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CP Algebra 2

Unit 2B Writing and Graphing Quadratics

Name: ___________________________________ Period: __________

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Unit 2B Writing and Graphing Quadratics Learning Targets:

Modeling with

Quadratic Functions

1. I can identify a function as quadratic given a table, equation, or graph.

2. I can determine the appropriate domain and range of a quadratic equation or event.

3. I can identify the minimum or maximum and zeros of a function with a calculator.

4. I can apply quadratic functions to model real-life situations, including quadratic regression models from data.

Graphing

5. I can graph quadratic functions in standard form (using properties of quadratics).

6. I can graph quadratic functions in vertex form (using basic transformations).

7. I can identify key characteristics of quadratic functions including axis of symmetry, vertex, in/max, y-intercept, domain and range.

Writing Equations of Quadratic Functions

8. I can rewrite quadratic equations from standard to vertex and vice versa.

9. I can write quadratic equations given a graph or given a vertex and the y-intercept (without a calculator).

10. I can write quadratic equations given the roots/zeros/intercepts/solutions.

11. I can write quadratic equations in vertex form by completing the square.

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Modeling Quadratic Functions After this lesson and practice, I will be able to …

¨ identify a function as quadratic given a table, equation, or graph. (LT 1) ¨ determine the appropriate domain and range of a quadratic equation or event. (LT 2) ¨ identify the minimum or maximum, and zeros of a function with a calculator. (LT 3) ¨ apply quadratic functions to model real-life situations, including quadratic regression models. (LT 4)

--------------------------------------------------------------------------------------------------------------------------------------------------- QuadraticFunction:Afunctionthatcanbewrittenintheform____________________where____________.ThisiscalledStandardForm!IdentifyQuadratics(LT1)Example1:Determinewhethereachfunctionisquadraticornot.Ifso,writeitinstandardform.a. b. c. d. d e. f. g. PropertiesofQuadraticFunctionsLet’sgraph𝑦 = 𝑥!,whichiscalledtheparentfunction.Theshapeofthegraphiscalledaparabola.Thepointatwhichthegraphcomestoamaximumorminimumiscalledthevertex.Theverticallinethatpassesthroughthevertexoftheparabolaiscalledthelineofsymmetry.

!!y = x(x −5) !!x2 −(5x +2+ x2) !!(2x +3)(x −7)

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DomainandRange(LT2)Example2:Supposeaquadraticfunctionhasavertexof(-3,-5)andcontainsthepoints(-4,-3)and(-2,-3).Identifytheequationfortheaxisofsymmetryanduseittohelpyousketchagraphofthefunction.Thenidentifythedomainandrange.Example3:Supposeaquadraticfunctionhasavertexof(5,6)andcontainsthepoints(7,2)and(2,-3).Identifytheequationfortheaxisofsymmetryanduseittohelpyousketchagraphofthefunction.Thenidentifythedomainandrange.UsingYourCalculator(LT3)Reminder:Theplaceswherethegraphcrossesthex-axisarecalled______________________ In some situations, you want to quickly approximate the zeros or the minimum/maximum of a quadratic equation. Thankfully, your graphing calculator is very helpful in accomplishing this feat. Example 4: Approximate the zeros and min or max of the equation !!3x2 +5x =6 using a graphing calculator. Round to four decimal places.

1) Put the equation you want to graph in standard form. 2) Enter the equation in your calculator as Y1 =. Press GRAPH. 3) Press 2nd TRACE, then press 2: ZERO. 4) Move your cursor just to the “left” of the first point of intersection. Press ENTER. 5) Move your cursor just to the “right” of the first point of intersection. Press ENTER. 6) The screen will show “Guess”. Press ENTER again. The first zero in this example is _________. 7) Repeat steps 3-6 to obtain the second root. The second zero is __________. 8) Press 2nd TRACE, then press MIN or MAX (depending on the shape of your parabola). 9) Move your cursor just to the “left” of the min or max. Press ENTER. 10) Move your cursor just to the “right” of the min or max. Press ENTER. 11) The screen will show “Guess”. Press ENTER again. The __________ in this example is _________.

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Example 5: Approximate the zeros and min or max of the equation below using a graphing calculator. Round to four decimal places.

a) −2x2 +10x−1= y b) f (x)= x2 +8x+16 Zero(s) :_________________________ Zero(s) :_________________________

Min Max is ________ at __________ Min Max is ________ at __________

ApplicationsofQuadraticFunctions(LT4)Quadraticfunctionshavemanyapplicationstoreal-worldscenarios.Forexample,quadraticscanbeusedtomodeltheheightofaprojectileovertime.Theycanalsobeusedtohelpbusinessespredictrevenue.Example6:Theheightofapuntedfootballcanbemodeledbyh=−0.01x2 +1.18x+2 .Thehorizontaldistanceinfeetfromthepunter’sfootisx,andhistheheightoftheballinfeet.Calculatehowfarthepunterkickedtheballusingyourgraphingcalculator. ____________________feet ___________________yardsExample7:Abakeryknowsfrommarketresearchthat-6p+180modelsthenumberofpiessoldonaverageday,wherepisthepriceofapieindollars.a.Writeanequationtorepresenttherevenuethebakerycanexpectonanaverageday.R(p)=_______________________________b.Whatpricewillmaximizetherevenue? ____________________c.Whatisthemaximumrevenue? ____________________

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Example8:Thedatainthetablerepresentstheamountofrainfallforeachmonth,ininches.a.Graphascatterplotofthedatainyourcalculator.Whattypeoffunctionbestrepresentsthedata?-Enteryourdata(2NDSTAT,Edit…)-TurnonStatPlot1(2nd,STATPLOT)-Graphthedatab.Findthe____________regressionlinethatmodelsthedatausingyourcalculator.-STAT,right,QuadReg-YourcommandshouldlooklikeQuadReg_________________.c.Usingyourequation,predicttherainfallinNovember.

Month Rainfall(inches)

Jan 5.7Feb 4.2Mar 3.8Apr 2.4May 1.7Jun 1.6Jul 0.8Aug 1Sep 1.8Oct 2.1Nov ???Dec 5.4

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Graphing with Properties of Quadratics After this lesson and practice, I will be able to …

¨ graph quadratic functions in standard form (using properties of quadratics). (LT 5) ¨ identifykeycharacteristicsofquadraticfunctionsincludingaxisofsymmetry,vertex,min/max,y-intercept,

domainandrange.(LT7)--------------------------------------------------------------------------------------------------------------------------------------------------- Oneofthemostimportantthingswewilldothisyearisgraphfunctions.Thesimplest(butnotalwayseasiestway)ofgraphingistomakeatable,butthattakesalotoftimeandisfrustratingifyoucan’tfindthevertexrightaway.Solet’sexploresomeeasierwaysofgraphing!Startwiththeparentquadraticfunction:________________Gotowww.desmos.comandclickStartGraphing.Inthefirstblank,typeintheparentfunction(usethecarrotkey^forthesquaredpartoftheequation).Youshouldseeaparabolaappearwithitsvertexat(0,0).Followthedirectionsbelowandfillintheblanks.Intheboxwhereyoujustwrotetheparentfunction,addan“a”beforethex.Youshouldseeabluebuttonappearnexttothewords“addslider”.Clickthebluebutton.Youshouldnowbeabletochangethevalueofabyusingtheslider.Experimentwiththeslider,thenfillintheblanks… When𝑎 > 0,theparabolaopens__________. When𝑎 < 0,theparabolaopens__________.Gobacktotheparentfunctionandadd“+bx”.Thefirstboxshouldnowread:𝑦 = 𝑎𝑥! + 𝑏𝑥.Addasliderforblikeyoudidwitha,thenexperimentwithmovingbaround.InBox4,addthefollowingequation:𝑥 = − !

!!.Thefractionwillautomaticallyformwhenyoutypein“/”.

Youshouldseeadifferentcoloredlineappearonthegraph.Nowexperimentwiththeaandbsliders. 𝑥 = − !

!!isthe___________________________________.

Sothex-coordinateofthevertexis− !

!!andthey-coordinateiswhatyougetwhenyouplugin− !

!!.

Inotherwords,thevertexis: − !!!

, 𝑓(− !!!

) Finally,add“+c”totheparentfunctioninBox1.Itshouldnowread:𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐.Addasliderforc,thenexperimentwithmovinga,b,andc. Thepoint(0,c)isthe__________________________.Thesepropertiesmakegraphingmucheasier

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Example2:Graph Stepstographingquadraticfunctions:1.Findandgraphtheaxisofsymmetry.2.Findandgraphthevertex.3.Identifythey-intercept.Graphthepointanditspointofreflectionacrosstheaxisofsymmetry.4.Evaluatethefunctionforanothervalueofx.Plotthepointanditspointofreflectionacrosstheaxisofsymmetry.5.Drawa_________________throughthegraphedpoints.Example3:Graph𝑦 = −3𝑥! + 6𝑥 + 2AxisofSymmetry:__________y-intercept:________Doesthisfunctionhaveaminormax(circleone)?Whatistheminimumormaximumfunctionvalue?y=_______________Wheredoestheminormaxoccur?x=__________Whatarethedomainandrange?Domain:___________________Range:___________________

!!y = x2 −2x −3

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Examples4and5:4.Graph AxisofSymmetry:__________y-intercept:________Doesthisfunctionhaveaminormax(circleone)?Whatistheminimumormaximumfunctionvalue?y=_______________Wheredoestheminormaxoccur?x=__________Whatarethedomainandrange?Domain:___________________Range:___________________

!!y = −x2 +4x −6 5.Graph

AxisofSymmetry:______________y-intercept:________Doesthisfunctionhaveaminormax(circleone)?Whatistheminimumormaximumfunctionvalue?y=_______________Wheredoestheminormaxoccur?x=__________Whatarethedomainandrange?Domain:___________________Range:___________________

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Graphing with Transformations Afterthislessonandpractice,Iwillbeableto…

¨ graphquadraticfunctionsinvertexform(usingbasictransformations).(LT6)¨ identifykeycharacteristicsofquadraticfunctionsincludingaxisofsymmetry,vertex,min/max,y-intercept,

domainandrange.(LT7)---------------------------------------------------------------------------------------------------------------------------------------------------Inadditiontographingfromstandardform,wecanalsographbyperformingtransformations.Thisworksverywellwhenthequadraticiswritteninvertexform.

StandardForm𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙+ 𝒄 −

𝒃𝟐𝒂

VertexForm𝒚 = 𝒂(𝒙− 𝒉)𝟐 + 𝒌

h

𝑦 = 𝑥! − 4𝑥 + 4 𝑦 = (𝑥 − 2)!

𝑦 = 𝑥! + 6𝑥 + 8 𝑦 = (𝑥 + 3)! − 1

𝑦 = −3𝑥! − 12𝑥 − 8 𝑦 = −3(𝑥 + 2)! + 4

𝑦 = 2𝑥! + 12𝑥 + 19 𝑦 = 2(𝑥 + 3)! + 1

Comparethevaluesof− !!!andhineachrow.Writeaformulatoshowtherelationshipbetween− !

!!andh.

VertexFormofaQuadraticEquation:Thevertexis_________________________andtheaxisofsymmetrylineis____________________Nowlet’stransformtheparentfunctiony=x2.Gotowww.desmos.comandclickStartGraphing.Inthefirstbox,typetheparentfunction.Youcanusethecarrotbutton(^)forsquare.Youshouldseearedparabolaappearonthegraph.Inthesecondbox,writethefollowingequation:𝑦 = 𝑎𝑥!.Clickthebluebuttonthatappearstomakeaslider.Experimentwiththeslider,thenfillintheblanksbelow… Whenaisnegative,theparabola______________________________(thisiscalledareflection) When𝑎 > 0,theparabola_____________________________________________

When0 < 𝑎 < 1,theparabola_____________________________________________

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AddparenthesisandtheletterhtoBox2sothatitsaysthis:𝑦 = 𝑎(𝑥 − ℎ)!.

Whenℎ > 0,theparabolashifts_________________________Whenℎ < 0,theparabolashifts_________________________

NowgobacktoBox2andadd+ksothatBox1saysthis:𝑦 = 𝑎(𝑥 − ℎ)! + 𝑘.

When𝑘 > 0,theparabolashifts_________________________

When𝑘 < 0,theparabolashifts_________________________Example1:Describethetransformationsappliedtotheparentquadraticfunction.Thendeterminethevertex,axisofsymmetry,andwhethertheparabolaopensupordown.

a. !!y = (x −4)2 −5 b. !!y = −2(x +3)

2 +6 c.!!y = −12 x

2 −4

Example2:Graph!!y = −2(x −4)2 +3 .

1.Determinehowfarup/down/right/leftthevertexshifts,then

graphit.

2.Graphtheaxisofsymmetry.

2.Determinewhethertheparabolaopensupordown.

4.Whena=1,thepointsaregraphedlikethisfromthevertex:over-1-up-1,over-2-up-4,over-3-up-9.Ifaisnot1,thenwehaveaverticalstretchofa.Multiplythe“up”partofthispatternbya.Thengraphthreepointsoneithersideoftheparabola.5.Connectthepointswithasmoothcurve.

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Example3:Graph!!y = 12(x +2)

2 −4 Example4:Graph𝑦 = −2 𝑥 − 3 ! + 9

Example5:Graph y =−3(x+4)2 +2 Example6:Graph𝑦 = .5 𝑥 + 1 ! − 5

Vertex:____________AxisofSymmetry:______________Domain:_____________________Range:______________________Whatistheminimumormaximumfunctionvalue?

Vertex:____________AxisofSymmetry:______________Domain:_____________________Range:______________________Whatistheminimumormaximumfunctionvalue?

Vertex:____________AxisofSymmetry:______________Domain:_____________________Range:______________________Whatistheminimumormaximumfunctionvalue?

Vertex:____________AxisofSymmetry:______________Domain:_____________________Range:______________________Whatistheminimumormaximumfunctionvalue?

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Writing Quadratic Functions Afterthislessonandpractice,Iwillbeableto…

¨ rewritequadraticequationsfromstandardtovertexandvieversa.(LT8)¨ writequadraticequationsfromagraphorgivenavertex/pointwithoutacalculator.(LT9)¨ writequadraticequationsgiventheroots/zeros/intercepts/solutions.(LT10)

---------------------------------------------------------------------------------------------------------------------------------------------------Wehaveexploredseveralpropertiesofquadraticequationswritteninstandardformandvertexform.Let’sthinkabouttheadvantagesofeachequationform:AdvantagestoStandardForm AdvantagestoVertexFormRewritingQuadraticEquations(LT8)Sinceeachequationformhasadvantages,itisimportanttobeabletorewriteequationsinmultipleforms.Example1:Rewritethevertexformequationinstandardformandviceversa.a. !!y =2x

2 +10x +7 b. !!y = −3(x +2)2 −6

c. y = x2 + 2x + 4 d. y = 3x2 −12x +17

e. y = − 12x − 4( )

2+10 f. y = 2 x +5( )

2−30

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WritingQuadraticEquationsfromaGraph(LT9)Anotherimportantskillisbeingabletowritequadraticequationswhengiventhegraphorpoints.Example2:Writetheequationforeachparabolainvertexform.a. b.Example3:Writetheequationoftheparabolawiththegivenvertex,y-interceptorpoint.Writeinvertexformthenrewriteinstandardform.a.vertexat(-3,6)anday-interceptof33.Vertexform:__________________________ standardform:___________________________b..vertex(-2,1)andthegraphcontainsthepoint(4,11)

Vertexform:__________________________ standardform:___________________________

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Wecanalsowritetheequationofquadraticsfromthesolutions(orrootsorzerosorintercepts!).GivenTheRoots/Zeros/Intercepts/Solutions(LT10)Example4:Writeaquadraticequationinintercept/factoredformwiththefollowingzeros:

A)2and-4 B)4and4 C)and-7

GraphtheequationfromExample1AinyourgraphingcalculatororonDesmos.Whatpointappearstobethevertexofthisequation?_________Howdoesthex-coordinateofthevertexrelatetothetwointercepts?Example2:Writeaquadraticequationinx-interceptorfactoredformwiththefollowingzeros.Thenfindthevertexofeachequation.

A.-1and3 B.-4iand4i C.0and8 Example3:Writeaquadraticequationinstandardformthathasthefollowingroots.

A.-3iand3i B.0and2 C.3and-5 Example4:Writeaquadraticequationinstandardformthathasthefollowingx-intercepts.

A.(0,0)and(-3,0) B.(5,0)and(-2,0)

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Return of Completing the Square! Afterthislessonandpractice,Iwillbeableto…

¨ writequadraticequationsinvertexformbycompletingthesquare.(LT11)---------------------------------------------------------------------------------------------------------------------------------------Inthelastunit,wecompletedthesquareasawaytosolvequadraticequations.Nowwewillusethesamestrategytorewriteequationsintovertexform.WarmUp:RewritethisequationbyCompletingtheSquare: y = x2 −10x + 22 Thatformofquadraticequationshouldlookfamiliar…whatformdoesitlooklike?So in addition to being a nice way to solve quadratics, Completing thee Square is also a nice way to convert Standard Form equations to Vertex form. Let’s practice! Example 5: Write each equation in vertex form. Identify the vertex.

A) y = 2x2 −8x +1 B) y = x2 + 4x − 7 C) 22 28 99y x x= − +

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