GeometricTransformationsNotes

1

AdvancedMath

Mar19:23AM

Atranslationisnothingmorethanageometrictransformationthatslideseachpointinafigurethesamedistanceinthesamedirection

Inthistranslation,CDEisbeingtranslatedtotherightbythesamelengthassegmentAB.

WhatdoyouthinkistrueaboutCDEandC'D'E'?

CDEC'D'E'

Mar19:34AM

Afigurecanbetranslatedinanydirectionandanydistance.

Acartravelingdowntheroadisagoodexampleofatranslationinaction.Theshapeofthecarisnotbeingalteredinanyway,itissimplybeingmovedfromonepointtoanother.

TranslationExample.gsp

GeometricTransformationsNotes

2

AdvancedMath

Aug87:45AM

Translation means SLIDE

Aug138:58AM

TranslationRules

*Totranslateafigureaunitstotheright,increasethexcoordinateofeachpointaunits.

*Totranslateafigureaunitstotheleft,decreasethexcoordinateofeachpointaunits.

*Totranslateafigureaunitsup,increasetheycoordinateofeachpointaunits.

*Totranslateafigureaunitsdown,decreasetheycoordinateofeachpointaunits.

GeometricTransformationsNotes

3

AdvancedMath

Mar110:27AM

Areflectionisatransformationthatflipsafigureacrossalinetocreateit'simage.

Anotherwaytothinkaboutitisthateachpointinareflectedimageisthesamedistanceawayfromthelineofreflectionastheoriginalpointwas.

Whatistrueaboutafigureandit'sreflectedimage?

Aug139:03AM

ReflectionRule

*Thereflectionofthepoint(a,b)acrossthexaxisisthepoint(a,b)

*Thereflectionofthepoint(a,b)acrosstheyaxisisthepoint(a,b)

GeometricTransformationsNotes

4

AdvancedMath

Mar110:32AM

Rotationsareprobablythemostdifficulttypeofgeometrictransformationtounderstand.

Arotationisatransformationthatturns,orspins,afigurearoundapoint.

Mar111:43AM

Afigurecanberotatedaroundapointonthefigureitself.

Inthiscasethetrianglewasrotatedmultipletimesaroundacenterpointthatwasalsoonevertexofthetriangle.

Orafigurecanberotatedaroundapointthatiscompletelyseparatefromthefigureitself

Inthiscase,thesametrianglewasrotatedaroundapointseparatefromthetriangle.

GeometricTransformationsNotes

5

AdvancedMath

Mar111:48AM

Rotationsaregenerallymeasuredbytheangleofrotation.

Thisfigurewouldrepresenta90orotationbecausetheanglecreatedbythecorrespondingverticesandthecenterofrotationisa90oangle.

Thisfigurerepresentsa135orotationforthesamereason.

90o

135o

RotationExample.gsp

Aug138:50AM

1)Therotationofthepoint(x,y)90degreesclockwiseabouttheorigin,isthepoint(y,x).

2)Therotationofthepoint(x,y)180degreesclockwiseabouttheoriginisthepoint(x,y).

3)Therotationofthepoint(x,y)90degreescounterclockwiseabouttheoriginisthepoint(y,x)

RotationRules

GeometricTransformationsNotes

6

AdvancedMath

Aug1311:31AM

A

B C

D

RotateFigureABCD90degreesand180degrees

Aug1311:47AM

Rotatethetriangle90degreesand180degreesusingtherule

GeometricTransformationsNotes

7

AdvancedMath

Mar111:54AM

Dilationsareatypeoftransformationduringwhichwegroworshrinkafigure.

We'veactuallyalreadystudiedthistypeoftransformationquiteindepth?Doesanyonerememberwhen?

ThemajorityofourStretchingandShrinkingunitwasdevotedtosimilarfigurescreatedbydilatingaplanefigure,soI'mnotgoingtospendalotoftimetalkingaboutthem.

Mar18:02PM

Dilationsareusuallymeasuredbytheirscalefactor.

Reminder:Ascalefactorthatisgreaterthanonecreatesalargerimage.

Ascalefactorbetween0and1createsasmallerfigure.

ScaleFactor=3

ScaleFactor=1/2

GeometricTransformationsNotes

8

AdvancedMath

Aug138:55AM

DilationRule

Todilateafigurewithrespecttotheorigin,multiplythecoordinatesofeachofitspointsbythepercentofdilation.

Mar19:17AM

Therearefourbasictransformationsthatwewillbeconcernedwiththisyear.Talktothepeopleatyourableandtrytodefineeachoftheseinyourownwords

Translation Reflection

Rotation Dilation

GeometricTransformationsNotes

9

AdvancedMath

Mar18:24PM

Recap

Translation Reflection

Rotation Dilation

Ashiftoraslidethatcreatesacongruentimage

Aflipthatcreatesacongruentimage.

Aspinorturnthatcreatesacongruentimage

Growingorshrinkingafiguretocreateasimilarfigure.

Mar18:07PM

Inadditiontorecognizinganddescribingthese4basictranformations,therearealsosomerulesthatgovernhowtranslationsbehaveonacoordinategrid.ConsiderquadrilateralCDEFbelow.Whatarethecoordinatesofthe4vertices?

(2,1)

(4,2)

(3,6)

(2,4)

WhatwouldhappentothosecoordinatesifItranslatedthefigure7unitstotheleft?

GeometricTransformationsNotes

10

AdvancedMath

Mar18:17PM

(2,1)

(4,2)

(3,6)

(2,4)

Iwouldhavetosubtract7fromthexvalueofeachcoordinatepair!

(5,1)

(3,2)

(4,6)

(5,4)

Aug138:42AM

Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation

A B

CD

A'

B'C'

D'

GeometricTransformationsNotes

11

AdvancedMath

Aug138:42AM

Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation

A B

CD

A' B'

C'D'

Aug138:42AM

Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation

A B

CD

A' B'

C'D'

GeometricTransformationsNotes

12

AdvancedMath

Aug138:42AM

Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation

A B

CDA' B'

C'D'

Aug1412:08PM

Doyouknowtherules?

Horizontaltranslation:(x,y)=(xa,y)

Verticaltranslation:(x,y)=(x,ya)

Diagonaltranslation:(x,y)=(xa,ya)

Reflectionaboutyaxis:(x,y)=(x,y)

Reflectionaboutxaxis:(x,y)=(x,y)

Rotation90degreesclockwise:(x,y)=(y,x)

Rotation180degrees:(x,y)=(x,y)

Rotation90degreescounterclockwise:(x,y)=(y,x)

Dilationby.5:(x,y)=(x/2,y/2)

Dilationby2:(x,y)=(x2,y2)

Attachments

TranslationExample.gsp

RotationExample.gsp

SMART Notebook

SMART Notebook

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