2.1 Transformers: Shifty y’s - MR. CONGLETON · 2.1 Transformers: Shifty y’s A Develop...
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SECONDARY MATH II // MODULE 2 STRUCTURES OF EXPRESSIONS – 2.1 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 2.1 Transformers: Shifty y’s A Develop Understanding Task Optima Prime is designing a robot quilt for her new grandson. She plans for the robot to have a square face. The amount of fabric that she needs for the face will depend on the area of the face, so Optima decides to model the area of the robot’s face mathematically. She knows that the area A of a square with side length x units (which can be inches or centimeters) is modeled by the function, ! ! = ! ! square units. 1. What is the domain of the function ! ! in this context? 2. Match each statement about the area to the function that models it: Matching Equation (A,B, C, or D) Statement Function Equation The length of each side is increased by 5 units. A !(!) = 5! ! The length of each side is multiplied by 5 units. B ! = (! + 5) ! The area of a square is increased by 5 square units. C ! = (5!) ! The area of a square is multiplied by 5. D ! = ! ! + 5 Optima started thinking about the graph of ! = ! ! (in the domain of all real numbers) and wondering about how changes to the equation of the function like adding 5 or multiplying by 5 affect the graph. She decided to make predictions about the effects and then check them out. CC BY Lily Mulholland https://flic.kr/p/5AdC7o Page 1 x
Transcript of 2.1 Transformers: Shifty y’s - MR. CONGLETON · 2.1 Transformers: Shifty y’s A Develop...
SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.1 Transformers: Shifty y’s
A Develop Understanding Task
OptimaPrimeisdesigningarobotquiltforhernewgrandson.Sheplansfortherobottohaveasquareface.Theamountoffabricthatsheneedsforthefacewilldependontheareaoftheface,soOptimadecidestomodeltheareaoftherobot’sfacemathematically.SheknowsthattheareaAofasquarewithsidelengthxunits(whichcanbeinchesorcentimeters)ismodeledbythefunction, ! ! = !!squareunits.1. Whatisthedomainofthefunction! ! inthiscontext?
2. Matcheachstatementabouttheareatothefunctionthatmodelsit:MatchingEquation
(A,B,C,orD)
Statement FunctionEquation
Thelengthofeachsideisincreasedby5units.
A) AB) !(!) = 5!!
Thelengthofeachsideismultipliedby5units.
C) BD) ! = (! + 5)!
Theareaofasquareisincreasedby5squareunits.
E) CF) ! = (5!)!
Theareaofasquareismultipliedby5. G) DH) ! = !! + 5
Optimastartedthinkingaboutthegraphof! = !!(inthedomainofallrealnumbers)andwonderingabouthowchangestotheequationofthefunctionlikeadding5ormultiplyingby5affectthegraph.Shedecidedtomakepredictionsabouttheeffectsandthencheckthemout.
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3. Predicthowthegraphsofeachofthefollowingequationswillbethesameordifferentfromthegraphof! = !!.
Similaritiestothegraphof! = !!
Differencesfromthegraphof! = !!
! = 5!!
! = (! + 5)!
! = (5!)!
! = !! + 5
4. Optimadecidedtotestherideasusingtechnology.Shethinksthatitisalwaysagoodideatostartsimple,soshedecidestogowith! = !! + 5.Shegraphsitalongwith! = !!inthesamewindow.Testityourselfanddescribewhatyoufind.
5. Knowingthatthingsmakealotmoresensewithmorerepresentations,Optimatriesafewmoreexampleslike! = !! + 2 and! = !! − 3,lookingatbothatableandagraphforeach.Whatconclusionwouldyoudrawabouttheeffectofaddingorsubtractinganumberto! = !!?Carefullyrecordthetablesandgraphsoftheseexamplesinyournotebookandexplainwhyyourconclusionwouldbetrueforanyvalueofk,given,! = !! + !.
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. Afterheramazingsuccesswithadditioninthelastproblem,Optimadecidedtolookatwhathappenswithadditionandsubtractioninsidetheparentheses,orasshesaysit,“addingtothe!beforeitgetssquared”.Usingyourtechnology,decidetheeffectofhintheequations:! = (! + ℎ)!and! = (! − ℎ)!.(Choosesomespecificnumbersforh.)Recordafewexamples(bothtablesandgraphs)inyournotebookandexplainwhythiseffectonthegraphoccurs.
7. Optimathoughtthat#6wasverytrickyandhopedthatmultiplicationwasgoingtobemorestraightforward.Shedecidestostartsimpleandmultiplyby-1,soshebeginswith! = −!!.Predictwhattheeffectisonthegraphandthentestit.Whydoesithavethiseffect?
8. Optimaisencouragedbecausethatonewaseasy.Shedecidestoendherinvestigationforthedaybydeterminingtheeffectofamultiplier,a,intheequation:! = !!!.Usingbothpositiveandnegativenumbers,fractionsandintegers,createatleast4tablesandmatchinggraphstodeterminetheeffectofamultiplier.
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Left ofright depending on H1 I
left right
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Dilate
if a is negative Rx axis
SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.1 2.1
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Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
SET
Topic:TransformationsonquadraticsMatching:Choosetheareamodelthatisthebestmatchfortheequation.
______11.x! + 4 ______12. x + 4 ! ______13. 4x ! _______14.4x!a. b.
c. d.
x -3 -2 -1 0 1 2 3f(x)=x2 9 4 1 0 1 4 9
15a)g(x)= b)x -3 -2 -1 0 1 2 3g(x) 2 -3 -6 -7 -6 -3 2
c) Inwhatwaydidthegraphmove?d) Whatpartoftheequationindicatesthismove?
Atableofvaluesandthegraphforf(x)=x2isgiven.Comparethevaluesinthetableforg(x)tothoseforf(x).Identifywhatstaysthesameandwhatchanges.a)Usethisinformationtowritethevertexformoftheequationofg(x).b)Graphg(x).c)Describehowthegraphchangedfromthegraphoff(x).Usewordssuchasright,left,up,anddown.d)Answerthequestion.
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.2
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.2 Transformers: More Than
Meets the y’s
A Solidify Understanding Task
Writetheequationforeachproblembelow.Useasecondrepresentationtocheckyourequation.1. Theareaofasquarewithsidelengthx,wherethesidelengthisdecreasedby3,theareaismultipliedby2andthen4squareunitsareaddedtothearea.
2.
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.2
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3.
4.
x f(x)
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.2
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Grapheachequationwithoutusingtechnology.Besuretohavetheexactvertexandatleasttwocorrectpointsoneithersideofthelineofsymmetry.
5. ! ! = −!! + 3
6. ! ! = (! + 2)! − 5
7. ℎ ! = 3(! − 1)! + 2
8. Given:! ! = !(! − ℎ)! + !a. Whatpointisthevertexoftheparabola?
b. Whatistheequationofthelineofsymmetry?
c. Howcanyoutelliftheparabolaopensupordown?
d. Howdoyouidentifythedilation?
9. Doesitmatterinwhichorderthetransformationsaredone?Explainwhyorwhynot.
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