functions and graphs

5/24/2018 Functions and Graphs

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FunctionsndGraphs

s the Earth's populationcontinues o grow, the sowaste generatedby the population grows withGovernmentmust plan for disposaland recyclingever-growingamountsof solid waste.Plannerscan use dafrom the past to predict future waste generationand plan fenough acilities for disposingof and recycling the waste.Spage151.


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64 CHAPTER FunctionsndGraphs

What you,ll learnaboutr NumericalModelsr Algebraic odelsI Craphical odelsr TheZeroFactor roperty: r Problem olvingr lrapherFailurendHiddentsehaviorr AWordAboutproof

. . . and whyNum_erical,.lgebrac,andgraphicalmodelsprovide ifferentmethodsto vis.ualize,nalyze,andunoer_srand ata.

Chapter Overview $In this chapterwe begin the study of functionsthebook.y"* pr*l;;; courses avenrroduc"",t11.T11:"ntinTheseundioni canbe"i.,,;;;;:,:_':"^":"":l

you o somebaerties anbedescribed:::*:o ":ing a graphingalculator,na,".0" ,, ;;,fi ffii ll:,ffi1',f# l"';, i","rv,rin later hapterstr",.*" exploreroperries"rfr::.1Y will se.unctlons n grea

Year MinimumHourlyWagePurchasinsPowern-2001Dollars1940

19451950I y)5196019651970197519801985199019952000

0.300.300.750.751.001.251.602.103.10

4.255 ts

3.682.88\ /.14.795.826.846.886.805.484.574.94s.34

EXAMPIE Trackinghe minimum wage;h", ?|Tr;rff?f"i"",,3,1^,T^:.:p_:ftheminimurom e40o2000.rh";';i;";#;:;1,11 -nr-um houpurchasingpower of 2oo1 t^11--^ .-^:,^, ., flmum wageadjusfeurchasing owerof zoor aolars ui;;;;:l""um waseadjustAnswer he ollowinsoresrioncroi-^^.^r_, ?un onsumerricenswerheollowinguesrions"rr", "ir, irJffi:;T"T;l:(a) In what five_year period did the actual mjmost? ouruar rrunlfllum wage increost?

sorrr",http,tt **66816)-mn /m umwage/mn wagesa s_h so

(b)In whatyeardid a workerearning hemininpurchasing ower? ursuurllmumwageenjoy heg,f:rH:lrr^ the ongestperiodduringwhich theminimumwaged(d) A worker on minimum wroc in I oa^ .-._

ANDEQUATTONorvtNc, ' , , ' , , ,

t . ,.NumericafModelsscientistsand engineers avealwaysusedmathematicsto modolld an thereby o unravel r, rnyrr"rio.mathematicartt u"tur" rhat approximur",on1-^T:]oematicalml,l,1lhq,- predictingr,.i.o"r,uuto;l;u:i:Y::i r:' thepnorogv,herocess"ra"'i'ine-"ir,#ffi

I}:::ff;HTT,Hstudyrsetf,mathematical_o?"r^ii*1""""",we w'l be concernedprimarily with three types of mathematicalhis book: numericat ry*t:,;rr;;;;;;;ilr, una raphicat modypeofmodel ivesnsighr-tlr;"1 ;;ffiems, but hebestnftengained yswitchingromoneina or?oi"r toanother. evebility to do thatwill beoneof thegoalsof thiscourse.Perhapshemostbasic indof mathematicarmoders the numerican whichnumbersor data) ur" unulyr"ii".*r"-.rr*nts intophen;,"lffi#J,T::#,H:,?Tff *,fr{::r3"s;ibconomy. lY'vr^ ur utelrelated numbers hatmeasure h


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h m

ffiMFemaleear Total

SECTION.t Modeling and Equation Solving

soLUTloil(a) In the peri


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66 CHAPTER Functions ndGraphs

EXAfinpH3 Comparing pizzasA pizzeia sellsarectarround izza12+,,aiu^lgular 18"by 24"pizza{:i,l" same ricoption'giu"i;;;",iili?; i:l,T:ffi$are rhe'u*" i,iSOLUTIOilWeneed o","*ilffi ;";;ffl"ilffitr::\:ilXff,l",:

For the rectanguTarizza:Area=lXw:Ig

For the circular pizza:X 24 : 432 squareinches.

Area: "' : '(T)' - ,ooo- 452.4squa.reincheThe round pizza is larger and thereforegives more fbr the moneyThe algebraicmodels n Example3 comefrom pably encounteredalgebraicmodels frn- *^-.. ::,ot"tty' but you hand scienceaourrar.l?f,:f":':"":,:"rs"b;;;;;;#"#T3,n you

The ability to generatenumbers from formula, -;ffi:',,".':#:trffHJi*#l*ffiTff-"1#iis o fir anargebraicnoo"r o""r;l;;;;;";:,Y:l

moderinghen:dll)'3 ana)yzehy t works.Notail modets 1*T moreoptipredictions. used o make

z5 .ott hemarkedrice.re ;;;";;;il;;'rvmng tsdiscoilj#:T statearesaxf6.5vo,;J;;###,*#l;::;:::i: The discountprice d is related::;;:::;:ffi ;;":1"::il1",::#,ffi^fiif{ol,rh' Theactuar are rices is reratedo thediscount riced by the,Yil;f, *,r,r,*n"r" , u"on.tunt;j;;;. rheotarar

{i U-singheanswersror1,r,",,?r""#"#:,,;',T;?:,:;Jlffii""":nada.con,.,r_.^_- , Pm.What is p?il,Tl:t onlvhave 30, an ouaffordobuyashirtmarked36.. ;iil;:ffi:;:;:*.:*,.0:,:1:,0",:*ned tospendomoreh100, what is the mar usrcrrrunsdto spendno more thJ.r; ;;;;;;:il1ilHh':'9,Jffi":l yourarkedu."r


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For accpredion'For r

[-1,18] y -8,s61Flcunt 1.1 A scatter lotof the data roma Galileo ravity xperiment.Example)

SECTION .1 Modeling and Equation Solving

If numerical data do behave easonably enoughto suggest hat an algmodel rhight be found, it is often helpful to look at apicture first. That us to graphicalmodels.

GraphicalModelsA graphical model is a visible representationof a numerical modealgebraic model that gives insight into the relationships between vaquantities.Learning to interpret and usegraphs s a major goal of this

EXAMPLE VisualizingGalileo's ravityexPerimenGalileo Galilei (1564-1642) spent a good deal of time rolling ballsinclined planescarefully recording the distance hey traveled as a fuof elapsed ime. His experimentsare commonly repeated n physicsctoday,so it is easy o reproduce a typical table of Galilean data.

Elapsed ime (seconds) 0 I 2 J 4 5 6 '7Distance traveled (inches) 0 0.75 3 6.75 12 18.75 27 36.

What graphicalmodel fits the data? Can you find an algebraicmodel thSOIUTIOIU A scatterplot of the data s shown n Figure 1.1.Galileo's experiencewith quadratic unctions suggestedo him that thure was a parabolawith its vertex at the origin, he therefore modeeffect of gravity as a quadraticfunction:

d: ktz.Because he orderedpair (1, 0.75) must satisfy the equation, t followk: 0.75,yielding the equation

d: 0.75t2You can verify numerically that this algebraicmodel correctly predrest of the datapoints. We will have much more to say aboutChapter 2. l

This insight led Galileo to discover severalbasic laws of motion thateventually be named after Isaac Newton. While Galileo had found thbraic model to describe he path of the ball, it would take Newton's cto explain why it worked.

EXAMPLE Fittinga curve o dataWe showed n Example 2lhat thepercentageof females n the U.S.population has been steadily growing over the years. Model thisgraphically and use the graphical model to suggestan algebraicmo


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o6 C||APTERr FunctionsndGraphs

t-s,2slby 0,8lFtcunr 1.2 A scatter lotofthe datainTable .4.Example)

Flcune 1.3 The inewith equationy : 0.145x 3.9 sa good model orthe data n Table .4. Example )

SO|.UTfON Let t be the number of years after 19g0, and let Fcentageof females n theprison population from year 0 to year 2data n Table1.3we ger he conesponOing ;in Table1.4:ffitrt:

j:.r.ltli{

206.7Soarce: ./.5.,/utticeDepartment

A scatter lot of the data s shown n Figure 1.2.This pattern looks linear. If we use a line as our graphical modfind an algebraic model by finding tt " "quution of the linedescribe n chapter 2how a statistician rouid fina the best linedata, but we can get a pretty good fit for now by finding the linthe points (0, 3.8) and,20,6.j).The slope is (6.7 - 3.8)/(20 - 0) : 0.145 and the y_intercTherefore,he ine hasequation : 0.145x* 3.g.you ; ,"" i;;1.3 that this line doesa very nice ob of modeling the data.

:^t^TTl1]".4,1T0,,.,T:r" rom u*orpr,yri",n", "",,.tulli.og bje*s, whichshouldnspire o." ronfid"n;;;ffi""]I#::1..1,*^::lLrl r y" canepeat;i,dk;"perimentmany" ln ot*::::tty-slopedamps,ifferent""i" Ji*ffi ;;r;T*';l:1".::tlt:i:T 'T*:, and quadraticodet ill it t "u",vpu?9:e of this Exploration is to think more deeply about thmodel in the prison example.The linear model we found will not continue to predict the pe:"nr1C",:f Pm^aleprisoners n the U.S. indefinitety. Wtry _urieventually fail?Do you think that our linear model will give an accurateestimof the percentageof female prisonersn ifr" U.S. in 2009?Whwhy not?The linear model is sucha good fit thdt it actually calls our att1t::^,"^*.. riu.sual jump-in ihe percentag"oi f"_ur" prisoners1990.Statisticianswould look for ro_""unrruul ,.confoundinfactol 11 1990 that might explain the ump. Slhat sort of factoryou think might explain it?

Does Thble J.1 suggesta possible factor thatmight influencefemale crime statistics?

i 1,.,i+.

t-5,2s1 y 0,8l


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Pnenrqursrreneprenln the Prerequisite hapterwe definedsolutionof an equation, olvingan equa-tion,x-intercept, ndgraphof an equationin x andy.

SICTIONt.l Modeling and Equation Solving

There are other ways of graphing numerical data that are particularlyfor statistical studies.We will treat some of them in Chapter 9. The splot will be our choice of data graph for the time being, as it providclosestconnection o graphsoffunctions in the Cartesianplane.

TheZeroFactor ropertyThe main reason for studying algebra through the ageshas been toequations.We develop algebraicmodels for phenomenaso that we canproblems, and the solution to the problems usually come down to fisolutions f algebraic quations.If we are fortunate enough o be solving an equation n a single variabmight proceedas n the following example.

ffi rmr*PLE SolvingnequationlgebraicatlFind all real numbersx for which 6x3 : llxz * 10-t.SOLUTIONWe begin by changingthe form of the equation o6x3- l rxz-10x:0.We can then solve this equationalgebraically by factoring:

6x3- l lxz-10x:0x(6x2-l1x-10):ax(2x 5)(3x 2) :0

x:0 or 2x-5:0 orx:0 or

In Example 6, we used he important ZeroFactor Property of real num

It is this property that algebra students use to solve equations n whexpression s set equal to zero.Modern problem solvers are fortunate toan alternativeway to find suchsolutions.If we graph the expression, hen the x-intercepts of the graph of the esion will be the valuesfor which the expressionequals 0.

5x:t or3x 2:0

2A--li$ ffidq-@*


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7o (h{APfKR1 Functions nd Graphs

t-8,6lby -20,201Flcunr 1.4 The raph f| : x2+ 4x -'tO. Example)

SorvlrueEqu*rrerusmnrnT:eHffctocyExample showsone methodof solving nequationwith technology. omegrapherscouldalsosolve he equation n Exampleby finding the intersection fthe graphsofY : xz andY :10 - 4x.Somegraphershavebuilt- inequation olvers. achmethodhas ts advantages nd disadvantages ,utwe recommendhe "f inding he x-inter-cepts"techniqueor now becauset mostclosely aral le lshe classicallgebraicech-niques or f inding rootsof equations, ndmakes he connection etween he alge-braic nd graphicalmodels asiero fol lowand appreciate.

it; ffiK&ffiPltr Solvingan equation: omparingmethSolve he equation 2 : l0 - 4x.sotufloIrISolveAlgebraicallyThe given equation s equivalent o xz + 4x - l0:0.This quadratic equation has irrational solutions that can be fouquadraticormula' -4 + \,16 i nx: 2 : l '1416574and -4 - f l6 + 40 - -5.1416514while the decimal answersare certainly accurateenough for aipulposes, t is important to note that only the expressions ound bdratic formula give the exact teal number answers.The tidinesanswers is a worthy mathematical goal. Realistically, howeanswersare often impossible to obtain, even with the most sopmathematicalools.SolveGraphieallyWe first find an equivalent equationx2 + 4x - 10:0. We then graph theshown n Fisure 1.4.We then use the grapherto locatethe x-interceptsof the graph:

x : 1.7 165'14nd - -5.1 41651..NPwt4;Exel'We used the graphing utility of the calculator to solve graphExample 7. Most calculatorsalso have solvers hat would enableusnumerically for the samedecimal approximations without considgraph. Some calculators have computer algebra systems that wnumerically to produce exact answers n certain cases. n this boodistinguish between these two technological methods and the tpencil-and-papermethodsusedto solve algebraically.Every method of solving an equationusually comesdown to findian expressionequals zero. Ifwe use/(x) to denotean algebraicexpthe variablex, the connectionsare as follows.

with 0 on the right-hequat ion :xz *4x-

Fundarnentalonnection:i"ffiil', ---{fJ;1il:"t"s

the quation(x) 0' thenhei r. fh" numbei is a root (or solution)of the equation(x)zero fy:f(n.


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S#{Yl#ru' ,1 Modelingand EquationSolving

ProblemSolvingGeorgeP6lya(1887-l985) is sometimes alled he fatherof modernprsolving, not only because e was good at it (as he certainly was) bubecausehe published the most famous analysis of the problem-sprocess:How to Solve t: A NewAspectof MathematicalMethod. Hissteps"arewell known to mostmathematicians:

I.2.,.4.

Understand he problemDevise a plan.Carry out the plan.I-ook back.

The problem-solvingprocess hat we recommendyou use throughocoursewill be the following versionof P6lya's our steps.

Step -Understand the problem.. Read the problem as stated,several imes if necessary.. Be sureyou understandhe meaningof each erm used.. Restate he problem n your own words.Discuss he problem with

othersfyou can.. Identify clearly the information that you need to solve the problem. Find the information you need from the given data.Step2-Develop a mathemati cal model of the problem.. Draw a picture o visualize he problemsituation. t usuallyhelps. Introduce a variable to represent he quantity you seek.(In somec

theremay be more hanone.). Use the statementof the problem to find an equationor inequality

relates he variablesyou seek o quantities that you know.Step 3-Solve the mathematical model and support or confirmsolution.. Solve algebraically using traditional algebraicmethodsand suppgraphicatly or support numerically using a graphing utility.. Solve graphically or numerically using a graphing utility andconfirm algebraically using traditional algebraicmethods.. Solve graphically or numerically because here is no other way

possible.


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72 {HAPTER FunctionsndGraphs

t$ EXArnp$ Applying he problem-solvingrocej 3.:,iCireers at anauromanufactureraystudenrs0.0gperm. perday oroad est heirnewvehicles.I

$ (:) Y"r muchdidtheautomanufacrureraysally rodrive440mI dav?lifi (b)Johnearned$93 test-driving a new car in one day.How far di$ SotuTtolu: 1",1.f I lt.,:.". of a caror of Sallyor Johnwouldnorbehel; go olrecily^to^designinghemodel.BothJohnandSallyearned; ouy'prusgLr.08ermile.Murtiplydollars/mile y miles o gerd$ so ifz representshepay or driving miles n oneday,ouralgeb*is$ p :25 * 0.08x.

Solve lgebraically(a)TogetSally'spaywe etx : 440andsolveor p:p:25+0.08(440)

:60.20(b)TogetJohn'smileagewe et p : 93 andsolve br _r:

93 : 25 -F0.08r68 : 0.08x

680.08x: 850

SupportGraphicallyb)Flcuns 1.5 Craphicalupportor healgebraicolutionsnExample.

Figure 1.5a shows hat the point (440,60.20) is on the gy-

25.* 0.08x,supporring ur answero (a).Figure .5b showpoint (850,93) is on the graphof y : 25 + 0.0g;, supporring uto (b)' (we couldalsohavesupportedouranswernumericanybsubstitutingn for each andconfrmingthevalueof p.)

It is not really necess ry to show written supportas part of an algebrtion, but it is good practice to support unrr"r, whereverpossib"lereducethe chance or error. we wih often show written ,oppo.t of otions in this book in order to highlight the connection, u-orrg the algraphical, and numeripal models.

tsai:ia.:

t0,eaOl y 0,1s0l

t0,9a0l y 0, s0l


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Trcnnolocv NorrCneway o get he able n Figure .6bstouse he'Ask"featurefyourgraphingcalculatorndenter ach value eparately.

SE{T|ON X,I Modeling and Equation Solving

Grapher ailure nd HiddenBehaviorWhile the graphsproduced by computersand graphing calculators are woful tools for understandingalgebraic modelsand their behavior, t is importkeep in mind that machineshave limitations. Occasionally hey can prographical models that misrepresent he phenomenawe wish to study, a prowe call grapher failure. Sometimes he viewing window will be tooobscuringdetailsof thegraphwhich we call hidden behavior. We will gexampleof each ust to illustrate what can happen,but rest assured hat thesficulties rarely occur with graphical modelsthat arise from real-world prob

HXS.lKpi.ffiSeeing rapher ailureLook at the graph of y : 3/(2x - 5) on a graphing calculator. Is thex-intercept?SOtUnOil A graph s shown n Figure 1.6a.

X il2U?.t{9p.qgs,?.5 'e.581i-:r/ ,o

.1g.. ', , i . .r .-1501'.-*i580,.;,.,EBBtrH,,.:150O1,:,rl5o:, ,,15: , : . , : iYE 3/(ax-sl

t-3,6l by -3,3l(a) (b)

Ftcu*r I.6 (a)A graphwith a mysterious-intercept.b)Asx approachethevalue f 3/(2x 5)approacheso. (Example)The graph seems o show an x-interceptabout halfway between2 andconfirm this algebraically,we would sety : 0 and solve for x:

o: 32x-50(2x-s):3

0:3The statement0 : 3 is false for all x, so there can be no value that my : 0, and hence herecan be no x-intercept or the graph.What went wThe answer s a simple form of grapher ailure: The vertical line shoube there As suggestedby the table in Figure 1.6b, the actual gra, : l/(2x - 5) approaches oo to the left of x : 2.5, and comesdownf oo o the right of x : 2.5 (moreon this later). The expression3/(2x -undefrnedat x : 2.5, but the graph n Figure 1.6a does not reflect thigrapherplots points at regular increments rom left to right, connectpoints as t goes. t hits some ow point off the screen o the left of 2.lowed immediatelyby somehigh point off the screen o the right of 2.it connects hem with that unwanted line. Now try Exercise

\


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i'1

a,a ' 1III

:;a:'::

74 {}lAPTfRl FunctionsndGraphs

[-10. 0] by -10,10](a )

[-10,10] y -500, 001(b)Flcunr 1.7 Thegraph f : x3- 1.1x2 65.4x 229.5n twoviewing indows.Example0)

[4.95,.15] y -0.1, .11Ftcunr ,8 A closerook t hegraphofy : rt - 1.1x2 65.4x 229.5.(Example0)

mX&ffdtpl-f0 Not seeinghiddenbehaviorSolvegraphically: 3 - l.lx2 - 65.4x+ 229.5 0.SOtUTlOtf Figure 1.7a shows the graph in the standard -10[- 10, l0] window, an inadequate choice because tof the graph is off the screen. Our horizontal dimensions loowe adjust our vertical dimensions o [-500, 500], yielding thFigure 1.7b.We use the grapher to locate an x-intercept near -9 (whichbe -9) and then an x-intercept near 5 (which we find to be 5).leads us to believe that we are done. However, if we zoom inobserve he behaviorneatx:5, the graph ells a new story (FigIn this graph we see hat there are actually two x-interceptsneawe find to be 5 and 5.1). There are therefore hree roots (or zeequation 3 - I . Ixz - 65.4x+ 229.5 0: r : -9, x :5, and

:isYou might wonder if there could be strll more hiddenx-intercepts10 We will learn in Chapter 2 how the Fundamental Theoremguaranteeshat there are not.

AWord About ProofWhile Example 10 is still fresh on our minds, let us point out a very important, considerationabout our solution.We solvedgraphically to find two solutions, then eventually threeto the given equation.Although we did not show the steps, t is eafirm numerically that the three numbers ound are actually solutiostituting them into the equation. But the problem asked us tsolutions. While we could explore that equation graphically in more viewing windows and never find another solution, our failuthem would notprove that they are not out there somewhere.That FundamentalTheorem of Algebra is so mportant. It tells us that that most three real solutions to any cubic equation,so we know forthereare no more.Exploration is encouraged hroughoutthis book because t is howical progress s made. Mathematiciansare neversatisfied,howeverhave proved their results.We will show you proofs in later chaptwill ask you to produceproofsoccasionally n the exercises. hatime for you to set the technologyaside,get out a pencil, and showcal sequenceof algebraic steps hat something s undeniably and utrue. This process s called deductive reasoning.

,J:iiti;):t

alirj?ir:tiltf,:lrit*ii.i:


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OUICKREVIEW.1Fa;tor the following expressionscompletely over the realnumtrers.llL r: - 16n h.:-4

2.x2+l}x+254.3x3-15x2+l8x

SECTION.1 Modeling nd EquationSolving

EXAMPTEl Proving peculiar umber actProve that 6 id a factor of n3 - n for everypositive integern.SOIUTIOI{ You can explore this expression or various values of n ocalculator.Table 1.5 shows t for the frst 12 values of n.

nln3-n 0

56t20 210

78336 504

l0 11 12990 13201716

2346 24 60

9720

All of thesenumbersare divisible by 6, but that doesnot prove that thcontinue to be divisible by 6 for all values of n. In fact, a table withlion values, all divisible by 6, would not constitutea proof. Here is aLet nbe any positive integer.. We can factor n3 - n as the product of three numbers:(n-r)(n)(n+r) .. The factorization shows hat n3 - n is always the product of three

secutive ntegers.. Every set of three consecutive ntegers must contain a multiple of. Since 3 divides a factor of 13 - n, it follows that 3 is a factor of

n3 - nitself.. Every set of threeconsecutive ntegers must contain a multiple of. Since 2 divides a factor of n3 - n, it follows that 2 is a factor ofn3 - n itself.. Sinceboth 2 and 3 are factorsof n3 - n, we know that 6 is a fact

n3-n.End of proof

(For help, go to SectionA.2.)

tq{fftr$-'Sfub?

5. l6h4 8lt .x2+3x-49. 2x2 11x+ 5

6.x2+2xh*h28.x2-3x+4

10.x4+x2-20


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In Exercises1-10, match the numerical model to the corre_sponding graphical model (a-j) and algebraicmodel (ft_r).1.

3.

Ia,4olby t-1,71(c )

l-1,6lby -2(b)

t-s,aOl y t-10,(j )( t ly:40 - x2bly : rG - z(p)y:3x-2(r)y=*2+2

r- ?( r ly : ;

I- I ,7lby Ia,40l(el

(i)(k)v: x2+ x(m)y:(x+l)(" r-1)(o)y:100-2x

'(q)v : 2x(slY:2x+3

x 3 5 7 9 t2 15v 6 10 14 l8 24 300 I 2 4 5v 2 3 6 l1 18 27 l-2, r4lby 4,36]2 4 6 8 10 12v A 10 l6 22 28 345 10 l5 20 25 30v 90 80 70 60 50 40 t-3, l9lby l-2I 2 -1 A 6v 39 36 31 )A 15 AI 2 3 A 5 ov 5 7 o l t l- l l. )

t 5 7 9 11 13 15v I z 3 4 5 6 t-r,7lby 14,4 8 t2 t4 l8 24v 20 72 156 210 342 600

9.

10.

A 5 6 7 8v 8 15 .A 35 48 63t-l, 161by-l, el t-s,301 y -5,4 7 t2 t9 28 39v I 2 -l 4 5 6

t-3,9l bvt-2,601


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Exercises11-18 refer to the data n Table 1'6 below showing thepercentageof the female and male populations in the United 'Statesemployed in the civilian work force in certain years'Throughout,measure ime in years from 1955'

Year Women (7o) Men (Vo)

SECTIOIf.l Modeling and EquationSolving

re. Doing Arithmetic with Lists Enter the datafrom the"Totat' column of Table 1.2 of Example 2 into list L1your calculator. F,nterthe data from the "Female" coluinto list L2. Check a few computations to seethat thecedures in (a) and (b) cause the calculator to divide eelement of L2 by the corresponding entry in L1, multiby 100, and store the resulting list of percentagesn L(a)On the home screen,enter the commandloo x L, / L1--> L3.(b)Go to the top of list L3 and enterL: : 100(LzlT-r

20. ComParing Cakes A bakery sells a 9" by 13" cake fosameprice as an 8" diameter round cake. If the roundis twice the height of the rectangular cake, which optgives the most cake for the moneY?

zt. Stepping Stones A garden shop sells 12" by 12" sqstepping stones or the sameprice as 13" round stonall of the steppingstonesare the same hickness,whioption gives the most rock for the money?

22. Free Fall of a Smoke Bomb At the Oshkosh,WI, aJake Trouper drops a smoke bomb to signal the offrcbeginning ofthe show. gnoring air resistance,an obfree fall will fall dfeetin t seconds'where d and t ared by the algebraicmodeld : 16t2.(a)How long will it take the bomb to fa-ll 180 feet?(b) If the smoke bomb is in free fall for 12'5 secondsis dropped, how high was the airplane when the smobomb was dropped?

23. Physics Equipment A physics studentobtainsthe foing data involving a ball rolling down an inclined plawhere / is the elapsedtime in secondsand y is the dtraveled in inches.

Find an algebraicmodel hat fits thedata'U.S.AirTravelThenumber frevenue assengerin the U.S.over he L$-year eriod rom 1987 o 20shown n the tablebelow.

195519601965t970r97s198019851990r9952000

35.737.739.3.+5.J46.351.5J4.)f / .J58.960.2

85.483.380.779.777.977.476.376.475.074.7

12.

Source:J.5. ureau flabor Statistics.(a)According to the numerical model, what has been thetrend in females joining the work force over the years since1955?

(b) In what 5-year interval did the percentage of women whowere emploYed change the most?(a)According to the numerical model, what has been thetrend in males oining the work force since 1955?(b) In what s-year interval did the percentage of men whowere emploYedchange the most?

13. Model the data graphically with two scatter plots on thesamegraph, one showing the percentageof women"mpto "0 as a function of time and the other showing thesame or men. Measure ime in years since 1955'

14. Are the male percentages falling faster than the female per-centagesare rising, or vice versa?

15. Model the data algebraically with linear equations of theform y : mx * b'Write one equationfor the women's dataand another equation for the men's data' Use the 1955 and1995datato computethe sloPes'

16. If the percentages ontinueto follow the linear modelsyoufound in Exercise.15,what will the employmentpercent-agesfor women and for men be in the year 2005?If the percentages continue to follow the linear models youfoundln Exercise 15,when will the percentages f womenand men in the civilian work force be the same?What per-centagewill that be?Writing to Learn Explain why the percentages annot con-tinue indefinitely to follow the linear models that you wrotein Exercise 15.

24.

Year Passengers(millions) Year Passe(milli17. 1987

198819891990t99lt992t993

44't;7454.6453.7465.6452.3475.1488.5

1994t995t99619971998t9992000

52545859616366

0 z J 4 5v 0 t .2 4.8 10.8 t9.2 3

t8.

Source:Air TransportAsSocEtion.


16/18

78 CHAPTER FunctionsndGraphs

(a)Grapha scatter lot of thedata.Let.r be henumber fyears ince1997.9 t:1"] thedataalgebraicallywirh theequationp : t.t3* r 3.lx * 4 3,.wherer . tt. iur.i". "r pu*sengersn millions,and_r-ishenumber f yearsafter lSZ.Superimposehegraphof themodel", ,rr",r"lu". pr"r.(:lI:"_*rt to thealgebraicmodel,whenwill thenumberot passengerseach900million?(d)Do you think this algebraicmodelwilr still bevalid intheyear2007?rVhy iwhy not?

Exercises 25-28refer to the graphbelow,which shows he minimum

In Exercises29-3g, solve the equationalgebraicagraphically.29.v2-5:g-2v2 30. (x + 11)2: t

74.x(2x - 1) : tO

rr. 2x2- 5x * Z : (x _ 3)(x _ 2) + 3x?2. 2-n-] :orr. x(2x 5) = 12

salaries in major league baseballa recent l8-year period andtne averagesalaries n major leaguebasebatlover the sameperiod.Salaries re measuredn dollarsandtime is measured after the startinsyear (year 0).

75.x(x + 7) = 1496.x2-3x*4:2x2_7x_g37. * t-zf i iZ :o 38.,G* x:7In Exercises 9-46,solve he.1y1ationSraphicallyit to anequivalentequationwith 0 on ,h" fi;;;;othen inding the r-intercepts.?e.2x - 5: \G+ 4 ao. 3x_ 2l :2 t ;41. 2x -51=a -b_31 42. / ;T i :6_2.Vi_43.2x-3:x3 -5 44. x1_l :x3 _2x45. x + 1)-t : _r-1 * q6.x2 lxl47. SwanAutoRental harges 32perdayplus$0.1for anautomobileental.(a)Elaine ented car or onedayandshedroveHowmuchdid shepay?

(b)Ramon aid$69.g0o renta car or oneday.Hhedrive?48' connectingGraphs nd EquationsThecurvesgraphbeloware hegraphsof rh" rhr;;;;r-giu"ny1:4x*5

lz:x3 I2x2-x_t3ls:-x3-2x2+5x1_2.

(a)Write an equation that can be solved to find the pintersectionof the graphsof y1 andy2.

1,400,0@1,260,000t, l980,000840,000700,000560,000420.000280.000

t234soz--sl tz tz t+ ts rc i-tSource: alor eaguegasebal/layercArsociation-25. Which line is which, and how do you know?26. After peter Ueberroth,s esignation as baseballcommis_sioner n lggg and his zuccessor,sr,r."f, 0."* in 19g9,rhe eam wnersroke."eorp.euiourLrt"rilr. *oflsan aneraofcompedtivepenaing;;j;;"; salaries.Identiffwherehe19exprainowyou :r'":?ffi: appearn theeranh\27. The ownersartempted o halt the uncontrolled sp"naing b\fllp:1C a salary cap,-whichprompted " p;y;,strike irlree4.rhe strike caused he rsbs r.;;;;;J#;ortenedand lefr many fans angry. Identify ,il ;" ;;;5 salariesappear in the graph and explain horv you c.* ,po, ,fr"*.ZS. Writing to learn Analyzethe generalputt"_, io th"graphical model and give your ttrougtrts"i""i *fr",rr"long-term implication-s might be for(a) the players;(b) the team owners;(c) he baseball ans.


17/18

49.

(b) Write an equation that can be solved to find ther-intercepts of the graph of y3.(c)Writing to Learn How does the graphicalmodel reflectthe fact that the answers to (a) and (b) are equivalentalgebraically?

{d) Confirm numerically that the x-intercepts of y3 give thesame valueswhen substituted nto the expressions or y1and y2.Exploring Grapher Failure Let y : (x200)t/2w.(a)Explain algebraically why y : x for all x = 0.(b)Graph the equationy : (x20o1t/2o0n the window [0, 1]by [0, l] .(c)Is the graph different from the graph of y : x?(d) Can you explain why the grapher failed?Connecting Algebra and Geometry Explain how the alge-braic equation (x + b)2 : xz + zbx * b2models the areasofthe regions in the geometric figure shown below on the left:xbL

(Ex.50) (Ex.52)Exploring Hidden Behavior Solving graphically, find allreal solutions to the following equations.Watch out forhidden behavior.(a)y : tor: + 7.5x2 54.85x+ 37.95(b)y : x3 + x2 - 4.99x+ 3.03Connecting Algebra and Geometry The geometric figureshown on the right above s a large squarewith a smallsquaremissing.(a)Find the area of the figure.(b)What areamust be added to complete the large square?(c)Explain how the algebraic formula for completing thesquaremodels the completing of the square n (b).Proving a Theorem Provethat if n is a positive integer,thenn2 -t 2n is either odd or a multiple of 4. Compare yourproof with those of vour classmates.

SECTfON.1 Modeling nd Equation olving

54. Writing to Learn The graph below shows he distanhome against time for a jogger. Using information frograph,write a paragraphdescribingthejogger's workv

StandardizedestQuestions55. Ttue or False A product of real numbers s zero if anonly if every factor in the product is zero. Justify youanswer.56. True or False An algebraicmodel can always be usemakeaccurate redictions.In Exercises57-60, you may use a graphing calculator towhich algebraic model corresponds to the given graphicalnumerical model.

(blv=*+s(d lY:4x+z

MultipleChoice

(a)y=2x+3(cly= 12 zx(")v: ,4_"

5?.MultiPlechoice

'|.

52.t0,6lby -9, sl

t0,9 lby 0,6]MultipleChoice

I 2 J A 5 6v 6 9 t4 21 30 4lMultipleChoice

0 2 4 6 8 10v 3 7 l l 15 19 23


18/18

8o CHAPTER FunctionsndGraphs

Explorations61. Analyzing the Market Both Ahmad and LaToya watchthe stock market throughout the year for stocks that makesignificant jumps from one month to another. When theyspot one, eachbuys 100 shares.Ahmad's rule is to sell thestock if it fails to perform well for three months in a row.LaToya's rule is to sell in December f the stock has failedto perform well since ts purchase.

The graph below shows the monthly performance in dollars(Jan-Dec) of a stock rhat both Ahmad and LaToya havebeen watching.

t : :

(a)Both Ahmad and LaToya bought the stock early in theyear. n which month?(b)At approximarely what price did they buy the stock?(c)When did Ahmad sell the srock?(d)How much did Ahmad lose on the stock?(e)Writing to learn Explain why LaToya,s strategywasbetter than Ahmad's for this particular stock n this particu_lar year.(f) Sketch a l2-month graph of a stock's performance hatwould favor Ahmad's strategyover LaToya's.

62. croup Activity Creating Hidden BehaviorYou can create your own graphswith hidden behavior.Working in groups of two or three, try this exploration.(a)Graph the equarion y : (x + 2)(*2 - 4x + 4) in the win_dow [-4, a] by [-10, l0].(b) Confirm algebraically that this function has zeros only atx: -2and.x:2.(c)Graph the equation y = (x + 2)(x2 - 4x * 4.01) in thewindow -4,4) by [-10, t0](d) Confirm algebraically that this function has only onezero, at x : -2. (Use the discriminant.)(e)Graph the equation (x + 2)(x2 - 4x * 3.99) in the win_dow -4, a]by t-10, 101.(f) Confirm algebraically that this function has three zeros.(Use the discriminant.)

Extendinghe ldeas63. TheProliferationof Cell PhonesTable1.8shownumber f cellularphone ubscribersn theU.S.aaverageocalmonthlybill in theyearsrom 1988

Year Subscribers Averase Loclmillions) Monthly Bil1988198919901991r9921993r994r995199619971998199920002001

1.62.74.46.48.913.1

19.328.238.248.760.876.397.0118.4

95.0085.5283.9474.5668.5167.3158.6552.4548.8443.8639.8840.2445.1545.56

5ourrc:Ce//u/a/ elecoman cattbni& lnternetAss(a) Graph the scatter plots of the number of subscrithe average ocal monthly bill as functions of time,time I : the number of years after 1988.(b) One of the scatterplots clearly suggestsa quadrmodel with formula y : axz + &. Use the point at solve for &; then use the point at t = 9 to solve for(c) Superimpose the graph of the quadratic model oscatterplot. Does the fit appear o be good?(d)The other scatterplot can be roughly approximaa linear model in the form y : mx + &. Use the po/ : I and t : 72 to find a linear model.(e)Superimpose he graph of the linear model ontoter plot. Does the fit appear o be good?(f) Do you think that a quadratic model might be bethe linear model? Explain.(g) If your calculator doesquadratic regression,usequadratic curve to the linearlooking data. Superimpgraph onto the scatter plot. Does the fit appear to bthan the line?

Group Activity (Continuation of Exercise63) Disceconomic forces suggestedby the two models in Ex63 and speculate about the future by analyzing the64.

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