GraphsandEquationsofPolynomialFunctions
Youaregettingmorecomfortablewritingtheequationforapolynomialinfactoredform:
𝑓 𝑥 = 𝑎 𝑥 − 𝑧! 𝑥 − 𝑧! 𝑥 − 𝑧! …whengivenagraph.Youarealsoprettyconfident
sketchingthegraphofapolynomialusingit’sfactoredform.Todaywearegoingtoreinforce
thoseskills,whileintegratingtheuseoftechnologyandourgeneralknowledgeofthe
behaviorofpolynomials.
1. GraphtoEquationGiventhegraphofthe
polynomialfunction,answerthefollowing.
a. Justbylookingatthegraph,whatdegreeshouldit
haveandwhy?3,3zerosand2turningpoints
b. Justbylookingatthegraph,whatshouldthesign
(+or-)oftheleadingcoefficient,“a”,be?Why?
Negative,positivexvalueshavenegativeyvalues.
c. Usingthezeros,writeanequationforthegraph.Be
suretoincludeanunknown‘a’value.
y=a(x+2)(x–1)(x–0)
d. Usingthefactthat(−1, −4)isonthegraph,findthevalueoftheleadingcoefficient
(the“a”value).a=-2
y=-2(x+2)(x–1)(x–0)
e. UsingyourTI-84calculator,findtheexactcoordinatesofthemaximumandminimum
values(theturningpoints).
Max:(0.549,1.262)
Min:(-1.215,-4.225)
2.
EquationtoGraphConsiderthepolynomialfunction𝑓 𝑥 = 𝑥 − 1 𝑥 + 1 𝑥 − 2 𝑥 + 2 .
a. Findthey-interceptofthefunction
(0,4)
b. ACCURATELYsketchagraphofthefunctionusing
thexandy-intercepts.
VerifyusingDesmosorcalculator
3.
c. Usingyourcalculator,findtheexactcoordinatesofalltheturningpoints.Pleasespecify
whethertheyareamaxoramin.
Max:(0,4)
Min:(-1.581,-2.25)and(1.581,-2.25)
FactoringapolynomialConsiderthepolynomial𝑦 = 𝑥 ! − 𝑥 ! − 6𝑥.
a. Findthey-interceptofthefunction.
(0,0)
b. Beforeyoumakeanycalculations,howmanyx-intercepts(atmost)couldthispolynomial
haveandwhy?3,it’sa3rddegreepolynomial.
c. Completelyfactorthepolynomial.Hint:StartbyfirstfactoringoutaGCFfromallthreeterms.
y=x(x–3)(x+2)
d. Findallx-interceptsofthepolynomial.
(0,0),(3,0),(-2,0)
e. Usingthexandyintercepts,sketchagraphofthe
polynomial.VerifywithDesmos.
Theremainingproblemsaretobecompletedforhomework.
4. Giventhegraphofthepolynomialfunction,answer
thefollowing.
a. Justbylookingatthegraph,whatdegreeshould
ithaveandwhy?4,4zerosand3turns.
b. Justbylookingatthegraph,whatshouldthesign
(+or-)oftheleadingcoefficientbe?Why?
+,thepositivexvalueshavepositiveyvalues.
c. Usingthezeros,writeanequationforthegraph.
Besuretoincludeanunknown‘a’value.
y=a(x+3)(x+1)(x–1)(x–3)
d. Usingthefactthat(0, 9)isthey-intercept,findthevalueoftheleadingcoefficient(the“a”
value).
y=(x+3)(x+1)(x–1)(x–3)
e. UsingyourTI-84calculator,findtheexactcoordinatesofthemaximumandminimumvalues
(theturningpoints).
Max:(0,9) Min:(-2.236,-16)and(2.236,-16)
5.
Considerthepolynomialfunction𝑝 𝑥 = 𝑥 ! − 2𝑥 ! − 8𝑥.
a. Findthey-interceptofthefunction
(0,0)
b. Completelyfactorthepolynomial.Hint:Startbyfirst
factoringoutaGCFfromallthreeterms.
p(x)=x(x–4)(x+2)
c. ACCURATELYsketchagraphofthefunctionusingthexandy-intercepts.
VerifywithDesmos
d. Usingyourcalculator,findtheexactcoordinatesofalltheturningpoints.Please
specifywhethertheyareamaxoramin.
Max:(-1.097,5.049) Min:2.431,-16.901)
6.
a.
b.
c.
d.
EquationtoGraphConsiderthepolynomialfunction𝑔 𝑥 = −𝑥(𝑥 + 2)(𝑥 − 2)(𝑥 − 3).
Isthegraphgoingopenuporopendown?Why?
Down,thereisanegative“a”value.
Findthey-interceptofthefunction.
(0,0)
ACCURATELYsketchagraphofthefunctionusingthex
andy-intercepts.Rescaletheaxesasnecessary.
VerifywithDesmos
Usingyourcalculator,findtheexactcoordinatesofalltheturningpoints.Pleasespecify
whethertheyareamaxoramin.
Min:(0.93,-6.035)Max(-1.254,12.949)and(2.574,2.879)