ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam

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Workthemultiplechoicequestionsfirst,choosingthesinglebestresponsefromthechoicesavailable.Indicateyouranswerhereandonyouranswersheet.Then,attemptthetiebreakerquestionsattheendstartingwithTieBreaker#1,then#2,andfinally#3.Turninyouranswersheetandthetiebreakerpageswhenyouarefinished.Youmaykeepthepageswiththemultiple-choicequestions.Figuresaren’tnecessarilydrawntoscale.Anglesaregiveninradiansunlessotherwisestated.

1. Given𝑓 𝑥 = 𝑥$ − 6𝑥,expand𝑓 𝑥 − '$.

a. 𝑥$ − 6𝑥 − '

$

b. 𝑥$ − 5𝑥 − '')

c. 𝑥$ − 6𝑥 + '')

d. 𝑥$ − 7𝑥 + ',)

2. Considerthegraphsandtheirequationsbelow.

Whichcoefficient(A,B,C,orD)hastheleastvalue?

a. Ab. Bc. Cd. De. Unabletodetermine

3. Identifytheverticalasymptote(s)ofthefunction𝑓 𝑥 = -../01)

a. 𝑥 = 0b. 𝑥 = 8c. 𝑥 = ±8d. 𝑥 = 64e. Therearenoverticalasymptotes.

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4. Given𝑓 𝑥 = 𝑥$ − 4and𝑔 𝑥 = '.,whatisthedomainofthefunction𝑓 𝑔 𝑥 .

a. −'

$, '$

b. −2,2 c. −'

$, 0 ∪ 0, '

$

d. −∞, 0 ∪ 0,∞ e. −∞,−2 ∪ 2,∞

5. AhomeownermowsthelawneveryWednesdayafternoon.Letℎbeafunctionthatdeterminestheheightofthegrass(ininches)intermsoftheamountoftimethathaspassedsinceWednesdayafternoon(indays).Whichofthefollowingisthecorrectinterpretationoftheequationℎ 2 = 5?

a. OnMondayafternoon,thegrassis2inchestall.b. OnFridayafternoon,thegrassis5inchestall.c. OnWednesdayafternoon,thegrasshasjustbeencut.d. Twotimestheheightofthegrassis5inches.

6. Determinethevalueofsin 𝜃 iftan 𝜃 = − '$-andcos 𝜃 > 0.

a. sin 𝜃 = − -'$

b. sin 𝜃 = − ','$

c. sin 𝜃 = − -',

d. sin 𝜃 = − '$',

e. Cannotbedetermined

7. Thegraphof𝑓(𝑥)isshowntotheright.Statetheinterval(s)of𝑥onwhich𝑓isincreasing.

a. −2.12,0.786 b. −∞, 1.015 ∪ −2.052,∞ c. 1.015,−2.052 d. −∞,−2.12 ∪ 0.786,∞ e. Unabletodetermine

ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam

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8. Findallvalidvaluesfor𝜃inthefunctionbelow.Assumethat𝑘isaninteger.

𝑊 𝜃 =sin 𝜃

cos 𝜃 + 2

a. 𝜃 ≠ −2b. 𝜃 = −2c. 𝜃 ≠ 𝑘𝜋d. 𝜃 = 𝑘𝜋e. Noneoftheabove.

9. Forthefunction𝑓 𝑥 = 2 − $,𝑥,findtheaveragerateofchangeof𝑓(𝑥)as𝑥increases

from−3to1.

a. 2b. −$

,

c. 4d. )

,

e. Unabletodetermine

10. Markworksasacookandspendsallthemoneyheearnsongraphictees.Let𝑤beafunctionthatdeterminestheamountofmoneyMarkhasearned(𝑚)intermsofthenumberofhourshehasworked(ℎ).Let𝑔beafunctionthatdeterminesthenumberofgraphicteesMarkcanbuy(𝑛)intermsoftheamountofmoneyMarkhasearned(𝑚).Whichofthefollowingcompositionsoffunctionsmakessense?

a. 𝑤 𝑚 𝑛 b. 𝑤 𝑔 𝑚 c. 𝑔 𝑤 ℎ d. ℎ 𝑚 𝑛 e. Unabletodeterminewiththeinformationgiven.

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11. Thegraphof𝑓isshowntotheright.Choosethegraphclosestto𝑓0'.

a. b.

c. d.

e. Noneoftheabove.

12. Determineifthefunction𝑓 𝑥 = $.Q,-.Q'

isone-to-one.Ifitisone-to-one,find𝑓0' 𝑥 .

a. 𝑓0' 𝑥 = − $.Q,-.Q'

b. 𝑓0' 𝑥 = ,.Q$.Q-

c. 𝑓0' 𝑥 = 0,.Q'-.Q$

d. 𝑓0' 𝑥 = 0.Q,-.0$

e. Thisfunctionisnotone-to-one

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Forproblems13and14,considerthefollowingrationalfunction.

𝑅 𝑥 =3𝑥$

𝑥 + 𝑥$+ 50

13. Finishthestatement:As𝑥increaseswithoutbound,𝑅 𝑥 approaches…

a. −1, 0b. 3c. 50d. 53e. +∞

14. Finishthestatement:As𝑥approaches−1fromtheleft,𝑅(𝑥)approaches…

a. +∞b. −∞c. 50d. 0e. Unabletodetermine.

15. Maryboughtausedcarfor$5000andthesalespersontoldherthatthevalueofthecarisexpectedtodepreciateatarateof8%peryear.WhichofthefollowingmodelsthevalueofMary’scarwithrespecttothenumberofyearsthathavepassedsincesheboughtit?

a. 𝑀 𝑛 = 5000 − 0.08𝑛b. 𝑀 𝑛 = 5000 0.92 Wc. 𝑀 𝑛 = 5000 0.08 Wd. 𝑀 𝑛 = 5000 − 0.92𝑛e. Unabletodetermine

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Usethefollowingtrianglefigureinproblems16and17.

16. Thelawofcosinesandthelawofsinescanbeusedtofindthelengthofthesidesofanarbitrary(non-right)triangle.Applyoneoftheselawstofindthelengthofside𝑎intheabovefigure.

a. 17.5𝑢𝑛𝑖𝑡𝑠b. 21.5𝑢𝑛𝑖𝑡𝑠c. 306.0𝑢𝑛𝑖𝑡𝑠d. 460.9𝑢𝑛𝑖𝑡𝑠e. Unabletodetermine

17. Determinetheareaoftheabovetriangle.

a. 55.3𝑢𝑛𝑖𝑡𝑠$b. 67.5𝑢𝑛𝑖𝑡𝑠$c. 96.6𝑢𝑛𝑖𝑡𝑠$d. 161.0𝑢𝑛𝑖𝑡𝑠$e. Theareacannotbefound.

18. Definelog_ 𝐴 = 10andlog_(𝐵) = 2.Usethesetoevaluatelog_b/

c.

a. 𝑥 = 5b. 𝑥 = 8c. 𝑥 = 18d. 𝑥 = 50e. 𝑥 = 98

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19. Anangleof2.4radiansinterceptsacircleofradius3.2,asshowninthefiguretotheright.Findthelengthofthearc𝑆.

a. 1.33b. 2.00c. 7.49d. 7.68

20. Whichofthefollowingisapointontheunitcircleandtheterminalsideoftheangle−)e

,?

a. −'$, − 𝟑

𝟐

b. −'$, ,$

c. − 𝟑𝟐, − '

$

d. − 𝟑𝟐, '$

e. ,$, − '

$

21. Usethepropertiesoflogarithmstocombinethefollowingexpressionintoasinglelogarithm.2 log 𝑧 − 3 log 𝑥 + 1

a. log 𝑧$ 𝑥 + 1 ,

b. log i/

.Q' j

c. log $i,.Q,

d. log 2𝑧 − 3𝑥 − 3 e. Noneoftheabove.

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22. Solvethefollowingequation.Giveallsolutionsintheinterval0 ≤ 𝜃 < 2𝜋.

tan 𝜃 +𝜋3

= 1

a. 𝜃 = ''e

'$, $,e'$

b. 𝜃 = $e,, -e,

c. 𝜃 = e)

d. 𝜃 = − e'$

e. Noneoftheabove.

23. Afiremanhasa20-footladderthatshecanusetorescuesomeonefromawindow.Thesafetyofficerdeterminesthatthisladdercanbeleanedagainstaverticalwallwithaminimumangleofelevationof50°andamaximumangleofelevationof75°.Giventhesetwoconstraints,whatisthedifferenceinminimumandmaximumheightsthattheladdercansafelyreach?

a. 4.0feetb. 8.5feetc. 15.3feetd. 19.3feete. Unabletodetermine.

24. Let𝜃beanangleinquadrantIVsuchthatcos 𝜃 = pq.Findtheexactvalueofcsc 𝜃.

a. csc 𝜃 = − 'rq

b. csc 𝜃 = q'r

c. csc 𝜃 = 'rq

d. csc 𝜃 = − q'r

25. Usetrigonometricidentitiestosimplifythefollowingexpression.sin 𝑥 sec 𝑥tan$ 𝑥

a. cot 𝑥b. 1c. cos 𝑥d. tan$ 𝑥e. Noneoftheabove.

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TieBreaker#1Name:____________________School:___________________Given𝑓 𝑥 = 3𝑥$ − 5𝑥 + 1,fullysimplifythedifferencequotient:t uQv 0t u

v

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TieBreaker#2Name:____________________School:___________________Yourgrandmotherhasdecidedtopushlifetotheedgeandtrybungeejumping!Sheisstandinginthecenterofabridgeandjumpswithnoinhibitions.Inthefunofitall,younoticethattherelationshipofherheightwithrespecttotheamountoftimethathaspassedsinceshejumpedformsaperfectparabolicshape.Thebridgeis100feetintheair.ThreesecondsafterjumpingGrandma’sbungeecordiscompletelytaughtandsheisatherminimumheightof40feetfromtheground.WriteaquadraticfunctionthatwouldrelateGrandma’sheightaboveground(infeet)withrespecttothenumberofsecondsthathavepassedsinceshejumped.

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TieBreaker#3Name:____________________School:___________________Verifythefollowingtrigonometricidentity:

1sin 𝑥

−sin 𝑥

1 + cos 𝑥= cot 𝑥

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ACTMPrecalculusKey20191. D

2. B

3. C

4. C

5. B

6. D

7. D

8. E

9. B

10. C

11. D

12. D

13. D

14. A

15. B

16. B

17. A

18. C

19. D

20. B

21. B

22. A

23. A

24. D

25. A

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TieBreakerNumber1SolutionGiven𝑓 𝑥 = 3𝑥$ − 5𝑥 + 1,simplifythedifferencequotient:t uQv 0t u

v

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TieBreakerNumber2SolutionYourgrandmotherhasdecidedtopushlifetotheedgeandtrybungeejumping!Sheisstandinginthecenterofabridgeandjumpswithnoinhibitions.Inthefunofitall,younoticethattherelationshipofherheightwithrespecttotheamountoftimethathaspassedsinceshejumpedformsaperfectparabolicshape.Thebridgeis100feetintheair.ThreesecondsafterjumpingGrandma’sbungeecordiscompletelytaughtandsheisatherminimumheightof40feetfromtheground.WriteaquadraticfunctionthatwouldrelateGrandma’sheightaboveground(infeet)withrespecttothenumberofsecondsthathavepassedsinceshejumped.

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TieBreakerNumber3SolutionVerifythefollowingtrigonometricidentity:

1sin 𝑥

−sin 𝑥

1 + cos 𝑥= cot 𝑥