Lesson 15: Structure in Graphs of Polynomial...
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Lesson 15 M1 ALGEBRA II Lesson 15: Structure in Graphs of Polynomial Functions Student Outcomes § Students graph polynomial functions and describe end behavior based upon the degree of the polynomial. Lesson Notes So far in this module, students have practiced factoring polynomials using several techniques and examined how they can use the factored form of the polynomial to identify interesting characteristics of the graphs of these functions. In this lesson, students continue exploring graphs of polynomial functions in order to identify how the degree of the polynomial influences the end behavior of these graphs. They also discuss how to identify -intercepts of the graphs of polynomial functions and are given an opportunity to construct viable arguments and critique the reasoning of others in the Opening Exercise (MP.3). Opening Exercise (8 minutes) Opening Exercise Sketch the graph of = . What will the graph of = look like? Sketch it on the same coordinate plane. What will the graph of = look like? Have students recall and sketch the graph of = , . Discuss the characteristics of the graph, where the -intercept is, and why the graph stays above the -axis on either side of the -intercept. In pairs or in groups, have them discuss or write what they think the graph of = . will look like and how they think it compares to the graph of = , . Once they do so, they should sketch their idea of the graph of on top of the graph of . Discuss with students what they have sketched, and emphasize the similarities between the two graphs. Since = , () will increase faster as increases than does. Both graphs pass through , . The basic shapes are the same, but near the origin the graph of is flatter than the graph of . Finally, in pairs or in groups, have students discuss or write what they think the graph of ℎ = 5 will look like and how they think it will compare to graphs of and . Once they do so, students should sketch on the same graph the previous two graphs. Again, discuss graphs with students, and emphasize the similarities between graphs. Since = ⋅ ⋅ , the graph of again passes through the origin. Since we are squaring and multiplying by squares, the graph of should look about the same as the graphs of and but increase even faster and be even flatter near the origin. 150
Transcript of Lesson 15: Structure in Graphs of Polynomial...
Lesson15
M1ALGEBRAII
Lesson15:StructureinGraphsofPolynomialFunctions
StudentOutcomes
§ Studentsgraphpolynomialfunctionsanddescribeendbehaviorbaseduponthedegreeofthepolynomial.
LessonNotesSofarinthismodule,studentshavepracticedfactoringpolynomialsusingseveraltechniquesandexaminedhowtheycanusethefactoredformofthepolynomialtoidentifyinterestingcharacteristicsofthegraphsofthesefunctions.Inthislesson,studentscontinueexploringgraphsofpolynomialfunctionsinordertoidentifyhowthedegreeofthepolynomialinfluencestheendbehaviorofthesegraphs.Theyalsodiscusshowtoidentify𝑦-interceptsofthegraphsofpolynomialfunctionsandaregivenanopportunitytoconstructviableargumentsandcritiquethereasoningofothersintheOpeningExercise(MP.3).
OpeningExercise(8minutes)OpeningExercise
Sketchthegraphof𝒇 𝒙 = 𝒙𝟐.Whatwillthegraphof𝒈 𝒙 = 𝒙𝟒looklike?Sketchitonthesamecoordinateplane.Whatwillthegraphof𝒉 𝒙 = 𝒙𝟔looklike?
Havestudentsrecallandsketchthegraphof𝑓 𝑥 = 𝑥,.Discussthecharacteristicsofthegraph,wherethe𝑥-interceptis,andwhythegraphstaysabovethe𝑥-axisoneithersideofthe𝑥-intercept.
Inpairsoringroups,havethemdiscussorwritewhattheythinkthegraphof𝑔 𝑥 = 𝑥.willlooklikeandhowtheythinkitcomparestothegraphof𝑓 𝑥 = 𝑥,.Oncetheydoso,theyshouldsketchtheirideaofthegraphof𝑔ontopofthegraphof𝑓.Discusswithstudentswhattheyhavesketched,andemphasizethesimilaritiesbetweenthetwographs.
Since𝒈 𝒙 = 𝒙𝟐 𝟐,𝒈(𝒙)willincreasefasteras𝒙increasesthan𝒇 𝒙 does.Bothgraphspassthrough 𝟎, 𝟎 .Thebasicshapesarethesame,butneartheoriginthegraphof𝒈isflatterthanthegraphof𝒇.
Finally,inpairsoringroups,havestudentsdiscussorwritewhattheythinkthegraphofℎ 𝑥 = 𝑥5willlooklikeandhowtheythinkitwillcomparetographsof𝑓and𝑔.Oncetheydoso,studentsshouldsketchonthesamegraphtheprevioustwographs.Again,discussgraphswithstudents,andemphasizethesimilaritiesbetweengraphs.
Since𝒉 𝒙 = 𝒙𝟐 ⋅ 𝒙𝟐 ⋅ 𝒙𝟐,thegraphof𝒉againpassesthroughtheorigin.Sincewearesquaringandmultiplyingbysquares,thegraphof𝒉shouldlookaboutthesameasthegraphsof𝒇and𝒈butincreaseevenfasterandbeevenflatterneartheorigin.
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§ Usingagraphingutility,havestudentsgraphallthreefunctionssimultaneouslytoconfirmtheirsketches.
Discussion(5minutes)
UsethegraphsfromtheOpeningExercisetoframethefollowingdiscussionaboutendbehavior.
Askstudentstocompareanddescribethebehaviorofthevalue𝑓(𝑥)astheabsolutevalueof𝑥increaseswithoutbound.Introducethetermendbehaviorasawaytotalkaboutthefunctionandwhathappenstoitsgraphbeyondtheboundedregionofthecoordinateplanethatisdrawnonpaper.Thatis,theendbehaviorisawaytodescribewhathappenstothefunctionas𝑥approachespositiveandnegativeinfinitywithouthavingtodrawthegraph.
Notetoteacher:Itisimportanttonotethatendbehaviorcannotbegivenaprecisemathematicaldefinitionuntiltheconceptofalimitisintroducedincalculus.Togetaroundthisdifficulty,mosthighschooltextbooksdrawpicturesandstatethingslike,“As𝑥 → ∞,𝑓 𝑥 → ∞.”Wedothisalso,butitisimportanttocarefullydescribetostudentsthemeaningofthephrase,“As𝑥approachespositiveinfinity,”beforeusingthephrase(oritssymbolversion)todescribeendbehavior.Thatisbecausethephraseappearstomeanthatthesymbol𝑥isliterally“movingalongthenumberlinetotheright.”Nottrue!Recallthatavariableisjustaplaceholderforwhichanumbercanbesubstituted(thinkofablankorboxusedinGrade2equations)and,therefore,doesnotactuallymoveorvary.
Thephrase,“As𝑥 → ∞,”canbeprofitablydescribedasaprocessbywhichtheuserofthephrasethinksofrepeatedlysubstitutinglargerandlargerpositivenumbersinfor𝑥,eachtimeperformingwhatevercalculationisrequiredbytheproblemforthatnumber(whichinthislessonisfindingthevalueofthefunction).
Thisishowmathematiciansoftenusethephraseeventhoughtheprecisedefinitionoflimitremovesanyneedtothinkofalimitasaprocess.
§ ENDBEHAVIOR(description):Let𝑓beafunctionwhosedomainandrangearesubsetsoftherealnumbers.Theendbehaviorofafunction𝑓isadescriptionofwhathappenstothevaluesofthefunction
• as𝑥approachespositiveinfinity,and• as𝑥approachesnegativeinfinity.
Helpstudentsunderstandthedescriptionofendbehaviorusingthefollowingpicture.
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As𝑥 → −∞, As𝑥 → ∞,𝑓 𝑥 → −∞ 𝑓 𝑥 → ∞
Askstudentstomakeageneralizationabouttheendbehaviorofpolynomialsofevendegreeinwritingindividuallyorwithapartner.Theyshouldconcludethatanevendegreepolynomialfunctionhasthesameendbehavioras𝑥 → ∞andas𝑥 → −∞.Afterstudentshavegeneralizedtheendbehavior,havethemcreatetheirowngraphicorganizerlikethefollowing.
𝑥 → ∞
𝑥 → −∞
𝑓(𝑥) → −∞
𝑓(𝑥) → ∞
Graphof𝑦 = 𝑓(𝑥)
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Ifstudentssuspectthatendbehaviorofapolynomialfunctionwithevendegreewillalwaysincrease,thensuggestexaminingthegraphsof𝑓 𝑥 = 1 − 𝑥,and𝑔 𝑥 = −𝑥..
Example1(8minutes)
StudentsarenowgoingtolookatanewsetoffunctionsbutasksimilarquestionstothoseaskedintheOpeningExercise.
Example1
Sketchthegraphof𝒇 𝒙 = 𝒙𝟑.Whatwillthegraphof𝒈 𝒙 = 𝒙𝟓looklike?Sketchthisonthesamecoordinateplane.Whatwillthegraphof𝒉 𝒙 = 𝒙𝟕looklike?Sketchthisonthesamecoordinateplane.
Havestudentsrecallandsketchthegraphof𝑓 𝑥 = 𝑥>.Discussthecharacteristicsofthegraph,wherethe𝑥-interceptis,andwhythegraphisabovethe𝑥-axisfor𝑥 > 0andbelowthe𝑥-axisfor𝑥 < 0.
Inpairsoringroups,havestudentsdiscussorwritewhattheythinkthegraphof𝑔 𝑥 = 𝑥Bwilllooklikeandhowitwillrelatetothegraphof𝑓 𝑥 = 𝑥>.Theyshouldsketchtheirresultsontopoftheoriginalgraphof𝑓.Discusswithstudentswhattheyhavesketched,andemphasizethesimilaritiestothegraphof𝑓 𝑥 = 𝑥>.
Finally,inpairsoringroups,havestudentsdiscussorwritewhattheythinkthegraphofℎ 𝑥 = 𝑥Cwilllooklikeandhowitwillrelatetothegraphsof𝑓and𝑔.Theyshouldsketchonthesamegraphtheyusedwiththeprevioustwographs.Again,discussthegraphswithstudents,andemphasizethesimilaritiesbetweengraphs.
Usingagraphingutility,havestudentsgraphallthreefunctionssimultaneouslytoconfirmtheirsketches.
Even-DegreePo
sitiv
eLead
ingCo
efficient
NegativeLead
ingCo
efficient
𝑓(𝑥) = 𝑥,
𝑓(𝑥) = −𝑥,
As𝑥 → ∞,𝑓(𝑥) → ∞
As𝑥 → −∞,𝑓(𝑥) → ∞
As𝑥 → ∞,𝑓(𝑥) → −∞
As𝑥 → −∞,𝑓(𝑥) → −∞
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Askstudentstocompareanddescribethebehaviorofthevalueof𝑓(𝑥)astheabsolutevalueof𝑥increaseswithoutbound.Guidethemtousetheterminologyoftheendbehaviorofthefunction.
Askstudentstomakeageneralizationabouttheendbehaviorofpolynomialsofodddegreeindividuallyorwithapartner.Afterstudentshavegeneralizedtheendbehavior,havethemcreatetheirowngraphicorganizerlikethefollowing.
Ifstudentssuspectthatpolynomialfunctionswithodddegreealwayshavethevalueofthefunctionincreaseas𝑥increases,thensuggestexaminingafunctionwithanegativeleadingcoefficient,suchas𝑓 𝑥 = 4 − 𝑥or𝑔 𝑥 = −𝑥>.
§ HowdothesegraphsdifferfromthoseintheOpeningExercise?Whyaretheydifferent?ú StudentsmaytalkabouthowtheOpeningExercisegraphsstayabovethe𝑥-axiswhileinthisexamplethegraphscut
throughthe𝑥-axis.Guidestudentsasnecessarytoconcludingthattheendbehaviorofeven-degreepolynomialfunctionsisthatbothendsbothapproachpositiveinfinityorbothapproachnegativeinfinitywhiletheendbehaviorofodd-degreepolynomialfunctionsisthatthebehavioras𝑥 → ∞isoppositeofthebehavioras𝑥 → −∞.
𝑓(𝑥) = 𝑥>
𝑓(𝑥) = −𝑥>
Odd-Degree
Positiv
eLead
ingCo
efficient
NegativeLead
ingCo
efficient
As𝑥 → ∞,𝑓(𝑥) → ∞
As𝑥 → −∞,𝑓(𝑥) → −∞
As𝑥 → ∞,𝑓(𝑥) → −∞
As𝑥 → −∞,𝑓(𝑥) → ∞
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Exercise1(8minutes)
Keepingtheresultsoftheexamplesaboveinmind,havestudentsworkwithpartnersoringroupstoanswerthefollowingquestions.
Exercise1
a. Considerthefollowingfunction,𝒇 𝒙 = 𝟐𝒙𝟒 + 𝒙𝟑 − 𝒙𝟐 + 𝟓𝒙 + 𝟑,withamixtureofoddandevendegreeterms.PredictwhetheritsendbehaviorwillbelikethefunctionsintheOpeningExerciseormorelikethefunctionsfromExample1.Graphthefunction𝒇usingagraphingutilitytocheckyourprediction.
Studentsseethatthisfunctionactsmoreliketheeven-degreemonomialfunctionsfromtheOpeningExercise.
b. Considerthefollowingfunction,𝒇 𝒙 = 𝟐𝒙𝟓 − 𝒙𝟒 − 𝟐𝒙𝟑 + 𝟒𝒙𝟐 + 𝒙 + 𝟑,withamixtureofoddandevendegreeterms.PredictwhetheritsendbehaviorwillbelikethefunctionsintheOpeningExerciseormorelikethefunctionsfromExample1.Graphthefunction𝒇usingagraphingutilitytocheckyourprediction.
Studentsseethatthisfunctionactsmorelikeodd-degreemonomialfunctionsfromExample1.Theycandrawaconclusionsuchasthatthefunctionbehaveslikethehighestdegreeterm.
c. Thinkingbacktoourdiscussionof𝒙-interceptsofgraphsofpolynomialfunctionsfromthepreviouslesson,sketchagraphofaneven-degreepolynomialfunctionthathasno𝒙-intercepts.
Studentsmaydrawthegraphofaquadraticfunctionthatstaysabovethe𝒙-axissuchasthegraphof𝒇 𝒙 = 𝒙𝟐 + 𝟏.
d. Similarly,canyousketchagraphofanodd-degreepolynomialfunctionwithno𝒙-intercepts?
Havestudentsworkinpairsorgroupsanddiscoverthatbecauseofthe“cutthrough”natureofgraphsofodd-degreepolynomialfunctionthereisalwaysan𝒙-intercept.
Conclusion:Graphsofodd-poweredpolynomialfunctionsalwayshavean𝒙-intercept,whichmeansthatodd-degreepolynomialfunctionsalwayshaveatleastonezero(orroot)andthatpolynomialfunctionsofodd-degreealwayshaveoppositeendbehaviorsas𝒙 → ∞and𝒙 → −∞.
Havestudentsconcludethatthegraphsofodd-degreepolynomialfunctionsalwayshaveatleastone𝑥-interceptandsothefunctionsalwayshaveatleastonezero.Thegraphsofeven-degreepolynomialfunctionsmayormaynothave𝑥-intercepts.
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Exercise2(8minutes)
Inthisexercise,studentsusewhattheylearnedtodayaboutendbehaviortodeterminewhetherornotthepolynomialfunctionusedtomodelthedatahasanevenorodddegree.
Exercise2
TheCenterforTransportationAnalysis(CTA)studiesallaspectsoftransportationintheUnitedStates,fromenergyandenvironmentalconcernstosafetyandsecuritychallenges.A1997studycompiledthefollowingdataofthefueleconomyinmilespergallon(mpg)ofacarorlighttruckatvariousspeedsmeasuredinmilesperhour(mph).Thedataarecompiledinthetablebelow.
FuelEconomybySpeed
Speed(mph) FuelEconomy(mpg)𝟏𝟓 𝟐𝟒. 𝟒𝟐𝟎 𝟐𝟕. 𝟗𝟐𝟓 𝟑𝟎. 𝟓𝟑𝟎 𝟑𝟏. 𝟕𝟑𝟓 𝟑𝟏. 𝟐𝟒𝟎 𝟑𝟏. 𝟎𝟒𝟓 𝟑𝟏. 𝟔𝟓𝟎 𝟑𝟐. 𝟒𝟓𝟓 𝟑𝟐. 𝟒𝟔𝟎 𝟑𝟏. 𝟒𝟔𝟓 𝟐𝟗. 𝟐𝟕𝟎 𝟐𝟔. 𝟖𝟕𝟓 𝟐𝟒. 𝟖
Source:TransportationEnergyDataBook,Table4.28.http://cta.ornl.gov/data/chapter4.shtml
a. Plotthedatausingagraphingutility.Whichvariableistheindependentvariable?
Speedistheindependentvariable.
b. Thisdatacanbemodeledbyapolynomialfunction.Determineifthefunctionthatmodelsthedatawouldhaveanevenorodddegree.
Itseemswecouldmodelthisdatabyaneven-degreepolynomialfunction.
c. Istheleadingcoefficientofthepolynomialthatcanbeusedtomodelthisdatapositiveornegative?
Theleadingcoefficientwouldbenegativesincetheendbehaviorofthisfunctionistoapproachnegativeinfinityonbothsides.
d. Listtwopossiblereasonsthedatamighthavetheshapethatitdoes.
Possibleresponses:Fueleconomyimprovesuptoacertainspeed,butthenwindresistanceathigherspeedsreducesfueleconomy;theincreasedgasneededtogohigherspeedsreducesfueleconomy.
Closing(3minutes)
§ Inthislesson,studentsexploredthecharacteristicsofthegraphsofpolynomialfunctionsofevenandodd-degree.Graphsofeven-degreepolynomialsdemonstratethesameendbehavioras𝑥 → ∞asitdoesas𝑥 → −∞,whilegraphsofodd-degreepolynomialsdemonstrateoppositeendbehavioras𝑥 → ∞asitdoesas𝑥 → −∞.Becauseofthisfact,graphsofodd-degreepolynomialfunctionsalwaysintersectthe𝑥-axis;therefore,odd-degreepolynomialfunctionshaveatleastonezeroorroot.
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As𝑥 → ∞,𝑓(𝑥) → ∞As𝑥 → −∞,𝑓(𝑥) → −∞
As𝑥 → ∞,𝑓(𝑥) → −∞As𝑥 → −∞,𝑓(𝑥) → ∞
As𝑥 → ∞,𝑓(𝑥) → ∞As𝑥 → −∞,𝑓(𝑥) → ∞
As𝑥 → ∞,𝑓(𝑥) → −∞As𝑥 → −∞,𝑓(𝑥) → −∞
Odd-DegreeEven-DegreePo
sitiveLead
ingCo
efficient
NegativeLead
ingCo
efficient
§ Studentsalsolearnedthatitisthehighestdegreetermofthepolynomialthatdeterminesifthegraphexhibitsodd-degreeendbehaviororeven-degreeendbehavior.Thismakessensebecausethehighestdegreetermofapolynomialdeterminesthedegreeofthepolynomial.
Havestudentssummarizethelessoneitherwithagraphicorganizerorawrittensummary.Agraphicorganizerisincludedbelow.
RelevantVocabulary
EVENFUNCTION:Let𝒇bea functionwhosedomainandrangeisasubset oftherealnumbers.Thefunction𝒇iscalledeveniftheequation𝒇(𝒙) = 𝒇(−𝒙)istrueforeverynumber𝒙inthedomain.
Even-degreepolynomialfunctionsaresometimesevenfunctions,like𝒇(𝒙) = 𝒙𝟏𝟎,andsometimesnot,like𝒈(𝒙) = 𝒙𝟐 − 𝒙.
ODDFUNCTION:Let𝒇beafunctionwhosedomainandrangeisasubsetoftherealnumbers.Thefunction𝒇iscalledoddiftheequation𝒇 −𝒙 =−𝒇(𝒙)istrueforeverynumber𝒙inthedomain.
Odd-degreepolynomialfunctionsaresometimesoddfunctions,like𝒇(𝒙) = 𝒙𝟏𝟏,andsometimesnot,like𝒉(𝒙) = 𝒙𝟑 − 𝒙𝟐.
ExitTicket(5minutes)
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Name Date
Lesson15:StructureinGraphsofPolynomialFunctions
ExitTicketWithoutusingagraphingutility,matcheachgraphbelowincolumn1withthefunctionincolumn2thatitrepresents.
a.
1. 𝑦 = 3𝑥>
b.
2. 𝑦 = 12 𝑥
,
c.
3. 𝑦 = 𝑥> − 8
d.
4. 𝑦 = 𝑥. − 𝑥> + 4𝑥 + 2
e.
5. 𝑦 = 3𝑥B − 𝑥> + 4𝑥 + 2
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ExitTicketSampleSolutions
Withoutusingagraphingutility,matcheachgraphbelowincolumn1withthefunctionincolumn2thatitrepresents.
a.
1. 𝒚 = 𝟑𝒙𝟑
b.
2. 𝒚 = 𝟏𝟐 𝒙
𝟐
c.
3. 𝒚 = 𝒙𝟑 − 𝟖
d.
4. 𝒚 = 𝒙𝟒 − 𝒙𝟑 + 𝟒𝒙 + 𝟐
e.
5. 𝒚 = 𝟑𝒙𝟓 − 𝒙𝟑 + 𝟒𝒙 + 𝟐
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ProblemSet
1. GraphthefunctionsfromtheOpeningExercisesimultaneouslyusingagraphingutilityandzoominattheorigin.
a. At𝒙 = 𝟎. 𝟓,orderthevaluesofthefunctionsfromleasttogreatest.
b. At𝒙 = 𝟐. 𝟓,orderthevaluesofthefunctionsfromleasttogreatest.
c. Identifythe𝒙-value(s)wheretheorderreverses.Writeabriefsentenceonwhyyouthinkthisswitchoccurs.
2. TheNationalAgriculturalStatisticsService(NASS)isanagencywithintheUSDAthatcollectsandanalyzesdatacoveringvirtuallyeveryaspectofagricultureintheUnitedStates.Thefollowingtablecontainsinformationontheamount(intons)ofthefollowingvegetablesproducedintheU.S.from1988–1994forprocessingintocanned,frozen,andpackagedfoods:limabeans,snapbeans,beets,cabbage,sweetcorn,cucumbers,greenpeas,spinach,andtomatoes.
VegetableProductionbyYear
Year VegetableProduction(tons)1988 𝟏𝟏, 𝟑𝟗𝟑, 𝟑𝟐𝟎1989 𝟏𝟒, 𝟒𝟓𝟎, 𝟖𝟔𝟎1990 𝟏𝟓, 𝟒𝟒𝟒, 𝟗𝟕𝟎1991 𝟏𝟔, 𝟏𝟓𝟏, 𝟎𝟑𝟎1992 𝟏𝟒, 𝟐𝟑𝟔, 𝟑𝟐𝟎1993 𝟏𝟒, 𝟗𝟎𝟒, 𝟕𝟓𝟎1994 𝟏𝟖, 𝟑𝟏𝟑, 𝟏𝟓𝟎
Source:NASSStatisticsofVegetablesandMelons,1995,Table191.http://www.nass.usda.gov/Publications/Ag_Statistics/1995-1996/agr95_4.pdf
a. Plotthedatausingagraphingutility.
b. Determineifthedatadisplaythecharacteristicsofanodd-oreven-degreepolynomialfunction.
c. Listtwopossiblereasonsthedatamighthavesuchashape.
3. TheU.S.EnergyInformationAdministration(EIA)isresponsibleforcollectingandanalyzinginformationaboutenergyproductionanduseintheUnitedStatesandforinformingpolicymakersandthepublicaboutissuesofenergy,theeconomy,andtheenvironment.ThefollowingtablecontainsdatafromtheEIAaboutnaturalgasconsumptionfrom1950–2010,measuredinmillionsofcubicfeet.
U.S.NaturalGasConsumptionbyYear
Year U.S.naturalgastotalconsumption(millionsofcubicfeet)
1950 𝟓. 𝟕𝟕1955 𝟖. 𝟔𝟗1960 𝟏𝟏. 𝟗𝟕1965 𝟏𝟓. 𝟐𝟖1970 𝟐𝟏. 𝟏𝟒1975 𝟏𝟗. 𝟓𝟒1980 𝟏𝟗. 𝟖𝟖1985 𝟏𝟕. 𝟐𝟖1990 𝟏𝟗. 𝟏𝟕1995 𝟐𝟐. 𝟐𝟏
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2000 𝟐𝟑. 𝟑𝟑2005 𝟐𝟐. 𝟎𝟏2010 𝟐𝟒. 𝟎𝟗
Source:U.S.EnergyInformationAdministration.http://www.eia.gov/dnav/ng/hist/n9140us2a.htm
a. Plotthedatausingagraphingutility.
b. Determineifthedatadisplaythecharacteristicsofanodd-oreven-degreepolynomialfunction.
c. Listtwopossiblereasonsthedatamighthavesuchashape.
4. Weusethetermevenfunctionwhenafunction𝒇satisfiestheequation𝒇 −𝒙 = 𝒇(𝒙)foreverynumber𝒙initsdomain.Considerthefunction𝒇 𝒙 = −𝟑𝒙𝟐 + 𝟕.Notethatthedegreeofthefunctioniseven,andeachtermisofanevendegree(theconstanttermisdegree𝟎).a. Graphthefunctionusingagraphingutility.
b. Doesthisgraphdisplayanysymmetry?
c. Evaluate𝒇 −𝒙 .
d. Is𝒇anevenfunction?Explainhowyouknow.
5. Weusethetermoddfunctionwhenafunction𝒇satisfiestheequation𝒇 −𝒙 = −𝒇(𝒙)foreverynumber𝒙initsdomain.Considerthefunction𝒇 𝒙 = 𝟑𝒙𝟑 − 𝟒𝒙.Thedegreeofthefunctionisodd,andeachtermisofanodddegree.
a. Graphthefunctionusingagraphingutility.
b. Doesthisgraphdisplayanysymmetry?
c. Evaluate𝒇(−𝒙).
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d. Is𝒇anoddfunction?Explainhowyouknow.
6. Wehavetalkedabout𝒙-interceptsofthegraphofafunctioninboththislessonandthepreviousone.The𝒙-interceptscorrespondtothezerosofthefunction.Considerthefollowingexamplesofpolynomialfunctionsandtheirgraphstodetermineaneasywaytofindthe𝒚-interceptofthegraphofapolynomialfunction.
𝑓 𝑥 = 2𝑥, − 4𝑥 − 3 𝑓 𝑥 = 𝑥> + 3𝑥, − 𝑥 + 5 𝑓 𝑥 = 𝑥. − 2𝑥> − 𝑥, + 3𝑥 − 6
