexample,11xmeansthesamething
as11·xor11×x. Wecanthereforealsowrite:
P=2l+2w
Areaislengthmultipliedbywidth. Inalgebraictermsweget:
A=l×w→A=l·w→A=lw
Note: 2l+2wbyitselfisanexampleofavariableexpression;P=2l+2wisanexampleofanequation. Themain
differencebetweenexpressionsandequationsisthepresenceofanequalssign(=).
Intheaboveexample,wefoundthesimplestpossiblewaystoexpresstheperimeterandareaofarectanglewhenwe
don’tyetknowwhatitslengthandwidthactuallyare. Now,whenweencounterarectanglewhosedimensionswedo
know, wecansimplysubstitute(orplugin)thosevaluesintheaboveequations. Inthischapter, wewillencounter
manyexpressionsthatwecanevaluatebyplugginginvaluesforthevariablesinvolved.
Evaluate Algebraic Expressions
Whenwearegivenanalgebraicexpression,oneofthemostcommonthingswemighthavetodowithitisevaluate
itforsomegivenvalueofthevariable. Thefollowingexampleillustratesthisprocess.
Example2
Letx=12. Findthevalueof2x−7.
Tofindthesolution,wesubstitute12forxinthegivenexpression. Everytimeweseex,wereplaceitwith12.
2x−7=2(12)−7
=24−7
=17
Note: Atthisstageoftheproblem,weplacethesubstitutedvalueinparentheses. Wedothistomakethewritten-out
problemeasiertofollow,andtoavoidmistakes. (Ifwedidn’tuseparenthesesandalsoforgottoaddamultiplication
sign,wewouldendupturning2xinto212insteadof2times12!)
Example3
Lety=−2.Findthevalueof 7−11y+2.
y
3
1.1. VariableExpressions www.ck12.org
Solution
7 1
−11(−2)+2=−3 +22+2
(−2) 2
1
=24−3
2
1
=20
2
Manyexpressionshavemorethanonevariableinthem. Forexample,theformulafortheperimeterofarectanglein
theintroductionhastwovariables: length(l)andwidth(w). Inthesecases,becarefultosubstitutetheappropriate
valueintheappropriateplace.
Example4
The area of a trapezoid is given by the equation A= h(a+b). Find the area of a trapezoid with bases a=10 cm
2
andb=15cmandheighth=8cm.
Tofindthesolutiontothisproblem,wesimplytakethevaluesgivenforthevariablesa, b,andh,andplugthemin
totheexpressionforA:
h
A= (a+b) Substitute10fora, 15forb, and8forh.
2
8 8
A= (10+15) Evaluatepiecebypiece.10+15=25; =4.
2 2
A=4(25)=100
Solution: Theareaofthetrapezoidis100squarecentimeters.
Evaluate Algebraic Expressions with Exponents
Many formulas and equations in mathematics contain exponents. Exponents are used as a short-hand notation for
repeatedmultiplication. Forexample:
2·2=22
2·2·2=23
4