3.5 throwing an interception

SECONDARY MATH II // MODULE 3

SOLVING QUADRATIC & OTHER EQUATIONS- 3.5

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

3.5 Throwing an Interception

A Develop Understanding Task

Thex-intercept(s)ofthegraphofafunction are

oftenveryimportantbecausetheyarethesolutiontotheequation .Inpasttasks,we

learnedhowtofindthex-interceptsofthefunctionbyfactoring,whichworksgreatforsome

functions,butnotforothers.Inthistaskwearegoingtoworkonaprocesstofindthex-

interceptsofanyquadraticfunctionthathasthem.We’llstartbythinkingaboutwhatwe

alreadyknowaboutafewspecificquadraticfunctionsandthenusewhatweknowto

generalizetoallquadraticfunctionswithx-intercepts.

1.Whatcanyousayaboutthegraphofthefunction ?

a. Graphthefunction

b. Whatistheequationofthelineofsymmetry?

c. Whatisthevertexofthefunction?

2.Nowlet’sthinkspecificallyaboutthex-intercepts.

a. Whatarethex-interceptsof ?

b. Howfararethex-interceptsfromthelineofsymmetry?

c. Ifyouknewthelineofsymmetrywasthelinex=h,andyouknowhowfarthex-interceptsarefromthelineofsymmetry,howwouldyoufindtheactualx-intercepts?

d. Howfarabovethevertexarethex-intercepts?

e. Whatisthevalueof!(!)atthex-intercepts?

�

f (x)

�

f (x) = 0

�

f (x) = x 2 − 2x − 3

�

f (x) = x 2 − 2x − 3

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Justtomakeitalittleeasiertotalkaboutsomeofthefeaturesthatrelatetotheintercepts,

let’snamethemwithvariables.Fromnowon,whenwetalkaboutthedistancefromthelineof

symmetrytoeitherofthexintercepts,we’llcallitd.Thediagrambelowshowsthisfeature.

Wewillalwaysrefertothelineofsymmetry

asthelinex=h,sothetwox-interceptswill

beatthepoints

(h–d,0)and(h+d,0).

3.So,let’sthinkaboutanotherfunction:

a. Graphthefunctionbyputtingtheequationintovertexform.

b. Whatisthevertexofthefunction?

c. Whatistheequationofthelineofsymmetry?

d. Whatdoyouestimatethex-interceptsofthefunctiontobe?

e. Whatdoyouestimatedtobe?

f. Whatisthevalueof atthex-intercepts?

�

f (x) = x 2 − 6x + 4

�

f (x)

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g. Usingthevertexformoftheequationandyouranswertopart“f”above,writeanequationandsolveittofindtheexactvaluesofthexintercepts.

h. Whatistheexactvalueofd?

i. Useacalculatortofindapproximationsforthex-intercepts.Howdotheycomparewithyourestimates?

4. Whataboutafunctionwithaverticalstretch?Canwefindexactvaluesforthex-interceptsthesameway?Let’stryitwith: .

a. Graphthefunctionbyputtingtheequationintovertexform.

b. Whatisthevertexofthefunction?

c. Whatistheequationofthelineofsymmetry?

d. Whatdoyouestimatethex-interceptsofthefunctiontobe?

e. Whatdoyouestimatedtobe?

f. Whatisthevalueof atthex-intercepts?

�

f (x) = 2x 2 − 8x + 5

�

f (x)

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g. Usingthevertexformoftheequationandyouranswerto“f”above,writeanequationandsolveittofindtheexactvaluesofthex-intercepts.

h. Whatistheexactvalueofd?

i. Compareyoursolutiontoyourestimateoftheroots.Howdidyoudo?

5. Finally,let’strytogeneralizethisprocessbyusing:torepresentanyquadratic

functionthathasx-intercepts.Here’sapossiblegraphof .

a. Starttheprocesstheusualwaybyputtingtheequationintovertexform.It’salittletricky,butjustdothesamethingwitha,b,andcaswhatyoudidinthelastproblemwiththenumbers.

b. Whatisthevertexoftheparabola?

�

f (x) = ax 2 + bx + c

�

f (x)

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c. Whatisthelineofsymmetryoftheparabola?

d. Writeandsolvetheequationforthex-interceptsjustasyoudidpreviously.

6. Howcouldyouusethesolutionsyoujustfoundtotellwhatthex-interceptsareforthefunction ?

7. Youmayhavefoundthealgebraforwritingthegeneralquadraticfunctioninvertexformabitdifficult.Hereisanotherwaywecanworkwith

thegeneralquadraticfunctionleadingtothesameresultsyoushouldhavearrivedatin5d.

a. Sincethetwox-interceptsaredunitsfromthelineofsymmetryx=h,ifthequadraticcrossesthex-axisitsx-interceptsareat(h–d,0)and(h+d,0).Wecanalwayswritethefactoredformofaquadraticifweknowitsx-intercepts.Thefactoredformwilllooklike

�

f (x) = a(x − p)(x − q) wherepandqarethetwox-intercepts.So,usingthisinformation,writethefactoredformofthegeneralquadratic using

thefactthatitsx-interceptsareath-dandh+d.

b. Multiplyoutthefactoredform(youwillbemultiplyingtwotrinomialexpressionstogether)togetthequadraticinstandardform.Simplifyyourresultasmuchaspossiblebycombiningliketerms.

�

f (x) = x 2 − 3x −1

�

f (x) = ax 2 + bx + c

�

f (x) = ax 2 + bx + c

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c. Younowhavethesamegeneralquadraticfunctionwritteninstandardformintwodifferentways,onewherethecoefficientsofthetermsarea,bandcandonewherethecoefficientsofthetermsareexpressionsinvolvinga,handd.Matchupthecoefficients;thatis,b,thecoefficientofxinoneversionofthestandardformisequivalentto________intheotherversionofthestandardform.Likewisec,theconstantterminoneversionofthestandardformisequivalentto_________intheother.

d. Solvetheequationsb=________andc=________forhandd.Workwithyourequationsuntilyoucanexpresshanddwithexpressionsthatonlyinvolvea,bandc.

e. Basedonthiswork,howcanyoufindthex-interceptsofanyquadraticusingonlythevaluesfora,bandc?

f. Howdoesyouranswerto“e”comparetoyourresultin5d?

8. Allofthefunctionsthatwehaveworkedwithonthistaskhavehadgraphsthatopen

upward.Wouldtheformulaworkforparabolasthatopendownward?Tellwhyorwhynotusinganexamplethatyoucreateusingyourownvaluesforthecoefficientsa,b,andc.

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SOLVING QUADRATICS & OTHER EQUATIONS – 3.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

3.5

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READY Topic:Convertingmeasurementofarea,areaandperimeter.Whileworkingwithareasissometimesessentialtoconvertbetweenunitsofmeasure,forexamplechangingfromsquareyardstosquarefeetandsoforth.Converttheareasbelowtothedesiredmeasure.(Hint:areaistwodimensional,forexample1yd2=9ft2because3ftalongeachsideofasquareyardequals9squarefeet.)1.7yd2=?ft2 2.5ft2=?in2 3.1mile2=?ft2

4.100m2=?cm2 5.300ft2=?yd2 6.96in2=?ft2

SET Topic:Transformationsandparabolas,symmetryandparabolas7a.Grapheachofthequadraticfunctions.

! ! = !!! ! = !! − 9

ℎ ! = (! + 2)! − 9b.Howdothefunctionscomparetoeachother?c.Howdog(x)andh(x)comparetof(x)?

d.Lookbackatthefunctionsaboveandidentifythex-interceptsofg(x).Whatarethey?e.Whatarethecoordinatesofthepointscorrespondingtothex-interceptsing(x)ineachoftheotherfunctions?Howdothesecoordinatescomparetooneanother?

READY, SET, GO! Name PeriodDate

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3.5 throwing an interception

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