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StudentP 1April2017 QuadraticFunctionsPerformanceTask Period4 Algebra1 Debel ParabolasintheRealWorld Parabolascanbefoundeverywhereinourday-to-daylives.Theyexistinarchitecture,art, objects,andthepathsofprojectiles.HerearesomeexamplesIfound: Theenormoussteelcablessupportingthefamous GoldenGateBridgeinSanFranciscoforma parabolathatopensupwards.Thedistance betweenthetowersofthebridgeis1,200meters! Thetopsofthetowersrearabout230metersabove sealevel.Eachofthetwomaincablessuspending thebridgeis2332meterslongs. (source:goldengatebridge.org) Justabouteveryobjectthatisthrownintotheair followsaparabolicpath.Thisappliestofootballs, bullets,cannonballs,arrows,Frisbees,androcks. (Somelightobjects,suchasballoonsorpaper airplanes,willnotfollowaparabolicpath,butwill bemovedaboutbytheair.)Someobjectswhich areveryfast,suchasbullets,willhaveaverywide parabolicpath(verticalcompression).Iwatched childrenthrowingballsthatfollowedparabolic paths,butwasn’tabletocapturethisinastill picture. ThislittleTiffany-stylelampsitsinmylivingroom atopthepiano.Itslampshadeisapproximately parabolic.(Ofcoursethelampshadeisthree dimensional,butitcrosssectionisaparabola.) Thevertexoftheparabolaisatthetopofthe lampshade.Thislittleparabolaisonlyaboutfive inchestall. StudentP QuadraticFunctionsPerformanceTask Algebra1 Debel TheEiffelTowerinParis, France,isaremarkable featofarchitecture,full offascinatinggeometry. Itslowerarchesare roughlyparabolicin shape.Thedimensionsin mymodelareinmeters, andapproximatethetrue measurementsofthe EiffelTower. Ihavealignedthe x-axiswiththe groundandthe y-axiswiththe axisofsymmetry oftheparabola. 1April2017 Period4 Vertexandy-intercept: (#, &#) Maximum:50 Axisofsymmetry: ! = # x-intercept (−)*, #) x-intercept ()*, #) StudentP 1April2017 QuadraticFunctionsPerformanceTask Period4 Algebra1 Debel Ihavealignedmyx-axiswiththeground.They-axisgoesthroughthecenterofthe towerandismyaxisofsymmetry. Allmeasurementsareapproximateandareinmeters. Thezeros(x-intercepts)oftheparabolaarethepointsatwhichtheinsideedgesof thetower’ssupportstouchtheground. Thecoordinatesofthezerosareat: (−37,0)/01(37,0) TheVertexoftheparabolaisat: (0,50) They-interceptisalsothevertexandisat: (0,50) Theaxisofsymmetryisthey-axis.Itsequationis: 3 = 0 Themaximumvalueoftheparabolaistheheightofthearch: 50 TableofValues: X Y (Ireadthesevalues -37 0 offmygraph.Iusedthe -10 46 pointsontheparabola,even 0 50 thoughtheywereslightly 25 27 differentfromthepointson 37 0 theEiffelTower.) StudentP 1April2017 QuadraticFunctionsPerformanceTask Period4 Algebra1 Debel FindingtheEquationofmyParabola: Tofindtheequationofmyparabolainfactoredform,Ifirstpluginthezeros(-37and37)for “p”and“q:” 4 3 =/ 3−5 3 =6 4 3 = /(3 + 37)(3 − 37) Ithenpluginthevertexforxandyinordertofindthevalueofa: 50 = / 0 + 37 0 − 37 50 = / ∙ −1369 50 /=− 1369 FromthisIcanassemblemyfunction: < ! =- &# (! + )*)(!-)*) >)?@ Icanconvertmyfunctiontogeneral/standardformbyapplyingFOILandsimplifyingtheresult: 50 4 3 =− (3 A − 373 + 373 − 1369) 1369 50 4 3 =− (3 A − 1369) 1369 < ! =- &# B ! + &# >)?@ FindingVertexFormisveryeasyinthiscase!Thatisbecauseinthiscase,generalformand vertexformareidentical.Icouldexpressmyfunctioninthefollowingwaytomakevertexform evenmorerecognizable: < ! = &# !-# >)?@ B + &# CℎEFGH/0FIℎHJHKIH3EF/I(0,50) StudentP 1April2017 QuadraticFunctionsPerformanceTask Period4 Algebra1 Debel Iwillnowpluginoneofthepointsfrommytableofvalues(25,27)toverifywhethermy functionisworking.Iwilltryallthreeforms.(Iwilluseacalculatortohelpwithmymath!) FactoredForm: 50 4 25 = − 25 + 37 25 − 37 1369 50 4 25 = − ∙ 62 ∙ (−12) 1369 4 25 = 27.17 GeneralForm/VertexForm: 50 4 25 = − (25)A + 50 1369 50 4 25 = − ∙ 625 + 50 1369 31250 4 25 = − + 50 1369 4 25 = 27.17 ThepointIreadoffofmygraphwasat(25,27).ThisisveryclosetothevaluesIcalculatedwith myfunctions!(Theonlydifferenceisthelevelofaccuracy.Graphspermitonlyan approximationofvalues,whilethefunctionwillprovideamoreaccuratevalue.) IndoingthisprojectIlearnedthatparabolasoccurineverydaylife,andthatIhavethetoolsto graphthemandwritefunctionstodescribeandmakecalculationsaboutthem.Therearemany potentialapplicationsforthisskill,suchaspredictingwhereathrownobjectmayland, calculatingtheheightofanarchthatistootallformetomeasuredirectly,ordescribingthe preciseshapeIwouldlikeafancywindowtohaveinabuildingImaydesignoneday. I’vestartedseeingparabolaseverywhere:Evensomeheadsareparabolic!
