GCSEMATHEMATICSKEYTOPIC

PRACTICESHEETS

DRAWINGQUADRATICGRAPHS

(AQAHIGHER)

ThesequestionsaresuitableforHigherTierstudents.

Allquestionsshouldbedonewithoutacalculator.

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1(a) Whichoftheseco-ordinatesisapointonthegraph𝑦 = 𝑥$ + 2𝑥 − 1?Circlethecorrectanswer.

[1mark]

(-1,0)(-4,7)(-3,14)(3,11)1(b) Whichoftheseistrueaboutthegraph𝑦 = 𝑥$ + 2𝑥 − 1? Tickyouranswer.

[1mark]

They-interceptis(0,1)

Thegraphhasaminimum

Theco-ordinate(0,-1)isapointonthecurve

Theshapeofthegraphiscalledaperabolar1(c) Whataretheco-ordinatesoftheturningpointofthegraph𝑦 = 𝑥$ + 2𝑥 − 1? Circlethecorrectanswer.

(1,-2)(-1,2)(-1,-2)(1,2)

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2(a) Completethetableofvaluesfor𝑦 = 5 − 𝑥 − 𝑥$

𝑥 -4 -3 -2 -1 0 1 2

𝑦 -7 5 5 -1

[2marks]2(b) Onthegridbelow,drawthegraphof𝑦 = 5 − 𝑥$ − 𝑥forvaluesof𝑥from-4to2

[2marks]

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3 Davinahasbeengiventhistableofvaluesfor𝑦 = 𝑥$ + 3𝑥 − 4

𝑥 -5 -4 -3 -2 -1 0 1

𝑦 6 0 -4 -6 -6 -4 0

Davina’steacherhasaskedhertodrawthegraphof𝑦 = 𝑥$ + 3𝑥 − 4forvaluesof𝑥from-5 to1.Davina’sgraphisshownbelow.

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WritedownthreemistakesDavinamadewhenshedrewhergraph.[3marks]

Mistake1…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

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4 Mayahasbeengiventhistableofvaluesfor𝑦 = −𝑥$ − 4𝑥 + 3

𝑥 -5 -4 -3 -2 -1 0 1

𝑦 -2 3 6 7 6 3 -2

Maya’steacherhasaskedhertodrawthegraphof𝑦 = −𝑥$ − 4𝑥 + 3forvaluesof𝑥from-5 to1.Maya’sgraphisshownbelow.

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WritedowntwomistakeswiththegraphMayahasdrawn.[2marks]

Mistake1…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

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5(a) Completethetableofvaluesfor𝑓(𝑥) = 𝑥$ + 2𝑥 − 7

𝑥 -4 -3 -2 -1 0 1 2

𝑓(𝑥) 1 -8 -4 1

[2marks]5(b) Onthegridbelow,drawthegraphof𝑓(𝑥)fortherange−4 ≤ 𝑥 ≤ 2

[2marks]

5(c) Useyourgraphtoestimatethevalueof𝑓(−2.5)

[1mark] 𝑓 −2.5 =………………………………………………………………………………………………………………………………………………….……

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6(a) Completethetableofvaluesfor𝑓(𝑥) = −𝑥$ + 4𝑥 − 3

𝑥 -1 0 1 2 3 4 5

𝑓(𝑥) -8 0 1 -3

[2marks]6(b) Onthegridbelow,drawthegraphof𝑓(𝑥) = −𝑥$ + 4𝑥 − 3forvaluesof𝑥from-1to5

[2marks]

6(c) Useyourgraphtoestimatethetwopossiblevaluesof𝑥forwhich𝑦 = −6

[2marks]

………………………………………………………………………and………………………………………………………………………

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7 Thegraphfor𝑦 = 2𝑥$ + 3𝑥 − 5isdrawnbelow.

7(a) Statewhethertheturningpointofthisgraphisaminimumoramaximum.

[1mark] Answer……………………………………………………………………………………………………………………………………………………………………………………………….7(b) Estimatetheco-ordinatesofthisturningpoint. Giveeachofyourestimatestoonedecimalplace.

[2marks]

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8(a) Istheturningpointofthegraphfor𝑦 = −𝑥$ + 𝑥 + 5aminimumoramaximum? Explainyouranswer.

[2marks]

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[1mark] …………………………………………,…………………………………………9 Whichofthesearecorrectstatementsaboutthegraph𝑦 = 𝑥$ − 5𝑥 + 1? Tickallcorrectstatements.

[3marks] Theco-ordinatesofthey-interceptare(0,1) Theturningpointisamaximum Theturningpointisaminimum Thegraphpassesthrough(3,-8)

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10 Theturningpointofthegraph𝑦 = 𝑥$ − 6𝑥 + 10is(a,b). aandbareintegers. Explain,withtheuseofadiagramorotherwise,whytheturningpointofthegraphcannotlie betweenthecoordinates(5,5)and(7,17).

[3marks]

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11 Graphsofthefollowingfunctionshavebeendrawnonthegridbelow Functions

𝑦 = 2𝑥 + 5 𝑦 = 𝑥$ + 6𝑥 + 10 𝑦 = −𝑥$ − 4𝑥 − 3

𝑦 = 2𝑥

EachgraphhasbeenlabelledGraphA,GraphB,GraphCorGraphD.

Matcheachofthefunctionswithitscorrespondinggraphbycompletingthetablebelow.

[3marks]

Graph Function

GraphA

GraphB

GraphC

GraphD

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12(a) Completethetableofvaluesfor𝑦 = 𝑥(3 − 𝑥)

𝑥 -2 -1 0 1 2 3 4

𝑦 -10 2 0 -4

[2marks]

12(b) Onthegridbelow,drawthegraphof𝑦 = 𝑥(3 − 𝑥)forvaluesof𝑥from-2to4

[2marks]

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12(c) Bydrawingtheline𝑦 = 2𝑥 − 2onthesamegrid,solvethesimultaneousequations

𝑦 = 3𝑥 − 𝑥$𝑦 = 2𝑥 − 2

[3marks] 𝑥 =………………………………………………………………………,𝑦 =……………………………………………………………………… and𝑥 =………………………………………………………………………,𝑦 =………………………………………………………………………

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13(a) Completethetableofvaluesfor𝑦 = 𝑥$ − 3𝑥 − 1

𝑥 -1 0 1 2 3 4 5

𝑦 -1 -1 3 9

[2marks]

13(b) Onthegridbelow,drawthegraphof𝑦 = 𝑥$ − 3𝑥 − 1forvaluesof𝑥from-1to5

[2marks]

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Thesolutionof𝑥$ − 3𝑥 − 1 < 2is𝑎 < 𝑥 < 𝑏13(c) Bydrawingasuitablelineonthesamegrid,findestimatesforthevaluesof𝑎and𝑏. Giveyouranswerstoonedecimalplace.

[3marks] 𝑎 =………………………………………………………………………………………………………………………………………………….…… 𝑏 =………………………………………………………………………………………………………………………………………………….……

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14(a) Drawthegraphfor𝑦 = 𝑥$ − 1onthegridbelowforvaluesintherange−3 < 𝑥 < 3

[4marks]

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14(b) Onthesamegrid,drawthegraphof𝑥$ + 𝑦$ = 4[2marks]

14(c) Usingyouranswerstoparts(a)and(b),estimatethesolutionstothesimultaneousequations

𝑥$ + 𝑦$ = 4𝑦 = 𝑥$ − 1

[3marks] 𝑥 =………………………………………………………………………,𝑦 =……………………………………………………………………… and𝑥 =………………………………………………………………………,𝑦 =………………………………………………………………………

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15(a) Completethetableofvaluesfor𝑦 = 2𝑥$ − 2𝑥 − 4

𝑥 -2 -1 0 1 2 3 4

𝑦 0 -4 8 20

[2marks]

15(b) Onthegridbelow,drawthegraphof𝑦 = 2𝑥$ − 2𝑥 − 4forvaluesof𝑥from-2to4

[2marks]

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Useyourgraphtoanswerthefollowingquestions15(c) Bydrawingasuitablelineonthegraph,estimatethegradientofthecurveatthepoint(3,8)

[2marks] Gradient=………………………………………………………………………………………………………………………………………………….……15(d) Estimatetheareaenclosedbetweenthecurveandthex-axis. Youmustshowallyourworkingout.

[4marks] Area=………………………………………………………………………………………………………………………………………………….……

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