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CP Algebra 2

Unit 2-2 Writing and Graphing Quadratics

Name: ___________________________________ Period: __________

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Unit 2-2 Writing and Graphing Quadratics Learning Targets:

Unit 2-1 One more

way to solve!

11. I can solve equations using the quadratic formula (with rationalized denominators).

12. I can use the discriminant to determine the number and type of solutions.

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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The Quadratic Formula Date: ____________ After this lesson and practice, I will be able to …

! solve using the quadratic formula (with rationalized denominators). (LT 11) ! use the discriminant to determine the number and type of solutions. (LT 12)

--------------------------------------------------------------------------------------------------------------------------------------------------- So far, you have learned three strategies for solving quadratic equations … they are: Each strategy has its own benefit depending on the characteristics of the equation. Today you’re going to see the fourth and final way of solving quadratic equations. It’s one of the most famous formulas in mathematics and it works on ____________ quadratic equation!

The Quadratic Formula: For any quadratic equation , the solutions are ... Example 1: Solve each equation. Write your solutions as exact values in simplest form (no decimals) A) 3x2 −5x = 2 B) !!2x2 +8x = −12 E) 9x2 +12x + 4 = 0

C) D) !!2x2 =7x −8 As we’ve seen, quadratic equations can have __________ or _________________ solutions. Let’s discover how to quickly determine which type of solution a given quadratic equation has …

Discriminant: Given , the discriminant is _________________.

24 8 1 0x x− + =

2 0ax bx c+ + =

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Fill out the table below with your group (split up the work if you want).

Equation Value of Discriminant Number and Type of

Solutions

!!8x2 −2x −3=0

!!2x2 +8x +12=0

!!x2 +10x +25=0

The discriminants of different quadratic equations are given below. Look for patterns in the table above, then predict how many real solutions each quadratic equation has.

Discriminant Number of Real Solutions

-5

16

0

Summary:

Value of Discriminant

Number and Types of Solutions

Positive

Zero

Negative

Example 2: Find the discriminant and give the number and type of solutions. Do not solve. A) B) C)

2 10 23 0x x+ + = 2 10 25 0x x+ + = 2 10 27 0x x+ + =

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Modeling Quadratic Functions Date: ____________ 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)

--------------------------------------------------------------------------------------------------------------------------------------------------- Let’s review the definition of a quadratic function … QuadraticFunction:Afunctionthatcanbewrittenintheform____________________where____________.Thedomainofaquadraticfunctionis_________________________________.StandardFormofaQuadraticFunction:

Example1:Determinewhethereachfunctionisquadraticornot.Ifso,writeitinstandardform.a. b. c. d.PropertiesofQuadraticFunctionsLet’sgraph𝑦 = 𝑥!,whichiscalledtheparentfunction.Theshapeofthegraphiscalleda____________________.Thepointatwhichthegraphcomestoamaximumorminimumiscalledthe___________________.Theverticallinelinethatpassesthroughthevertexoftheparabolaiscalledthe__________________________.

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

x y-3 9-2 4-1 10 01 12 43 9

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Example2:Supposeaquadraticfunctionasavertexof(-3,-5)andcontainsthepoints(-4,-3)and(-2,3).Identifytheequationfortheaxisofsymmetryanduseittohelpyousketchagraphofthefunction.Thenidentifythedomainandrange.Example3:Supposeaquadraticfunctionasavertexof(5,6)andcontainsthepoints(7,2)and(2,-3).Identifytheequationfortheaxisofsymmetryanduseittohelpyousketchagraphofthefunction.Thenidentifythedomainandrange.Theplaceswherethegraphcrossesthex-xisarecalled:UsingYourCalculator(LT3)In some situations, you want to quickly approximate the __________ and the _____________________/____________________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 first ________ 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:Smokejumpersarefirefighterswhoparachuteintoareasnearforestfires.Ajumper’sheightduringfreefall(beforetheparachuteopens)ismodeledby

!!h(t)= −16t2 +h0 wheretistimeinsecondsand!!h0 istheinitialheight.Ifajumper’splaneis

1500feetoffthegroundwhenhejumps,howlongisheinfreefallbeforehisparachuteopensat1200feet? ______________

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Example8:Anapartmentcomplexhas100two-bedroomunits.Themonthlyprofitindollarsrealizedfromrentingouttheapartmentsisgivenby:P(x)=-10x2+1500x–20,000a.Howmanyunitsmustberentedouttomaximizeprofits? ____________________b.Whatisthemaximumprofitrealizable? ____________________c.Howmanyapartmentsmustberentedbeforeaprofitismade? ___________________Example9:Abakeryknowsfrommarketresearchthat-6p+180modelsthenumberofpiessoldonaverageday,wherepisthepriceofapieindollars.a.Writeanequationtorepresenttherevenuethebakerycanexpectonanaverageday.R(p)=_______________________________b.Whatpricewillmaximizetherevenue? ____________________c.Whatisthemaximumrevenue? ____________________Example10: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 Date: ____________ After this lesson and practice, I will be able to …

! graph quadratic functions in standard form (using properties of quadratics). (LT 5) --------------------------------------------------------------------------------------------------------------------------------------------------- Oneofthemostimportantthingswewilldothisyearisgraphfunctions.Thesimplest(butnotalwayseasiestway)ofgraphingistomakeatable… Example1:Graphthequadraticfunctionbymakingatable,thenwritetheaxisofsymmetry,thedomain,andtherange:

Nowlet’sexploresomeeasierwaysofgraphing.Startwiththeparentquadraticfunction:________________Gotowww.desmos.comandclickStartGraphing.Inthefirstblank,typeintheparentfunction(usethecarrotkey^forthesquaredpartoftheequation).Youshouldseearedparabolaappearwithitsvertexat(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.

!!y = −2x2 +3

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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!Example2:Graph Stepstographingquadraticfunctions:1.Findandgraphtheaxisofsymmetry.2.Findandgraphthevertex.3.Identifythey-intercept.Graphthepointanditspointofreflectionacrosstheaxisofsymmetry.4.Evaluatethefunctionforanothervalueofx.Plotthepointanditspointofreflectionacrosstheaxisofsymmetry.5.Drawa_________________throughthegraphedpoints.

!!y = x2 −2x −3

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Examples3and43.Graph AxisofSymmetry:______________y-intercept:________Doesthisfunctionhaveaminormax(circleone)?Whatistheminimumormaximumfunctionvalue?y=_______________Wheredoestheminormaxoccur?x=__________Whatarethedomainandrange?Domain:___________________Range:___________________

!!y = −x2 +4x −6 4.Graph

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

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Example5:𝑦 = −𝑥! − 2𝑥 + 8 a=_____,b=_____,c=_____ Opens:___________because________AxisofSymmetry:____________Vertex:__________MinMaxof______at________Domain: Range:X-intercepts:__________________ Y-intercept:_________Example6:y= 1x2x5 2 +− a=_____,b=_____,c=_____

Opens:___________because________AxisofSymmetry:____________Vertex:__________MinMaxof______at________Domain: Range:X-intercepts:__________________ Y-intercept:_________

Example7:y=23xx

21 2 −+ a=_____,b=_____,c=_____

Opens:___________because________AxisofSymmetry:____________Vertex:__________MinMaxof______at________Domain: Range:X-intercepts:__________________ Y-intercept:_________

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Graphing with Transformations Date: ____________ Afterthislessonandpractice,Iwillbeableto…

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

domainandrange.(LT7)---------------------------------------------------------------------------------------------------------------------------------------------------Inadditiontographingfromstandardform,wecanalsographbyperforming________________________.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-3.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 Date: ____________ Afterthislessonandpractice,Iwillbeableto…

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

---------------------------------------------------------------------------------------------------------------------------------------------------Wehaveexploredseveralpropertiesofquadraticequationswrittenin_________________formand_______________form.Let’sdiscusstheadvantagesofeachequationform: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! Date: ____________ 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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