Geogebra1 Copy 1

8/2/2019 Geogebra1 Copy 1

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GeoGebraHelpOfficialManual3.2

MarkusHohenwarterandJudithHohenwarter

www.geogebra.org


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GeoGebraHelp3.2

Lastmodified:April22,2009

Authors

MarkusHohenwarter,[email protected]

JudithHohenwarter,[email protected]

GeoGebraOnline:http://www.geogebra.org


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Contents

1. WHATISGEOGEBRA?..........................................................................................................61.1. MultipleViewsforMathematicalObjects.......................................................................................6

1.1.1. GraphicsView...................................................................................................................................6

1.1.2. AlgebraView....................................................................................................................................7

1.1.3. SpreadsheetView.............................................................................................................................8

1.2. GeoGebraasaToolforTeachingandLearningMathematics...........................................................8

1.2.1. CustomizingtheUserInterface........................................................................................................8

1.2.2. ThePropertiesDialog..................................................................................................................... 10

1.2.3. TheContextMenu..........................................................................................................................10

1.3. GeoGebraasaPresentationTool..................................................................................................11

1.3.1. TheNavigationBar.........................................................................................................................111.3.2. TheConstructionProtocol..............................................................................................................11

1.3.3. CustomizetheSettings................................................................................................................... 12

1.4. GeoGebraasanAuthoringTool....................................................................................................13

1.4.1. PrintingOptions.............................................................................................................................13

1.4.2. CreatingPicturesoftheGraphicsView...........................................................................................13

1.4.3. CreatingInteractiveWebpages......................................................................................................14

2. GEOMETRICINPUT............................................................................................................162.1. GeneralNotes..............................................................................................................................16

2.2. ConstructionTools

........................................................................................................................

16

2.2.1. GeneralTools.................................................................................................................................17

2.2.2. PointTools.....................................................................................................................................18

2.2.3. VectorTools...................................................................................................................................19

2.2.4. SegmentTools................................................................................................................................19

2.2.5. RayTool..........................................................................................................................................20

2.2.6. PolygonTools.................................................................................................................................20

2.2.7. LineTools.......................................................................................................................................20

2.2.8. ConicSectionTools.........................................................................................................................21

2.2.9. ArcandSectorTools....................................................................................................................... 22

2.2.10. NumberandAngleTools............................................................................................................23

2.2.11. BooleanVariableTool................................................................................................................25

2.2.12. LocusTool..................................................................................................................................25

2.2.13. GeometricTransformation Tools...............................................................................................252.2.14. TextTool....................................................................................................................................26

2.2.15. ImageTool.................................................................................................................................28

3. ALGEBRAICINPUT.............................................................................................................303.1. GeneralNotes..............................................................................................................................30

3.2. DirectInput..................................................................................................................................32


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3.2.1. NumbersandAngles......................................................................................................................32

3.2.2. PointsandVectors..........................................................................................................................33

3.2.3. LinesandAxes................................................................................................................................33

3.2.4. ConicSections................................................................................................................................34

3.2.5. Functionsofx.................................................................................................................................34

3.2.6. PredefinedFunctionsandOperations...........................................................................................35

3.2.7. BooleanVariablesandOperations.................................................................................................36

3.2.8. ListObjectsandOperations............................................................................................................373.2.9. MatrixObjectsandOperations.......................................................................................................38

3.2.10. ComplexNumbersandOperations............................................................................................38

3.3. Commands...................................................................................................................................39

3.3.1. GeneralCommands........................................................................................................................40

3.3.2. BooleanCommands....................................................................................................................... 40

3.3.3. NumberCommands....................................................................................................................... 41

3.3.4. AngleCommand.............................................................................................................................45

3.3.5. PointCommands............................................................................................................................45

3.3.6. VectorCommands..........................................................................................................................47

3.3.7. SegmentCommand........................................................................................................................48

3.3.8. RayCommand................................................................................................................................48

3.3.9. PolygonCommand.........................................................................................................................493.3.10. LineCommands.........................................................................................................................49

3.3.11. ConicSectionCommands...........................................................................................................51

3.3.12. FunctionCommands..................................................................................................................52

3.3.13. ParametricCurveCommand......................................................................................................53

3.3.14. ArcandSectorCommands.........................................................................................................54

3.3.15. TextCommands.........................................................................................................................55

3.3.16. LocusCommand........................................................................................................................58

3.3.17. ListandSequenceCommands...................................................................................................58

3.3.18. GeometricTransformation Commands......................................................................................61

3.3.19. StatisticsCommands..................................................................................................................63

3.3.20. SpreadsheetCommands............................................................................................................67

3.3.21. MatrixCommands..................................................................................................................... 67

4. MENUITEMS........................................................................................................................694.1. FileMenu.....................................................................................................................................69

4.2. EditMenu.....................................................................................................................................71

4.3. ViewMenu...................................................................................................................................73

4.4. OptionsMenu..............................................................................................................................74

4.5. ToolsMenu..................................................................................................................................76

4.6. Window

Menu

..............................................................................................................................

77

4.7. HelpMenu...................................................................................................................................77

5. SPECIALGEOGEBRAFEATURES.....................................................................................795.1. Animation....................................................................................................................................79

5.1.1. AutomaticAnimation..................................................................................................................... 79

5.1.2. ManualAnimation..........................................................................................................................79


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5.2. ConditionalVisibility.....................................................................................................................80

5.3. UserDefinedTools.......................................................................................................................81

5.4. DynamicColors.............................................................................................................................82

5.5. JavaScriptInterface......................................................................................................................82

5.6. KeyboardShortcuts......................................................................................................................83

5.7. LabelsandCaptions......................................................................................................................87

5.8. Layers...........................................................................................................................................87

5.9. Redefine.......................................................................................................................................88

5.10. TraceandLocus............................................................................................................................88

6. INDEX.....................................................................................................................................90


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1. WhatisGeoGebra?GeoGebraisdynamicmathematicssoftwarethatjoinsgeometry,algebraandcalculus.ItisdevelopedforlearningandteachingmathematicsinschoolsbyMarkusHohenwarterandan

internationalteamofprogrammers.

1.1. MultipleViewsforMathematicalObjectsGeoGebraprovidesthreedifferentviewsofmathematicalobjects:aGraphicsView,a,

numericAlgebraView,andaSpreadsheetView.Theyallowyoutodisplaymathematical

objectsinthreedifferentrepresentations:graphically(e.g.,points,functiongraphs),

algebraically(e.g.,coordinatesofpoints,equations),andinspreadsheetcells.Thereby,allrepresentationsofthesameobjectarelinkeddynamicallyandadaptautomaticallyto

changesmadetoanyoftherepresentations,nomatterhowtheywereinitiallycreated.

1.1.1. GraphicsViewUsingtheconstructiontoolsavailableintheToolbaryoucandogeometricconstructionsin

theGraphicsViewwiththemouse.SelectanyconstructiontoolfromtheToolbarandread

theToolbarHelp(nexttotheToolbar)inordertofindouthowtousetheselectedtool.Any

objectyoucreateintheGraphicsViewalsohasanalgebraicrepresentationintheAlgebra

View.

AlgebraView

GraphicsView

Spreadsheet

View

InputBar

Toolbar Toolbar

Help


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Note:Afteractivatingthetool MoveyouareabletomoveobjectsintheGraphicsViewby

draggingthemwiththemouse.Atthesametime,theiralgebraicrepresentationsare

dynamicallyupdatedintheAlgebraView.

EveryiconintheToolbarrepresentsatoolboxthatcontainsaselectionofsimilar

constructiontools.Inordertoopenatoolbox,youneedtoclickonthesmallarrowinthe

lowerrightcorneroftheToolbaricon.

Hint:Constructiontoolsareorganizedbythenatureofresultingobjectsorthefunctionality

ofthetools.YouwillfindtoolsthatcreatedifferenttypesofpointsinthePointToolbox

(defaulticon )andtoolsthatallowyoutoapplygeometrictransformationsinthe

TransformationToolbox(defaulticon ).

1.1.2. AlgebraViewUsingtheInputBaryoucandirectlyenteralgebraicexpressionsinGeoGebra.Afterhitting

theEnterkeyyouralgebraicinputappearsintheAlgebraViewwhileitsgraphicalrepresentationisautomaticallydisplayedintheGraphicsView.

Example:Theinputf(x) = x^2givesyouthefunctionfintheAlgebraViewandits

functiongraphintheGraphicsView.

IntheAlgebraView,mathematicalobjectsareorganizedasfreeanddependentobjects.If

youcreateanewobjectwithoutusinganyotherexistingobjects,itisclassifiedasafree

object.Ifyournewlycreatedobjectwascreatedbyusingotherexistingobjects,itis

classifiedasadependentobject.

Hint:IfyouwanttohidethealgebraicrepresentationofanobjectintheAlgebraView,you

mayspecifytheobjectasanauxiliaryobject:Rightclick(MacOS:Ctrlclick)onthe

correspondingobjectintheAlgebraViewandselectPropertiesfromtheappearingContextMenu.OntabBasicofthePropertiesDialogyoumayspecifytheobjectasanAuxiliary

Object.Bydefault,auxiliaryobjectsarenotshownintheAlgebraView,butyoucanchange

thissettingbyselectingtheitemAuxiliaryObjectsfromtheViewmenu.

NotethatyouareabletomodifyobjectsintheAlgebraViewaswell:Makesurethatyou

activatethe MovetoolbeforeyoudoubleclickonafreeobjectintheAlgebraView.In

theappearingtextboxyoucandirectlyeditthealgebraicrepresentationoftheobject.After

hittingtheEnterkey,thegraphicalrepresentationoftheobjectwillautomaticallyadaptto

yourchanges.

IfyoudoubleclickonadependentobjectintheAlgebraView,adialogwindowappears

allowingyoutoRedefinetheobject.

GeoGebraalsooffersawiderangeofcommandsthatcanbeenteredintotheInputBar.You

canopenthelistofcommandsintherightcorneroftheInputBarbyclickingonthebutton

Command.Afterselectingacommandfromthislist(ortypingitsnamedirectlyintothe

InputBar)youcanpresstheF1keytogetinformationaboutthesyntaxandarguments

requiredtoapplythecorrespondingcommand.


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1.1.3. SpreadsheetViewInGeoGebrasSpreadsheetVieweverycellhasaspecificnamethatallowsyoutodirectly

addresseachcell.Forexample,thecellincolumnAandrow1isnamedA1.

Note:Thesecellnamescanbeusedinexpressionsandcommandsinordertoaddressthe

contentofthecorrespondingcell.

Inthespreadsheetcellsyoucanenternotonlynumbers,butalltypesofmathematical

objectsthataresupportedbyGeoGebra(e.g.,coordinatesofpoints,functions,commands).

Ifpossible,GeoGebraimmediatelydisplaysthegraphicalrepresentationoftheobjectyou

enteredinaspreadsheetcellintheGraphicsViewaswell.Thereby,thenameoftheobject

matchesthenameofthespreadsheetcellusedtoinitiallycreateit(e.g.,A5,C1).

Note:Bydefault,spreadsheetobjectsareclassifiedasauxiliaryobjectsintheAlgebraView.

YoucanshoworhidetheseauxiliaryobjectsbyselectingAuxiliaryObjectsfromtheView

menu.

1.2. GeoGebraasaToolforTeachingandLearningMathematics1.2.1. CustomizingtheUserInterfaceTheuserinterfaceofGeoGebracanbecustomizedbyusingtheViewmenu.Forexample,

youcanhidedifferentpartsoftheinterface(e.g.,theAlgebraView,SpreadsheetView,or

InputBar)bycheckingoruncheckingthecorrespondingmenuitemintheViewmenu.

ShowingandHidingObjects

YoumayshoworhideobjectsintheGraphicsViewindifferentways.

Youmayusetool Show/HideObjecttoshoworhideobjects. OpentheContextMenuandselectitem ShowObjecttochangethevisibility

statusoftheselectedobject.

IntheAlgebraView,theicontotheleftofeveryobjectshowsitscurrentvisibilitystate( shownor hidden).Youmaydirectlyclickonthelittlemarbleiconin

ordertochangethevisibilitystatusofanobject.

Youcanalsousethetool CheckBoxtoShow/HideObjectsinordertoshoworhideoneorseveralobjects.

Customizingthe

GraphicsView

InordertoadjustthevisiblepartofthedrawingpadintheGraphicsView,youcandragthe

drawingpadbyusingtool MoveDrawingPadandusethefollowingwaysofzooming:

Youmayusethetools ZoomInand ZoomOutinordertozoomintheGraphicsView.

Note:Thepositionofyourclickdeterminesthecenterofzoom.

YoumayusethescrollwheelofyourmouseinordertozoomintheGraphicsView.


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Youmayusekeyboardshortcutstozoomin(Ctrl+)andtozoomout(Ctrl). Afterrightclicking(MacOS:Ctrlclick)onanemptyspotonthedrawingpada

ContextMenuappearswhichallowsyoutoZoom.

Youmayspecifyazoomrectanglebyrightclicking(MacOS:Cmdclick)onanemptyspotintheGraphicsViewanddraggingthemousetotheoppositecornerofyour

desiredzoomrectangle.Releasethemousebuttoninordertofinishthezoom

rectangle,whichwillthenautomaticallyadjusttofillallthespaceintheGraphics

View.

YoucanalsoshoworhidethecoordinateaxesandacoordinategridintheGraphicsViewby

usingtheViewmenu.

Note:Anotherwayofshowingorhidingtheaxesandthegridisbyrightclicking(MacOS:

Ctrlclick)onthedrawingpadandselectingthecorrespondingitems Axesor Gridfrom

theappearingContextMenu.

CustomizingCoordinateAxesandGrid

ThecoordinateaxesandgridcanbecustomizedusingthePropertiesDialogoftheGraphicsView.Afterrightclicking(MacOS:Ctrlclick)onthedrawingpad,youcanopenthisdialog

windowbyselectingPropertiesfromtheappearingContextMenuoftheGraphicsView.

OntabAxes,youcan,forexample,changethelinestyleandunitsofthecoordinateaxes,andsetthedistanceofthetickmarkstoacertainvalue.Notethatyoucan

customizebothaxesindividually,byclickingontabsxAxisoryAxis.Furthermore,you

canalsochangetheratiobetweentheaxesandhideorshowtheaxesindividually.

OntabGrid,youcan,forexample,changethecolorandlinestyleofthecoordinategrid,andsetthedistanceforgridlinestoacertainvalue.Inaddition,youmayalso

setthegridtobeIsometric.

Note:ScalingtheaxesispossibleineverymodebypressingandholdingtheShiftkey(PC:alsoCtrlkey)whiledraggingtheaxis.

Note:ThePropertiesDialogoftheGraphicsViewisdifferentfromthePropertiesDialogfor

objects.

CustomizingtheToolbar

TheToolbarcanbecustomizedbyselectingCustomizeToolbarfromtheToolsmenu.

SelectthetoolortoolboxyouwanttoremovefromtheToolbarinthelistonthelefthand

sideoftheappearingdialogwindowandclickbuttonRemove>inordertoremovethe

tool/toolboxfromtheToolbar.

Note:YoucanrestorethedefaultToolbarbyclickingonthebuttonRestoreDefaultToolbar

intheleftlowercornerofthedialogwindow.


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1.2.2. ThePropertiesDialogThePropertiesDialogallowsyoutomodifypropertiesofobjects(e.g.,size,color,filling,line

style,linethickness,visibility).

YoucanopenthePropertiesDialoginseveralways:

Rightclick(MacOS:Ctrlclick)onanobjectandselect PropertiesfromtheappearingContextMenu.

Selectitem PropertiesfromtheEditmenu. Selectthe MovetoolanddoubleclickonanobjectintheGraphicsView.Inthe

appearingRedefinedialogwindow,clickonthebuttonProperties.

InthePropertiesDialogobjectsareorganizedbytypes(e.g.,points,lines,circles)inthelist

onthelefthandside,whichmakesiteasiertohandlelargenumbersofobjects.Youneedto

selectoneormoreobjectsfromthislistinordertochangeits/theirproperties.

Note:Byclickingonaheadinginthelistofobjects(e.g.,Point)youcanselectallobjectsof

thistypeandtherefore,quicklychangethepropertiesforalltheseobjects.

Youcanmodifythepropertiesofselectedobjectsusingthetabsontherighthandside(e.

g.,Basic,Color,Style,Advanced).

Note:Dependingontheselectionofobjectsinthelist,adifferentsetoftabsmaybe

available.

ClosethePropertiesDialogwhenyouaredonewithchangingpropertiesofobjects.

1.2.3. TheContextMenuTheContextMenuprovidesaquickwaytochangethebehaviororadvancedpropertiesofanobject.Rightclick(MacOS:Ctrlclick)onanobjectinordertoopenitsContextMenu.For

example,itallowsyoutochangetheobjectsalgebraicnotation(e.g.,polarorCartesian

coordinates,implicitorexplicitequation)andtodirectlyaccessfeatureslike Rename,

Delete, TraceOn,AnimationOn,or CopytoInputBar.

Note:IfyouopentheContextMenuforapointintheGraphicsView,itgivesyoutheoption

TracetoSpreadsheet(onlyiftheSpreadsheetViewisactive).Onceselected,thisfeature

allowsyoutorecordthecoordinatesofthepointintheSpreadsheetViewifitismoved.

Note:Selecting PropertiesintheContextMenuopensthePropertiesDialog,whereyou

canchangethepropertiesofallobjectsused.


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1.3. GeoGebraasaPresentationTool1.3.1. TheNavigationBarGeoGebraoffersaNavigationBarthatallowsyoutonavigatethroughtheconstruction

stepsofapreparedGeoGebrafile.SelectitemNavigationBarforConstructionStepsinthe

ViewmenuinordertodisplaytheNavigationBaratthebottomoftheGraphicsView.

TheNavigationBarprovidesasetofnavigationbuttonsanddisplaysthenumberof

constructionsteps(e.g.,2/7meansthatcurrentlythesecondstepofatotalof7

constructionstepsisdisplayed):

button:gobacktostep1 button:gobackstepbystep button:goforwardstepbystep button:gotothelaststep Play:automaticallyplaytheconstructionstepbystep

Note:Youmaychangethespeedofthisautomaticplayfeatureusingthetextboxto

therightofthe Playbutton.

Pause:pausetheautomaticplayfeatureNote:ThisbuttononlyappearsafteryouclickonthePlaybutton.

button:ThisbuttonopenstheConstructionProtocol.

1.3.2. TheConstructionProtocolYoucanaccesstheinteractiveConstructionProtocolbyselectingitemConstructionProtocol

fromtheViewmenu.Itisatablethatshowsallconstructionsteps.TheConstruction

ProtocolallowsyoutoredoapreparedconstructionstepbystepusingtheNavigationBaratthebottomoftheConstructionProtocoldialog.

NavigatingandModifyingtheConstructionProtocolYoumayusethekeyboardtonavigateintheConstructionProtocol:

Usetheuparrowofyourkeyboardtogotothepreviousconstructionstep. Usethedownarrowofyoukeyboardtogotothenextconstructionstep. UsetheHomekeytogotothebeginningoftheConstructionProtocol. UsetheEndkeytogototheendoftheConstructionProtocol. UsetheDeletekeyinordertodeletetheselectedconstructionstep.

Note:Thismayalsoaffectotherobjectsthatdependontheselectedobject/constructionstep.

YoumayalsousethemouseinordertonavigateintheConstructionProtocol:

Doubleclickarowinordertoselectaconstructionstep. DoubleclicktheheaderofanycolumninordertogotothestartoftheConstruction

Protocol.


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DraganddroparowinordertomoveaconstructionsteptoanotherpositionintheConstructionProtocol.

Note:Thisisnotalwayspossibleduetothedependenciesbetweendifferentobjects.

RightclickarowinordertoopentheContextMenufortheobjectofthisconstructionstep.

Note:Youcaninsertconstructionstepsatanyposition.Selecttheconstructionstepbelow

youwouldliketoinsertanewconstructionstep.LeavetheConstructionProtocolwindow

openwhileyoucreateanewobject.Thisnewconstructionstepisimmediatelyinsertedinto

theselectedpositionoftheConstructionProtocol.

UsingthecolumnBreakpointintheViewmenuoftheConstructionProtocolwindow,you

areabletodefinecertainconstructionstepsasBreakpoints.Thisallowsyoutogroupseveral

objectstogether.WhennavigatingthroughyourconstructionusingtheNavigationBar,

groupsofobjectsareshownatthesametime.

Note:YoumayswitchthedifferentcolumnsoftheConstructionProtocolonandoffby

usingtheViewmenuoftheConstructionProtocolwindow.

ExportingtheConstructionProtocolasaWebpageGeoGebraallowsyoutoexporttheConstructionProtocolasawebpage.First,youneedto

opentheConstructionProtocolusingtheViewmenu.Then,youcanopentheFilemenuof

theappearingConstructionProtocolwindowandselectitemExportasWebpage.

IntheexportwindowoftheConstructionProtocolyoucanenterTitle,Author,andaDate

fortheconstructionandchoosewhetherornotyouwanttoincludeapictureofthe

GraphicsViewandtheAlgebraView.Inaddition,youcanalsochoosetoexportaColorful

ConstructionProtocol.ThismeansthatobjectsintheConstructionProtocolwillmatchthecolorofthecorrespondingobjectsintheconstruction.

Note:TheexportedHTMLfilecanbeviewedwithanyInternetbrowser(e.g.Firefox,

InternetExplorer)andeditedwithmanytextprocessingsystems(e.g.OpenOfficeWriter).

1.3.3. CustomizetheSettingsGeoGebraallowsyoutochangeandsavesettingsusingtheOptionsmenu.Forexample,you

maychangetheAngleUnitfromDegreetoRadians,orchangethePointStyle,Checkbox

Size,andRightAngleStyle.Inaddition,youmaychangehowCoordinatesaredisplayedon

screenandwhichobjectsarelabeled(Labeling).

PleaseseethesectionabouttheOptionsmenuformoreinformation.

Youcansaveyourcustomizedsettingsbyselectingitem SaveSettingsfromtheOptions

menu.Afterdoingso,GeoGebrawillrememberyourcustomizedsettingsandusethemfor

everynewGeoGebrafileyoucreate.

Note:YoumayrestorethedefaultsettingsbyselectingRestoreDefaultSettingsfromthe

Optionsmenu.


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Note:IfyouuseGeoGebraasapresentationtool,youmightwanttoincreasetheFontSize

(Optionsmenu)soyouraudiencecaneasilyreadtextandlabelsofobjects.

1.4.

GeoGebra

as

an

Authoring

Tool

1.4.1. PrintingOptionsPrintingtheGraphicsViewGeoGebraallowsyoutoprinttheGraphicsViewofyourconstructions.Youcanfindthe

correspondingitemPrintPreviewintheFilemenu.IntheappearingPrintPreviewdialog

window,youcanspecifytheTitle,Author,andaDatefortheconstruction.Inaddition,you

cansettheScaleofyourprintout(incm)andchangetheOrientationofthepaperused

(portraitorlandscape).

Note:InordertoupdatethePrintPreviewafteryoumadechangestothetextorlayoutof

theprintout,youneedtopresstheEnterkey.

PrintingtheConstructionProtocol

IfyouwanttoprinttheConstructionProtocol,youfirstneedtoopentheConstruction

ProtocoldialogwindowbyusingtheViewmenu.Then,youcanopenthePrintPreview

windowoftheConstructionProtocolfromtheFilemenuofthisnewwindow.

Again,youmayenterTitle,Author,andaDateorchangetheScaleorpaperOrientation

beforeprintingyourConstructionProtocol.

Note:YoumayswitchthedifferentcolumnsName,Definition,Command,Algebra,and

BreakpointoftheConstructionProtocolonandoffbyusingtheViewmenuofthe

ConstructionProtocoldialogwindow.

1.4.2. CreatingPicturesoftheGraphicsViewSavingtheGraphicsViewasaPictureYoucansavetheGraphicsViewofyourconstructionsasapicturefileonyourcomputer.

Note:ThefullGraphicsViewwillbesavedasapicture.Ifyourconstructiondoesnotuseall

theavailablespaceintheGraphicsView,youmightwantto usetools MoveDrawingPad, ZoomIn,and/or ZoomOutinordertoplace

yourconstructionintheupperleftcorneroftheGraphicsView.Afterwards,youmay

reducethesizeoftheGeoGebrawindowbydraggingoneofitscornerswiththe

mouse.

usetheselectionrectangleinordertospecifywhichpartoftheGraphicsViewshouldbeexportedandsavedasapicture.


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YoumaycreatepointscalledExport_1andExport_2,whichwillbeusedtodefinediagonallyoppositecornersoftheexportrectangle.

Note:PointsExport1andExport2mustbewithinthevisibleareaoftheGraphics

View.

IntheFilemenu,selectitemExportbeforeclickingonitem GraphicsViewasPicture.In

theappearingdialogwindowyoumayspecifytheFormat,Scale(incm),andtheResolution

(indpi)oftheoutputpicturefile.

Note:Thetruesizeoftheexportedimageisshownatthebottomoftheexportwindowjust

abovethebuttons,bothincentimetersandpixel.

PleasefindmoreinformationaboutthedifferentpicturefilesavailableinsectionExport

GraphicsViewasPicture(png,eps).

CopyingtheGraphicsViewtoClipboardTherearedifferentwaysofcopyingtheGraphicsViewtoyourcomputersclipboard:

IntheEditmenu,youmayselectitem GraphicsViewtoClipboard. IntheFilemenu,youfirstneedtoselectitemExport,beforeyoucanclickonitem

GraphicsViewtoClipboard.

IntheExportGraphicsViewasPicturedialogwindow(menuFileExportGraphicsViewasPicture(png,eps))youmayclickonthebuttonClipboard.

ThisfeaturecopiesascreenshotoftheGraphicsViewtoyoursystem'sclipboardasaPNG

(seePNGformat)picture.Thispicturecanbepastedintootherdocuments(e.g.aword

processingdocument).

Note:InOrdertoexportyourconstructionatacertainscale(incm)pleaseusethemenu

item GraphicsViewasPictureintheFilemenu,Export.

1.4.3. CreatingInteractiveWebpagesGeoGebraallowsyoutocreateinteractivewebpages,socalledDynamicWorksheets,from

yourfiles.IntheFilemenu,youneedtoselectitemExportbeforeyoucanclickonitem

DynamicWorksheetasWebpage(html).Thisopenstheexportdialogwindowfor

DynamicWorksheets:

AtthetopoftheexportwindowyoucanentertheTitle,Author,andaDateforyourDynamicWorksheet.

TabGeneralallowsyoutoaddsometextaboveandbelowthedynamicconstruction(e.g.,adescriptionoftheconstructionandsometasks).Youcanalsodetermineif

theconstructionitselfmaybeincludeddirectlyintothewebpageorifitcanbe

openedbyclickingonabutton.

TabAdvancedallowsyoutochangethefunctionalityofthedynamicconstruction(e.g.,showareseticon,doubleclickshouldopentheGeoGebraapplicationwindow)as

wellastomodifytheuserinterfaceshownintheinteractiveapplet(e.g.,showthe

Toolbar,modifyheightandwidth).


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Note:Ifthesizeofyourappletistoobigtofitonacomputerscreenwithstandard

resolution(1024x768),youmaywanttoresizeitbeforetheactualexportasa

DynamicWorksheet.

Note:SeveralfilesarecreatedwhenyouexportaDynamicWorksheet:

HTMLfile(e.g.circle.html)thisfileincludestheworksheetitself GGBfile(e.g.circle.ggb)thisfileincludesyourGeoGebraconstruction JAR(severalfiles)thesefilesincludeGeoGebraandmakeyourworksheet

interactive

Allthesefiles(e.g.circle.html,circle.ggbandthegeogebra.jarfiles)havetobeinonefolder

(directory)toletthedynamicconstructionwork.

TheexportedHTMLfile(e.g.circle.html)canbeviewedwithanyInternetbrowser(e.g.

Mozilla,InternetExplorer,Safari).Inordertoletthedynamicconstructionwork,Javahasto

beinstalledonthecomputer.YoucangetJavafromhttp://www.java.comwithoutcharge.If

youwanttouseyourDynamicWorksheetinyourschool'scomputernetwork,askyourlocal

networkadministratortoinstallJavaonthecomputers.

Note:YoucanedittheDynamicWorksheet'stextwithmanywordprocessingsystems(e.g.

FrontPage,OpenOfficeWriter)byopeningtheexportedHTMLfile.Youmayalsoeditthe

DynamicWorksheet'sappletbyopeningtheGGBfileinGeoGebraandsavingitwiththe

samenameafterwards.


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2. GeometricInput2.1. GeneralNotesTheGraphicsViewshowsthegraphicalrepresentationofmathematicalobjects(e.g.,

points,vectors,segments,polygons,functions,curves,straightlines,conicsections).

Wheneverthemouseismovedoveroneoftheseobjectsadescriptionappearsasaroll

overtextandtheobjectishighlighted.

Thereareseveraltools/modestotellGeoGebrahowitshouldreacttomouseinputinthe

GraphicsView(seesectionConstructionTools).Forexample,clickingonthedrawingpad

cancreateanewpoint(seetool NewPoint),intersecttwoobjects(seetool Intersect

TwoObjects),orcreateacircle(see Circletools).

2.2. ConstructionToolsThefollowingconstructiontoolsormodescanbeactivatedbyclickingonthebuttonsofthe

Toolbar.Youcanclickonthesmallarrowinthelowerrightcornerofanicontoopena

Toolboxwithsimilarothertools.

Note:Withmostconstructiontoolsyoucaneasilycreatenewpointsbyclickingonempty

spacesonthedrawingpad.

SelectingObjects

Toselectanobjectmeanstoclickonitwiththemouseafterselectingthe Movetool.

Ifyouwanttoselectseveralobjectsatthesametime,youcoulddrawaselectionrectangle:

Selectthe Movetoolandclickonthepositionofthefirstcornerofyourdesiredselection

rectangle.Holdtheleftmousekeypresseddownandmovethepointertothepositionof

thediagonallyoppositecornerofyourdesiredselectionrectangle.Afterreleasingthemouse

button,allobjectswithintheselectionrectangleareselected.

Note:ByholdingtheCtrlkey(MacOS:Cmdkey)whileclickingondifferentobjects,youcan

selectseveralobjectsatthesametime.

FastRenamingofObjects

ToquicklyrenameaselectedornewlycreatedobjectjuststarttypingtoopentheRename

dialogforthisobject.Then,typeinthenewnameoftheselectedobjectandclickontheOK

button.


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2.2.1. GeneralToolsCopyVisualStyle

Thistoolallowsyoutocopyvisualproperties(e.g.,color,size,linestyle)fromoneobjectto

oneormoreotherobjects.Todoso,firstselecttheobjectwhosepropertiesyouwanttocopy.Then,clickonallotherobjectsthatshouldadopttheseproperties.

DeleteObject

Clickonanyobjectyouwanttodelete(alsoseecommandDelete).

Note:Youcanusethe Undobuttonifyouaccidentallydeletedthewrongobject.

Move

Draganddropfreeobjectswiththemouse.IfyouselectanobjectbyclickingonitinMovemode,youmay

deletetheobjectbypressingtheDeletekey movetheobjectbyusingthearrowkeys(seesectionManualAnimation)

Note:Youcanquicklyactivatethe MovetoolbypressingtheEsckeyofyourkeyboard.

MoveDrawingPad

DraganddropthedrawingpadintheGraphicsViewtochangeitsvisiblearea.

Note:

Youcan

also

move

the

drawingpadbypressing

the

Shiftkey

(MS

Windows:

also

Ctrlkey)anddraggingitwiththemouseinanymode.

Inthismodeyoucanalsoscaleeachoftheaxesbydraggingitwiththemouse.

RecordtoSpreadsheet

Thistoolallowsyoutomoveanobjectandtorecordasequenceofitsvaluesinthe

SpreadsheetView.Thistoolworksfornumbers,points,andvectors.

Note:GeoGebrawillusethefirsttwoemptycolumnsoftheSpreadsheetViewtorecordthe

valuesoftheselectedobjects.

Relation

Selecttwoobjectstogetinformationabouttheirrelationinapopupwindow(alsosee

commandRelation).


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RotatearoundPoint

Selectthecenterpointoftherotationfirst.Then,youmayrotatefreeobjectsaroundthis

pointbydraggingthemwiththemouse(alsoseecommandRotate).

Show/HideLabel

Clickonanobjecttoshoworhideitslabel.

Show/HideObject

Selecttheobjectyouwanttoshoworhideafteractivatingthistool.Then,switchtoanother

toolinordertoapplythevisibilitychangestothisobject.

Note:Whenyouactivatethistool,allobjectsthatshouldbehiddenaredisplayedinthe

GraphicsViewhighlighted.Inthisway,youcaneasilyshowhiddenobjectsagainby

deselectingthembeforeswitchingtoanothertool.

ZoomIn

Clickonanyplaceonthedrawingpadtozoomin(alsoseesectionCustomizingtheGraphics

View).


ZoomOut

Clickonanyplaceonthedrawingpadtozoomout(alsoseesectionCustomizingthe

GraphicsView).


2.2.2. PointToolsIntersectTwoObjects

Intersectionpointsoftwoobjectscanbecreatedintwoways(alsoseecommandIntersect).

Selectingtwoobjectscreatesallintersectionpoints(ifpossible).

Directlyclickingonanintersectionofthetwoobjectscreatesonlythissingleintersectionpoint.

Note:Forsegments,rays,orarcsyoumayspecifywhetheryouwanttoAllowoutlying

intersectionsontabBasicofthePropertiesDialog.Thiscanbeusedtogetintersection

pointsthatlieontheextensionofanobject.Forexample,theextensionofasegmentora

rayisastraightline.


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MidpointorCenter

Youmayclickoneithertwopointsoronesegmenttogetitsmidpoint.Youcanalsoclickon

aconicsection(circleorellipse)inordertocreateitscenterpoint(alsoseecommands

CenterandMidpoint).

NewPoint

ClickonthedrawingpadintheGraphicsViewinordertocreateanewpoint.The

coordinatesofthepointarefixedwhenthemousebuttonisreleased.

Note:

Byclickingonasegment,straightline,polygon,conicsection,function,orcurveyoucancreateapointonthisobject(alsoseecommandPoint).

Clickingontheintersectionoftwoobjectscreatesthisintersectionpoint(alsoseetool IntersectTwoObjectsandcommandIntersect).

2.2.3. VectorToolsVectorbetweenTwoPoints

Selectthestartingpointandthentheendpointofthevector(alsoseecommandVector).

VectorfromPoint

SelectapointAandavectorvtocreatethenewpointB=A+vaswellasthevectorfromA

toB(alsoseecommandVector).

2.2.4. SegmentToolsSegmentbetweenTwoPoints

SelecttwopointsAandBinordertocreateasegmentbetweenAandB(alsoseecommand

Segment).

Note:IntheAlgebraView,thesegment'slengthisdisplayed.

SegmentwithGivenLengthfromPoint

ClickonapointAthatshouldbethestartingpointofthesegment.Specifythedesired

lengthaofthesegmentintheappearingwindow(alsoseecommandSegment).

Note:ThistoolcreatesasegmentwithlengthaandendpointBwhichmayberotated

aroundthestartingpointAbyusingtool Move.


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2.2.5. RayToolRaythroughTwoPoints

SelectingtwopointsAandBcreatesaraystartingatAthroughB(alsoseecommandRay).

Note:IntheAlgebraViewtheequationofthecorrespondinglineisdisplayed.

2.2.6. PolygonToolsPolygon

Successivelyselectatleastthreepointswhichwillbetheverticesofthepolygon.Then,click

thefirstpointagaininordertoclosethepolygon(alsoseecommandPolygon).

Note:IntheAlgebraView,thepolygon'sareaisdisplayed.

RegularPolygon

SelecttwopointsAandBandspecifythenumbernofverticesinthetextfieldofthe

appearingdialogwindow.ThisgivesyouaregularpolygonwithnverticesincludingpointsA

andB(alsoseecommandPolygon).

2.2.7. LineToolsAngleBisector

Anglebisectorscanbedefinedintwoways(alsoseecommandAngleBisector):

SelectingthreepointsA,B,andCproducestheanglebisectoroftheenclosedangle,wherepointBistheapex.

Selectingtwolinesproducestheirtwoanglebisectors.Note:Thedirectionvectorsofallanglebisectorshavelength1.

BestFitLine

Createthebestfitlineforasetofpointsinthefollowingways(alsoseecommandFitLine):

Createaselectionrectanglethatcontainsallpoints. Selectalistofpointstocreatetheircorrespondingbestfitline.

LinethroughTwoPoints

SelectingtwopointsAandBcreatesastraightlinethroughAandB(alsoseecommand

Line).

Note:Thelinesdirectionvectoris(BA).


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ParallelLine

SelectingalinegandapointAdefinesastraightlinethroughAparalleltog(alsosee

commandLine).

Note:Thelinesdirectionisthedirectionoflineg.

PerpendicularBisector

ClickoneitherasegmentsortwopointsAandBinordertocreateaperpendicularbisector

(alsoseecommandPerpendicularBisector).

Note:ThebisectorsdirectionisequivalenttotheperpendicularvectorofsegmentsorAB

(alsoseecommandPerpendicularVector).

PerpendicularLine

SelectingalinegandapointAcreatesastraightlinethroughAperpendiculartolineg(also

seecommandPerpendicularLine).

Note:Thelinesdirectionisequivalenttotheperpendicularvectorofg(alsoseecommand

PerpendicularVector).

PolarorDiameterLine

Thistoolcreatesthepolarordiameterlineofaconicsection(alsoseecommandPolar).

Selectapointandaconicsectiontogetthepolarline. Selectalineoravectorandaconicsectiontogetthediameterline.

Tangents

Tangentstoaconicsectioncanbeproducedinseveralways(alsoseecommandTangent):

SelectingapointAandaconiccproducesalltangentsthroughAtoc. Selectingalinegandaconiccproducesalltangentstocthatareparalleltolineg. SelectingapointAandafunctionfproducesthetangentlinetofinx=x(A).

Note:x(A)representsthexcoordinateofpointA.IfpointAliesonthefunction

graph,thetangentrunsthroughpointA.

2.2.8. ConicSectionToolsCirclewithCenterandRadius

SelectthecenterpointMandentertheradiusinthetextfieldoftheappearingdialog

window(alsoseecommandCircle).


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CirclewithCenterthroughPoint

SelectingapointMandapointPdefinesacirclewithcenterMthroughP(alsosee

commandCircle).

Note:ThecirclesradiusisthedistanceMP.

CirclethroughThreePoints

SelectingthreepointsA,B,andCdefinesacirclethroughthesepoints(alsoseecommand

Circle).

Note:Ifthethreepointslieononestraightline,thecircledegeneratestothisline.

Compass

UKEnglish:Compasses

Selectasegmentortwopointstospecifytheradius.Then,clickonapointthatshouldbe

thecenterofthenewcircle.

ConicthroughFivePoints

Selectingfivepointsproducesaconicsectionthroughthesepoints(alsoseecommand

Conic).

Note:Iffourofthesefivepointslieonaline,theconicsectionisnotdefined.

Ellipse

Selectthetwofocioftheellipse.Then,specifyathirdpointthatliesontheellipse(alsosee

commandEllipse).

Hyperbola

Selectthetwofociofthehyperbola.Then,specifyathirdpointthatliesonthehyperbola

(alsoseecommandHyperbola).

Parabola

Selectapointandthedirectrixoftheparabola(alsoseecommandParabola).

2.2.9. ArcandSectorToolsNote:InGeoGebra,thealgebraicvalueofanarcisitslength.Thevalueofasectorisitsarea.


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CircularArcwithCenterbetweenTwoPoints

First,selectthecenterpointMofthecirculararc.Then,selectthestartingpointAofthe

arc,beforeyouselectapointBthatspecifiesthelengthofthearc(alsoseecommand

CircularArc).

Note:While

point

Aalways

lies

onthe

circular

arc,

point

Bdoes

not

have

tolieonit.

CircularSectorwithCenterbetweenTwoPoints

First,selectthecenterpointMofthecircularsector.Then,selectthestartingpointAofthe

sectorsarc,beforeyouselectapointBthatspecifiesthelengthofthesectorsarc(alsosee

commandCircularSector).

Note:WhilepointAalwaysliesonthesectorsarc,pointBdoesnothavetolieonit.

CircumcircularArcthroughThreePoints

SelectingthreepointsA,B,andCcreatesacirculararcthroughthesepoints.Thereby,point

Aisthestartingpointofthearc,pointBliesonthearc,andpointCistheendpointofthe

arc(alsoseecommandCircumcircularArc).

CircumcircularSectorthroughThreePoints

SelectingthreepointsA,B,andCcreatesacircularsectorthroughthesepoints.Thereby,

pointAisthestartingpointofthesectorsarc,pointBliesonthearc,andpointCisthe

endpointofthesectorsarc(alsoseecommandCircumcircularSector).

Semicircle

SelecttwopointsAandBtocreateasemicircleabovethesegmentAB(alsoseecommand

Semicircle).

2.2.10. NumberandAngleToolsAngle

Withthistoolyoucancreateanglesindifferentways(alsoseecommandAngle):

Clickonthreepointstocreateananglebetweenthesepoints.Thesecondpointselectedisthevertexoftheangle.

Clickontwosegmentstocreatetheanglebetweenthem. Clickontwolinestocreatetheanglebetweenthem. Clickontwovectorstocreatetheanglebetweenthem.


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Clickonapolygontocreateallanglesofthispolygon. Note:Ifthepolygonwascreatedbyselectingitsverticesincounterclockwise

orientation,theAngletoolgivesyoutheinterioranglesofthepolygon.

Note:Anglesarecreatedincounterclockwiseorientation.Therefore,theorderofselecting

theseobjectsisrelevantfortheAngletool.Ifyouwanttolimitthemaximumsizeofan

angleto180,uncheckAllowReflexAngleontabBasicofthePropertiesDialog.

AnglewithGivenSize

SelecttwopointsAandBandtypetheanglessizeintothetextfieldoftheappearing

window(alsoseecommandAngle).

Note:ThistoolcreatesapointCandanangle,whereistheangleABC.

Area

Thistoolgivesyoutheareaofapolygon,circle,orellipseasanumberandshowsadynamic

textintheGraphicsView(alsoseecommandArea).

DistanceorLength

Thistoolgivesyouthedistancebetweentwopoints,twolines,orapointandalineasa

numberandshowsadynamictextintheGraphicsView.Itcanalsogiveyouthelengthofa

segment,thecircumferenceofacircle,ortheperimeterofapolygon(alsoseecommands

DistanceandLength).

Slider

ClickonanyfreeplaceintheGraphicsViewtocreateasliderforanumberoranangle.The

appearingdialogwindowallowsyoutospecifytheName,Interval[min,max],and

Incrementofthenumberorangle,aswellastheAlignmentandWidthoftheslider(inpixel).

Note:IntheSliderdialogwindowyoucanenteradegreesymbolorpi()fortheinterval

andincrementbyusingthefollowingkeyboardshortcuts:

AltO(MacOS:CtrlO)forthedegreesymbol AltP(MacOS:CtrlP)forthepisymbol

ThepositionofaslidermaybeabsoluteintheGraphicsView(thismeansthattheslideris

notaffectedbyzooming,butalwaysremainsinthevisiblepartoftheGraphicsView)orrelativetothecoordinatesystem(seePropertiesDialogofthecorrespondingnumberor

angle).

Note:InGeoGebra,aslideristhegraphicalrepresentationofafreenumberorfreeangle.

Youcaneasilycreateasliderforanyexistingfreenumberoranglebyshowingthisobjectin

theGraphicsView(seeContextMenu;seetool Show/HideObject).


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Slope

ThistoolgivesyoutheslopeofalineandshowsaslopetriangleintheGraphicsView(also

seecommandSlope).

2.2.11. BooleanVariableTool

CheckBoxtoShow/HideObjects

ClickingintheGraphicsViewcreatesacheckbox(seesectionBooleanVariablesand

Operations)thatallowsyoutoshowandhideoneormoreobjects.Intheappearingdialog

windowyoucanspecifywhichobjectsshouldbeaffectedbythecheckbox.

Note:Youmayselecttheseobjectsfromthelistprovidedinthedialogwindoworselect

themwiththemouseinanyview.

2.2.12. LocusToolLocus

SelectapointBthatdependsonanotherpointAandwhoselocusshouldbedrawn.Then,

clickonpointAtocreatethelocusofpointB(alsoseecommandLocus).

Note:PointAhastobeapointonanobject(e.g.line,segment,circle).

Example:

Typef(x) = x^2 2 x 1intotheInputBarandpresstheEnterkey. PlaceanewpointAonthexaxis(seetool NewPoint;seecommandPoint). CreatepointB = (x(A), f'(x(A)))thatdependsonpointA. Selecttool LocusandsuccessivelyclickonpointBandpointA. DragpointAalongthexaxistoseepointBmovingalongitslocusline.

2.2.13. GeometricTransformationToolsThefollowinggeometrictransformationsworkforpoints,lines,conicsections,polygons,

andimages.

DilateObjectfromPointbyFactor

UKEnglish:EnlargeObjectfromPointbyFactor

Selecttheobjecttobedilated.Then,clickonapointtospecifythedilationcenterandenter

thedilationfactorintothetextfieldoftheappearingdialogwindow(alsoseecommands

Dilate(US)andEnlarge(UK)).


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ReflectObjectaboutLine

UKEnglish:ReflectObjectinLine

Selecttheobjectyouwanttoreflect.Then,clickonalinetospecifythemirror/lineof

reflection(also

see

command

Reflect).

ReflectObjectaboutPoint

UKEnglish:ReflectObjectinPoint

Selecttheobjectyouwanttoreflect.Then,clickonapointtospecifythemirror/pointof

reflection(alsoseecommandReflect).

ReflectPointaboutCircle

UKEnglish:ReflectPointinCircle

Thistoolallowsyoutoinvertapointinacircle.Selectthepointyouwanttoinvert.Then,

clickonacircletospecifythemirror/circleofinversion(alsoseecommandReflect).

RotateObjectaroundPointbyAngle

Selecttheobjectyouwanttorotate.Then,clickonapointtospecifythecenterofrotation

andentertherotationangleintothetextfieldoftheappearingdialogwindow(alsosee

commandRotate).

TranslateObjectbyVector

Selecttheobjectyouwanttotranslate.Then,clickonthetranslationvector(alsosee

commandTranslate).

2.2.14. TextToolInsertText

WiththistoolyoucancreatestaticanddynamictextorLaTeXformulasintheGraphicsView

(alsoseesectionTextCommands).

Atfirst,youneedtospecifythelocationofthetextinoneofthefollowingways:

ClickintheGraphicsViewtocreateanewtextatthislocation. Clickonapointtocreateanewtextthatisattachedtothispoint.

Then,adialogappearswhereyoumayenteryourtext.


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Note:Youmayspecifythepositionofatextasabsoluteonscreenorrelativetothe

coordinatesystemontabBasicofthePropertiesDialog.

Statictextdoesnotdependonanymathematicalobjectsandisusuallynotaffectedby

changesoftheconstruction.

Dynamictextcontainsvaluesofobjectsthatautomaticallyadapttochangesmadetothese

objects.

Mixedtextisacombinationofstaticanddynamictext.Inordertocreateamixedtextyou

mayenterthestaticpartofthetextusingthekeyboard(e.g.,Point A =).Then,clickon

theobjectwhosevalueyouwanttodisplayinthedynamicpartofthetext.

Note:GeoGebraautomaticallyaddsthesyntax("Point A = " + A)necessarytocreate

yourmixedtext:quotationmarksaroundthestaticpartofthetextandaplus(+)symbolto

connectthedifferentpartsofthetext.

Input Description

This is static text Statictext

A Dynamictext(ifpointAexists)

"Point A = " + A TwopartmixedtextusingthevalueofpointA

"a = " + a + "cm"Threepartmixed textusingthevalueof

numbera

Note:Ifanobjectwiththenamexxalreadyexistsandyouwanttocreateastatictextusing

theobjectsname,youneedtoenteritwithquotationmarks("xx").Otherwise,GeoGebra

willautomaticallycreateadynamictextthatgivesyouthevalueofobjectxxinsteadofits

name.However,youcantypeanytextthatdoesntmatchanyexistingobjectsname

withoutthequotationmarks.

Note:Withinamixedtext,thestaticpartneedstobeinbetweenapairofquotationmarks.

Differentpartsofatext(e.g.,staticanddynamicparts)needtobeconnectedusingplus(+)

symbols.

LaTeXFormulas

InGeoGebrayoucanwriteformulasaswell.Todoso,checktheboxLaTeXformulainthe

dialogwindowofthe InsertTexttoolandenteryourformulainLaTeXsyntax.

Note:InordertocreatetextthatcontainsaLaTeXformulaaswellasstatictextyoumay

enterthestaticpartofthetextandthenaddtheLaTeXformulainbetweenasetofdollar

symbols($).Example:The length of the diagonal is $\sqrt{ 2 }$.

Youcanselectthesyntaxforcommonformulasymbolsfromthedropdownmenunextto

theLaTeXcheckbox.ThisinsertsthecorrespondingLaTeXcodeintothetextfieldandplaces

thecursorinbetweenasetofcurlybrackets.Ifyouwouldliketocreatedynamictextwithin

theformula,youneedtoclickonanobjectcausingGeoGebratoinsertitsnameaswellas

thesyntaxformixedtext.


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SomeimportantLaTeXcommandsareexplainedinfollowingtable.Pleasehavealookatany

LaTeXdocumentationforfurtherinformation.

LaTeXinput Result

a \cdot b ba

\frac{a}{b}b

a

\sqrt{x} x

\sqrt[n]{x} n x

\vec{v} v

\overline{AB} ABx^{2} 2x

a_{1}1

a

\sin\alpha +\cos\beta

cossin

\int_{a}^{b} x dx b

a

xdx

\sum_{i=1}^{n} i^2 n

ii

1

2

2.2.15. ImageToolInsertImage

ThistoolallowsyoutoinsertanimageintotheGraphicsView.

First,specifythelocationoftheimageinoneofthefollowingtwoways: ClickintheGraphicsViewtospecifythepositionoftheimageslowerleftcorner. Clickonapointtospecifythispointasthelowerleftcorneroftheimage.

Then,afileopendialogappearsthatallowsyoutoselecttheimagefilefromthefilessaved

onyourcomputer.

Note:Afterselectingthetool InsertImage,youcanusethekeyboardshortcutAltclickin

ordertopasteanimagedirectlyfromyourcomputersclipboardintotheGraphicsView.

PropertiesofImages

Thepositionofanimagemaybeabsoluteonscreenorrelativetothecoordinatesystem.

YoucanspecifythisontabBasicofthePropertiesDialogoftheimage.

YoumayspecifyuptothreecornerpointsoftheimageontabPositionoftheProperties

Dialog.Thisgivesyoutheflexibilitytoscale,rotate,andevendistortimages(alsosee

commandCorner).

Corner1:positionofthelowerleftcorneroftheimage


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Corner2:positionofthelowerrightcorneroftheimage Note:ThiscornermayonlybesetifCorner1wassetbefore.Itcontrolsthewidthof

theimage.

Corner4:positionoftheupperleftcorneroftheimage Note:ThiscornermayonlybesetifCorner1wassetbefore.Itcontrolstheheightof

theimage.

Example:CreatethreepointsA,B,andCtoexploretheeffectsofthecornerpoints.

SetpointAasthefirstandpointBasthesecondcornerofyourimage.BydraggingpointsAandBin Movemodeyoucanexploretheirinfluence.

Now,removepointBasthesecondcorneroftheimage.SetpointAasthefirstandpointCasthefourthcornerandexplorehowdraggingthepointsnowinfluencesthe

image.

Finally,youmaysetallthreecornerpointsandseehowdraggingthepointsdistortsyourimage.

Example:Youalreadysawhowtoinfluencethepositionandsizeofyourimage.Ifyouwant

toattachyourimagetoapointAandsetitswidthto3anditsheightto4units,youcould

dothefollowing:

SetCorner1toA SetCorner2toA + (3, 0) SetCorner4toA + (0, 4)

Note:IfyounowdragpointAin Movemode,thesizeofyourimagedoesnotchange.

YoumayspecifyanimageasaBackgroundImageontabBasicofthePropertiesDialog.Abackgroundimageliesbehindthecoordinateaxesandcannotbeselectedwiththemouse

anymore.

Note:Inordertochangethebackgroundsettingofanimage,youmayopenthePropertiesDialogbyselecting PropertiesfromtheEditmenu.

TheTransparencyofanimagecanbechangedinordertoseeobjectsoraxesthatliebehind

theimage.YoucansetthetransparencyofanimagebyspecifyingaFillingvaluebetween0

%and100%ontabStyleofthePropertiesDialog.


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3. AlgebraicInput3.1. GeneralNotesThealgebraicrepresentationsofmathematicalobjects(e.g.,values,coordinates,equations)

areshownintheAlgebraView.YoucancreateandmodifyobjectsbyusingtheInputBarat

thebottomoftheGeoGebrawindow(seesectionsDirectInputandandCommands).

Note:AlwayspresstheEnterkeyaftertypingalgebraicinputintotheInputBar.

Note:PressingtheEnterkeyatanytimetogglesthefocusbetweentheInputBarandthe

GraphicsView.ThisallowsyoutoenterexpressionsandcommandsintotheInputBar

withouthavingtoclickonitwiththemousefirst.

NamingObjects

YoucanassignacertainnametoanobjectwhenyoucreateitusingtheInputBar:

Points:InGeoGebra,pointsarealwaysnamedusinguppercaseletters.Justtypeinthename(e.g.,A,P)andanequalsigninfrontofthecoordinatesorcommands.

Examples: C = (2, 4),P = (1; 180),Complex = 2 + i

Vectors:Inordertodistinguishbetweenpointsandvectors,vectorsneedtohavealowercasenameinGeoGebra.Again,typeinthename(e.g.,v,u)andanequalsign

infrontofthecoordinatesorcommands.

Examples:v = (1, 3),u = (3; 90),complex = 1 2i

Lines,circles,andconicsections:Theseobjectscanbenamedbytypinginthenameandacoloninfrontoftheirequationsorcommands.

Examples:g: y = x + 3,c: (x-1)^2 + (y 2)^2 = 4,hyp: x^2 y^2 = 2

Functions:Youcannamefunctionsbytyping,forexample,f(x) =org(x)=infrontofthefunctionsequationorcommands.

Examples:h(x) = 2 x + 4,q(x) = x^2,trig(x) = sin(x)

Note:

Ifyoudontmanuallyassignanametoanobject,GeoGebraassignsthenamesofnewobjectsinalphabeticalorder.

Youcan

create

indices

within

the

names

ofobjects

byusing

anunderscore.

For

exampleA1isenteredasA_1andsABisenteredass_{AB}.

ChangeValues

Therearetwowaysofmanipulatingafreeobjectsvalue:

ChangethevalueoftheobjectbyenteringitsnameandthenewvalueintheInputBar(seesectionDirectInput).


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Example:Ifyouwanttochangethevalueofanexistingnumbera=3,type

a = 5intotheInputBarandpresstheEnterkey.

Editthealgebraicrepresentation:Activatetool MoveanddoubleclickontheobjectintheAlgebraView.Thisopensatextboxwhereyoucanedittheobjects

value.PresstheEnterkeytoapplyyourchanges.

Note:Whilefreeobjectsvaluescanbechangeddirectly,thevaluesofdependentobjects

canonlybeinfluencedbychangingtheirparentobjectsorbyredefiningthedependent

object.

DisplayInputBarHistoryAfterplacingthecursorintheInputBaryoucanusetheupanddownarrowkeysof

yourkeyboardinordertonavigatethroughpriorinputstepbystep.

Note:Clickonthelittlequestionmark totheleftoftheInputBarinordertodisplaythe

helpfeaturefortheInputBar.

InsertName,Value,orDefinitionofanObjectintotheInputBarInsertthenameofanobject:Activatetool Moveandselecttheobjectwhosenameyou

wanttoinsertintotheInputBar.Then,presstheF5keyonyourkeyboard.

Note:ThenameoftheobjectisappendedtoanyexpressionyoutypedintotheInputBar

beforepressingtheF5key.

Insertthevalueofanobject:Therearetwowaysofinsertinganobjectsvalue(e.g.,(1,3),

3x5y=12)intotheInputBar.

Rightclick(MacOS:Ctrlclick)ontheobjectandselectitem CopytoInputBarfromtheappearingContextMenu. Activatetool MoveandselecttheobjectwhosevalueyouwanttoinsertintotheInputBar.Then,presstheF4keyonyourkeyboard.

Note:ThevalueoftheobjectisappendedtoanyexpressionyoutypedintotheInput

BarbeforepressingtheF4key.

Insertthedefinitionofanobject:Therearetwowaysofinsertinganobjectsdefinition(e.

g.,A=(4,2),c=Circle[A,B])intotheInputBar.

AltclickontheobjecttoinserttheobjectsdefinitionanddeletewhateverinputmighthavebeenintheInputBarbefore.

Activatetool MoveandselecttheobjectwhosedefinitionyouwanttoinsertintotheInputBar.Then,presstheF3keyonyourkeyboard.Note:ThedefinitionoftheobjectreplacesanyexpressionyoutypedintotheInput

BarbeforepressingtheF3key.


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3.2. DirectInputGeoGebracanworkwithnumbers,angles,points,vectors,segments,lines,conicsections,

functions,andparametriccurves.YoucanentertheseobjectsintotheInputBarbyusing

theircoordinatesorequationsandpressingtheEnterkey.

3.2.1. NumbersandAnglesNumbers

YoucancreatenumbersbyusingtheInputBar.Ifyouonlytypeinanumber(e.g.,3),

GeoGebraassignsalowercaseletterasthenameofthenumber.Ifyouwanttogiveyour

numberaspecificname,youcantypeinthenamefollowedbyanequalsignandthe

number(e.g.,createadecimalrbytypinginr = 5.32).

Note:InGeoGebra,numbersandanglesuseaperiod(.)asadecimalpoint.

YoucanalsousetheconstantandtheEulerconstanteforexpressionsandcalculationsby

selectingthemfromthedropdownmenunexttotheInputBarorbyusingkeyboard

shortcuts.

Note:Ifthevariableeisnotusedasanameofanexistingobjectyet,GeoGebrawill

recognizeitastheEulerconstantifyouuseitinnewexpressions.

Angles

Anglesareenteredindegree()orradians(rad).Theconstantisusefulforradianvalues

andcanalsobeenteredaspi.

Note:Youcanenteradegreesymbol()orthepisymbol()byusingthefollowingkeyboardshortcuts:

AltO(MacOS:CtrlO)forthedegreesymbol AltP(MacOS:CtrlP)forthepisymbol

Example:Youcanenteranangleindegree(e.g., = 60)orinradians(e.g.,

= pi/3).

Note:GeoGebradoesallinternalcalculationsinradians.Thedegreesymbol()isnothing

buttheconstant/180usedtoconvertdegreeintoradians.

Examples: Ifa=30isanumber,then = aconvertsnumberatoanangle=30,without

changingitsvalue.

Ifyoutypeinb = / ,theangleisconvertedbacktothenumberb=30,withoutchangingitsvalue.


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SlidersandArrowKeys

FreenumbersandanglescanbedisplayedasslidersintheGraphicsView(seetool

Slider).Usingthearrowkeys,youmaychangethevalueofnumbersandanglesinthe

AlgebraViewtoo(seesectionManualAnimation).

LimitValue

to

Interval

Freenumbersandanglesmaybelimitedtoaninterval[min,max]byusingtabSliderofthe

PropertiesDialog(seealsotool Slider).

Note:Fordependentanglesyoucanspecifywhethertheymaybecomereflexornotontab

BasicofthePropertiesDialog.

3.2.2. PointsandVectorsPointsandvectorsmaybeenteredinCartesianorpolarcoordinates(seesectionNumbers

andAngles).Note:Uppercaselabelsdenotepointswhereaslowercaselabelsrefertovectors.

Examples:

ToenterapointPoravectorvinCartesiancoordinatesyoumayuseP = (1, 0)orv = (0, 5).

InordertousepolarcoordinatestypeinP = (1; 0)orv = (5; 90).Note:Youneedtouseasemicolontoseparatethetwocoordinates.Ifyoudonttype

inthedegreesymbol,GeoGebrawilltreattheangleasifenteredinradians.

InGeoGebra,youcanalsodocalculationswithpointsandvectors.

Examples: YoucancreatethemidpointMoftwopointsAandBbyentering

M = (A + B) / 2intotheInputBar.

Youmaycalculatethelengthofavectorvusinglength = sqrt(v * v)

3.2.3. LinesandAxesLines

YoucanenteralineasalinearequationinxandyorinparametricformintotheInputBar.

Inbothcasespreviouslydefinedvariables(e.g.numbers,points,vectors)canbeusedwithintheequation.

Note:Youcanenteralinesnameatthebeginningoftheinputfollowedbyacolon.

Examples:

Typeing: 3x + 4y = 2toenterlinegasalinearequation. Defineaparametert(e.g.,t = 3)beforeenteringlineginparametricformusing

g: X = (-5, 5) + t (4, -3).


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Definetheparametersm = 2andb = -1.Then,youcanentertheequationh: y = m*x + btogetalinehinyinterceptform.

Axes

ThetwocoordinateaxesareavailableincommandsusingthenamesxAxisandyAxis.

Example:ThecommandPerpendicular[A, xAxis]constructstheperpendicularlinetothexaxisthroughagivenpointA.

3.2.4. ConicSectionsYoumayenteraconicsectionasaquadraticequationinxandy.Priordefinedvariables(e.

g.,numbers,points,vectors)canbeusedwithintheconicsequation.

Note:Theconicsectionsnamecanbeenteredatthebeginningoftheinputfollowedbya

colon.

Examples:

Ellipseell: ell: 9 x^2 + 16 y^2 = 144 Hyperbolahyp: hyp: 9 x^2 16 y^2 = 144 Parabolapar: par: y^2 = 4 x Circlec1: c1: x^2 + y^2 = 25 Circlec2: c2: (x 5)^2 + (y + 2)^2 = 25

Note:Ifyoudefinetwoparametersa = 4andb = 3inadvance,youmayenterfor

exampleanellipseasell: b^2 x^2 + a^2 y^2 = a^2 b^2.

3.2.5. FunctionsofxToenterafunctionyoucanusepreviouslydefinedvariables(e.g.numbers,points,vectors)

aswellasotherfunctions.

Examples:

Functionf: f(x) = 3 x^3 x^2 Functiong: g(x) = tan(f(x)) Namelessfunction: sin(3 x) + tan(x)

Note:Allavailablepredefinedfunctions(e.g.sin,cos,tan)aredescribedinsectionPre

definedFunctionsandOperations.

InGeoGebrayoucanalsousecommandstogetforexample,theIntegralandDerivativeofa

function.

Note:Youcanalsousethecommandsf'(x)orf''(x),inordertogetthederivativesofa

previouslydefinedfunctionf(x).

Example:Definefunctionfasf(x) = 3 x^3 x^2.Then,youcantypein

g(x) = cos(f' (x + 2))inordertogetfunctiong.


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Furthermore,functionscanbetranslatedbyavector(seecommandTranslate)andafree

functioncanbemovedwiththemousebyusingtool Move.

LimitFunctiontoInterval

Inordertolimitafunctiontoaninterval[a,b],youcanusethecommandFunction.

3.2.6. PredefinedFunctionsandOperationsTocreatenumbers,coordinates,orequations(seesectionDirectInput)youmayalsouse

thefollowingpredefinedfunctionsandoperations.

Note:Thepredefinedfunctionsneedtobeenteredusingparentheses.Youmustnotputa

spacebetweenthefunctionnameandtheparentheses.

Operation /Function Input

Addition +

Subtraction -Multiplication * orSpace key

Scalarproduct * orSpacekey

Division /

Exponentiation ^ or 2

Factorial !

Gammafunction gamma( )

Parentheses ( )

xcoordinate x( )

ycoordinate y( )

Absolutevalue abs( )

Sign sgn( )Squareroot sqrt( )

Cubicroot cbrt( )

Randomnumberbetween0and1 random( )

Exponentialfunction exp( ) orx

Logarithm(natural,tobasee) ln( ) or log( )

Logarithmtobase2 ld( )

Logarithmtobase10 lg( )

Cosine cos( )

Sine sin( )

Tangent tan( )

Arccosine acos( )

Arcsine asin( )

Arctangent atan( )

Hyperboliccosine cosh( )

Hyperbolicsine sinh( )

Hyperbolictangent tanh( )

Antihyperboliccosine acosh( )

Antihyperbolicsine asinh( )


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Operation /Function Input

Antihyperbolictangent atanh( )

Greatestintegerlessthanorequal floor( )

Leastintegergreaterthanorequal ceil( )

Round round( )

3.2.7. BooleanVariablesandOperationsYoucanusetheBooleanvariablestrueandfalseinGeoGebra.Justtype,forexample,

a = trueorb = falseintotheInputBarandpresstheEnterkey.

CheckBoxandArrowKeys

FreeBooleanvariablescanbedisplayedascheckboxesintheGraphicsView(seetool

CheckBoxtoShow/Hideobjects).Byusingthearrowkeysofyourkeyboardyoumayalso

changeBooleanvariablesintheAlgebraView(seesectionManualAnimation).

Note:YoumayalsouseBooleanvariableslikenumbers(value0or1).Thisallowsyoutouse

acheckboxasthedynamicspeedofananimatedsliderallowingyoutostartandstopthe

animation.Inthiscase,theanimationbuttonisonlyshownintheGraphicsViewifthereis

alsoananimatedsliderwithstatic(i.e.nondynamic)speed.

Operations

YoucanusethefollowingoperationsforBooleanvariablesandconditionsinGeoGebraby

eitherselectingthemfromthelistnexttotheInputBarorbyenteringthemusingthe

keyboard:

List

Keyboard Example Object

types

Equal == a b or a == bnumbers,points,

lines,conicsa,b

Unequal != a b or a != bnumbers,points,

lines,conicsa,b

Lessthan a > b numbersa,b

Lessorequal

than = b numbersa,b

And && a bora && b Booleansa,b

Or || a bora || b Booleansa,b

Not ! aor!a Booleana

Parallel a b linesa,b

Perpendicular a b linesa,b


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3.2.8. ListObjectsandOperationsUsingcurlybracesyoucancreatealistofseveralobjects(e.g.points,segments,circles).

Examples:

L = {A, B, C}givesyoualistconsistingofthreepriordefinedpointsA,B,andC. L = {(0, 0), (1, 1), (2, 2)} producesalistthatconsistsoftheentered

points,aswellasthesenamelesspoints.

Note:Bydefault,theelementsofthislistarenotshownintheGraphicsView.

CompareListsofObjects

Youcancomparetwolistsofobjectsbyusingthefollowingsyntax:

List1 == List2:Checksifthetwolistsareequalandgivesyoutrueorfalseasaresult.

List1 != List2:Checksifthetwolistsarenotequalandgivesyoutrueorfalseasaresult.

ApplyPre

defined

Operations

and

Functions

to

Lists

Note:Ifyouapplyoperationsandpredefinedfunctionstolists,youwillalwaysgetanew

listasaresult.

AdditionandSubtractionexamples:

List1 + List2:Addscorrespondingelementsoftwolists. Note:Thetwolistsneedtobeofthesamelength.

List + Number:Addsthenumbertoeveryelementofthelist. List1 List2:Subtractstheelementsofthesecondlistfromcorresponding

elementsofthefirstlist.

Note:Thelistsneedtobeofthesamelength. List Number:Subtractsthenumberfromeveryelementofthelist.

MultiplicationandDivisionexamples:

List1 * List2:Multipliescorrespondingelementsoftwolists. Note:Thelistsneedtobeofthesamelength.Ifthetwolistsarecompatible

matrices,matrixmultiplicationisused.

List * Number:Multiplieseverylistelementwiththenumber. List1 / List2:Divideselementsofthefirstlistbycorrespondingelementsofthe

secondlist.

Note:Thetwolistsneedtobeofthesamelength.

List / Number:Divideseverylistelementbythenumber. Number / List:Dividesthenumberbyeveryelementofthelist.

Examplesusingfunctions:

List^2:Squareseveryelementofthelist. sin(List):Appliesthesinefunctiontoeveryelementofthelist.


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3.2.9. MatrixObjectsandOperationsGeoGebraalsosupportsmatrices,whicharerepresentedasalistofliststhatcontainthe

rowsofthematrix.

Example:InGeoGebra,{{1,2,3},{4,5,6},{7,8,9}}representsthematrix

987

654

321

.

MatrixOperations

Additionandsubtractionexamples:

Matrix1 + Matrix2:Addsthecorrespondingelementsoftwocompatiblematrices.

Matrix1 Matrix2:Subtractsthecorrespondingelementsoftwocompatiblematrices.

Multiplicationexamples: Matrix * Number:Multiplieseveryelementofthematrixbythegivennumber. Matrix1 * Matrix2:Usesmatrixmultiplicationtocalculatetheresultingmatrix.

Note:Therowsofthefirstandcolumnsofthesecondmatrixneedtohavethesame

numberofelements.

Example:{{1, 2}, {3, 4}, {5, 6}} * {{1, 2, 3}, {4, 5, 6}}givesyouthematrix{{9,12,15},{19,26,33},{29,40,51}}.

2x2 Matrix * Point(orVector):Multipliesthematrixwiththegivenpoint/vectorandgivesyouapointasaresult.

Example:{{1, 2}, {3, 4}} * (3, 4)givesyouthepointA=(11,25).

3x3 Matrix * Point(orVector):Multipliesthematrixwiththegivenpoint/vectorandgivesyouapointasaresult. Example:{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * (1, 2)givesyouthe

pointA=(8,20).

Note:Thisisaspecialcaseforaffinetransformationswherehomogenous

coordinatesareused:(x,y,1)forapointand(x,y,0)foravector.Thisexampleis

thereforeequivalentto:

{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * {1, 2, 1}.

Otherexamples(seealsosectionMatrixCommands):

Determinant[Matrix]:Calculatesthedeterminantforthegivenmatrix. Invert[Matrix]:Invertsthegivenmatrix Transpose[Matrix]:Transposesthegivenmatrix

3.2.10. ComplexNumbersandOperationsGeoGebradoesnotsupportcomplexnumbersdirectly,butyoumayusepointstosimulate

operationswithcomplexnumbers.


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Example:Ifyouenterthecomplexnumber3 + 4iintotheInputBar,yougetthepoint

(3,4)intheGraphicsView.Thispointscoordinatesareshownas3+4iintheAlgebraView.

Note:YoucandisplayanypointasacomplexnumberintheAlgebraView.Openthe

PropertiesDialogforthepointandselectComplexNumberfromthelistofCoordinates

formatsontabAlgebra.

Ifthevariableihasnotalreadybeendefined,itisrecognizedastheorderedpairi=(0,1)or

thecomplexnumber0+1i.Thisalsomeans,thatyoucanusethisvariableiinordertotype

complexnumbersintotheInputBar(e.g.,q = 3 + 4i).

Additionandsubtractionexamples:

(2 + 1i) + (1 2i)givesyouthecomplexnumber31i. (2 + 1i) - (1 2i)givesyouthecomplexnumber1+3i.

Multiplicationanddivisionexamples:

(2 + 1i) * (1 2i)givesyouthecomplexnumber43i. (2 + 1i) / (1 2i)givesyouthecomplexnumber0+1i.

Note:Theusualmultiplication(2, 1)*(1, -2)givesyouthescalarproductofthetwo

vectors.

Otherexamples:

GeoGebraalsorecognizesexpressionsinvolvingrealandcomplexnumbers.

3 + (4 + 5i)givesyouthecomplexnumber7+5i. 3 - (4 + 5i)givesyouthecomplexnumber15i. 3 / (0 + 1i)givesyouthecomplexnumber03i. 3 * (1 + 2i)givesyouthecomplexnumber3+6i.

3.3. CommandsUsingcommandsyoucanproducenewandmodifyexistingobjects.

Note:Acommand'sresultmaybenamedbyenteringalabelfollowedbyanequalsign(=).

Intheexamplebelow,thenewpointisnamedS.

Example:Togettheintersectionpointoftwolinesgandhyoucanenter

S = Intersect[g, h](seecommandIntersect).

Note:Youcanalsouseindiceswithinthenamesofobjects:A1isenteredasA_1whileSABiscreatedusings_{AB}.

AutomaticCompletionofCommands

WhenyoutypeacommandintoGeoGebrasInputBar,thesoftwaretriestoautomatically

completethecommandforyou.Thismeansthatafteryoutypedinthefirsttwolettersof


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thecommandintotheInputBar,GeoGebradisplaysthefirstcommandofanalphabetically

sortedlistthatstartswiththeseletters.

Inordertoacceptthissuggestionandplacethecursorinbetweenthebrackets,hittheEnterkey.

Ifthesuggestedcommandisnottheoneyouwantedtotypein,justkeeptyping.GeoGebrawilladaptitssuggestionstothelettersyouenter.

3.3.1. GeneralCommandsConstructionStep

ConstructionStep[]:ReturnsthecurrentConstructionProtocolstepasanumber.

ConstructionStep[Object]:ReturnstheConstructionProtocolstepforthegivenobject

asanumber.

Delete

Delete[Object]:Deletestheobjectandallitsdependentsobjects.

Note:Alsoseetool DeleteObject

Relation

Relation[Object a, Object b]:Showsamessageboxthatgivesyouinformation

abouttherelationofobjectaandobjectb.

Note:Thiscommandallowsyoutofindoutwhethertwoobjectsareequal,ifapoint

liesonalineorconic,orifalineistangentorapassinglinetoaconic.

Note:Alsoseetool Relation

3.3.2. BooleanCommandsIf

If[Condition, Object]:Yieldsacopyoftheobjectiftheconditionevaluatestotrue,

andanundefinedobjectifitevaluatestofalse.

If[Condition, Object a, Object b]:Yieldsacopyofobjectaifthecondition

evaluatestotrue,andacopyofobjectbifitevaluatestofalse.

IsDefined

IsDefined[Object]:Returnstrueorfalsedependingonwhethertheobjectisdefinedor

not.


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IsInteger

IsInteger[Number]:Returnstrueorfalsedependingwhetherthenumberisaninteger

ornot.

3.3.3. NumberCommandsAffineRatio

AffineRatio[Point A, Point B, Point C]:Returnstheaffineratioofthree

collinearpointsA,B,andC,whereC=A+*AB.

Area

Area[Point A, Point B, Point C, ...]:Calculatestheareaofthepolygondefined

bythegivenpointsA,B,C,

Area[Conic c]:Calculatestheareaofaconicsectionc(circleorellipse).

Note:

Inordertocalculatetheareabetweentwofunctiongraphs,youneedtousethecommandIntegral.

Alsoseetool AreaAxisStep

AxisStepX[]:Returnsthecurrentstepwidthforthexaxis.

AxisStepY[]:Returnsthecurrentstepwidthfortheyaxis.

Note:TogetherwiththeCornerandSequencecommands,theAxisStepcommandsallowyou

tocreatecustomaxes(alsoseesectionCustomizingCoordinateAxesandGrid).

BinomialCoefficient

BinomialCoefficient[Number n, Number r]:Calculatesthebinomialcoefficient

nchooser.

Circumference

Circumference[Conic]:Returnsthecircumferenceofacircleorellipse.

CrossRatio

CrossRatio[Point A, Point B, Point C, Point D]:Calculatesthecrossratio

offourcollinearpointsA,B,C,andD,where

=AffineRatio[B,C,D]/AffineRatio[A,C,D].


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Curvature

Curvature[Point, Function]:Calculatesthecurvatureofthefunctioninthegiven

point.

Curvature[Point, Curve]:Calculatesthecurvatureofthecurveinthegivenpoint.

Distance

Distance[Point A, Point B]:YieldsthedistanceoftwopointsAandB.

Distance[Point, Line]:Yieldsthedistanceofthepointandtheline.

Distance[Line g, Line h]:Yieldsthedistanceoftheparallellinesgandh.

Note:Thedistanceofintersectinglinesis0.Thus,thiscommandisonlyinteresting

forparallellines.

Note:Alsoseetool DistanceorLength

GCD

UKEnglish:HCF

GCD[Number a, Number b]:Calculatesthegreatestcommondivisorofnumbersaandb

(UKEnglish:HCF=highestcommonfactor).

GCD[List of Numbers]:Calculatesthegreatestcommondivisorofthelistofnumbers

(UKEnglish:HCF=highestcommonfactor).

IntegerDivision

Div[Number a, Number b]:Calculatestheintegerquotientfordivisionofnumberaby

numberb.

Integral

Integral[Function, Number a, Number b]:Returnsthedefiniteintegralofthe

functionintheinterval[a,b].

Note:Thiscommandalsodrawstheareabetweenthefunctiongraphoffandthex

axis.

Integral[Function f, Function g, Number a, Number b]:Yieldsthedefinite

integralofthedifferencef(x)g(x)intheinterval[a,b].

Note:Thiscommandalsodrawstheareabetweenthefunctiongraphsoffandg.

Note:AlsoseecommandforIndefiniteIntegral

Iteration

Iteration[Function, Number x0, Number n]:Iteratesthefunctionntimesusing

thegivenstartvaluex0.

Example:Afterdefiningf(x) = x^2thecommandIteration[f, 3, 2]gives

youtheresult(32)2=81.


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LCM

LCM[Number a, Number b]:Calculatestheleastcommonmultipleoftwonumbersa

andb(UKEnglish:LCM=lowestcommonmultiple).

LCM[List of numbers]:Calculatestheleastcommonmultipleoftheelementsofthelist

(UKEnglish:LCM=lowestcommonmultiple).

Length

Length[Vector]:Yieldsthelengthofthevector.

Length[Point A]:Yieldsthelengthofthepositionvectorofthegivenpoint.

Length[Function, Number x1, Number x2]:Yieldsthelengthofthefunctiongraph

intheinterval[x1,x2].

Length[Function, Point A, Point B]:Yieldsthelengthofthefunctiongraph

betweenthetwopointsAandB.

Note:Ifthegivenpointsdonotlieonthefunctiongraph,theirxcoordinatesare

usedtodeterminetheinterval.

Length[Curve, Number t1, Number t2]:Yieldsthelengthofthecurvebetweenthe

parametervaluest1andt2.

Length[Curve c, Point A, Point B]:Yieldsthelengthofcurvecbetweentwo

pointsAandBthatlieonthecurve.

Length[List]:Yieldsthelengthofthelistwhichisthenumberofelementsinthelist.

Note:Alsoseetool DistanceorLength

LinearEccentricity

LinearEccentricity[Conic]:Calculatesthelineareccentricityoftheconicsection.

Note:Thelineareccentricityisthedistancebetweenaconic'scenteranditsfocus,or

oneofitstwofoci.

LowerSum

LowerSum[Function, Number a, Number b, Number n]:Yieldsthelowersumof

thegivenfunctionontheinterval[a,b]withnrectangles.

Note:Thiscommanddrawstherectanglesforthelowersumaswell.

MinimumandMaximum

Min[Number a, Number b]:Yieldstheminimumofthegivennumbersaandb.

Max[Number a, Number b]:Yieldsthemaximumofthegivennumbersaandb.

ModuloFunction

Mod[Integer a, Integer b]:Yieldstheremainderwhenintegeraisdividedbyinteger

b.


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Parameter

Parameter[Parabola]:Returnstheparameteroftheparabola,whichisthedistanceof

directrixandfocus.

Perimeter

Perimeter[Polygon]:Returnstheperimeterofthepolygon.

Radius

Radius[Circle]:Returnstheradiusofthecircle.

Randomcommands

RandomBetween[Min Integer, Max Integer]:Generatesarandomintegerbetween

minandmax(inclusive).

RandomBinomial[Number n of Trials, Probability p]:Generatesarandom

numberfromabinomialdistributionwithntrialsandprobabilityp.RandomNormal[Mean, Standard Deviation]:Generatesarandomnumberfroma

normaldistributionwithgivenmeanandstandarddeviation.

RandomPoisson[Mean]:GeneratesarandomnumberfromaPoissondistributionwith

givenmean.

SemiMajorAxisLength

SemiMajorAxisLength[Conic]:Returnsthelengthofthesemimajoraxis(halfofthe

majoraxis)oftheconicsection.

SemiMinorAxisLength

SemiMinorAxisLength[Conic]:Returnsthelengthofthesemiminoraxis(halfofthe

minoraxis)oftheconicsection.

Slope

Slope[Line]:Returnstheslopeofthegivenline.

Note:Thiscommandalsodrawstheslopetrianglewhosesizemaybechangedontab

StyleofthePropertiesDialog.

Note:Alsoseetool Slope

TrapezoidalSum

UKEnglish:TrapeziumSum

TrapezoidalSum[Function, Number a, Number b, Number n]:Calculatesthe

trapezoidalsumofthefunctionintheinterval[a,b]usingntrapezoids.

Note:Thiscommanddrawsthetrapezoidsofthetrapezoidalsumaswell.


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UpperSum

UpperSum[Function, Number a, Number b, Number n]:Calculatestheuppersum

ofthefunctionontheinterval[a,b]usingnrectangles.

Note:Thiscommanddrawstherectanglesoftheuppersumaswell.

3.3.4. AngleCommand

Angle

Angle[Vector v1, Vector v2]:Returnstheanglebetweentwovectorsv1andv2

(between0and360).

Angle[Line g, Line h]:Returnstheanglebetweenthedirectionvectorsoftwolinesg

andh(between0and360).

Angle[Point A, Point B, Point C]:ReturnstheangleenclosedbyBAandBC

(between0and360),wherepointBistheapex.

Angle[Point A, Point B, Angle]:ReturnstheangleofsizedrawnfrompointA

withapexB. Note:ThepointRotate[A,,B]iscreatedaswell.

Angle[Conic]:Returnstheangleoftwistofaconicsectionsmajoraxis(seecommand

Axes).

Angle[Vector]:Returnstheanglebetweenthexaxisandgivenvector.

Angle[Point]:Returnstheanglebetweenthexaxisandthepositionvectorofthegiven

point.

Angle[Number]:Convertsthenumberintoanangle(resultbetween0and2pi).

Angle[Polygon]:Createsallanglesofapolygoninmathematicallypositiveorientation(i.

e.,counterclockwise).

Note:Ifthepolygonwascreatedincounterclockwiseorientation,yougetthe

interiorangles.Ifthepolygonwascreatedinclockwiseorientation,yougettheexteriorangles.

Note:Alsoseetools Angleand AnglewithGivenSize

3.3.5. PointCommandsCenter

UKEnglish:Centre

Center[Conic]:Returnsthecenterofacircle,ellipse,orhyperbola.

Note:Alsoseetool MidpointorCenter

Centroid

Centroid[Polygon]:Returnsthecentroidofthepolygon.


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Corner

Corner[Number n of Corner]:CreatesapointatthecorneroftheGraphicsView

(n=1,2,3,4)whichisnevervisibleonscreen.

Corner[Image, Number n of Corner]:Createsapointatthecorneroftheimage

(n=1,2,3,4).

Corner[Text, Number n of Corner]:Createsapointatthecornerofthetext(n=1,2,3,4).

Note:Thenumberingofthecornersiscounterclockwiseandstartsatthelowerleftcorner.

Extremum

UKEnglish:TurningPoint

Extremum[Polynomial]:Yieldsalllocalextremaofthepolynomialfunctionaspointson

thefunctiongraph.

Focus

Focus[Conic]:Yields(all)focioftheconicsection.

InflectionPoint

InflectionPoint[Polynomial]:Yieldsallinflectionpointsofthepolynomialaspoints

onthefunctiongraph.

Intersect

Intersect[Line g, Line h]:Yieldstheintersectionpointoflinesgandh.

Intersect[Line, Conic]:Yieldsallintersectionpointsofthelineandconicsection

(max.2).Intersect[Line, Conic, Number n]:Yieldsthen

thintersectionpointofthelineand

theconicsection.

Intersect[Conic c1, Conic c2]:Yieldsallintersectionpointsofconicsectionsc1

andc2(max.4).

Intersect[Conic c1, Conic c2, Number n]:Yieldsthenthintersectionpointof

conicsectionsc1andc2.

Intersect[Polynomial f1, Polynomial f2]:Yieldsallintersectionpointsof

polynomialsf1andf2.

Intersect[Polynomial f1, Polynomial f2, Number n]:Yieldsthenth

intersectionpointofpolynomialsf1andf2.

Intersect[Polynomial, Line]:Yieldsallintersectionpointsofthepolynomialandtheline.

Intersect[Polynomial, Line, Number n]:Yieldsthenthintersectionpointofthe

polynomialandtheline.

Intersect[Function f, Function g, Point A]:Calculatestheintersectionpoint

offunctionsfandgbyusingNewton'smethodwithinitialpointA.

Intersect[Function, Line, Point A]:Calculatestheintersectionpointofthe

functionandthelinebyusingNewton'smethodwithinitialpointA.


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Note:Alsoseetool IntersecttwoObjects

Midpoint

Midpoint[Point A, Point B]:ReturnsthemidpointofpointsAandB.

Midpoint[Segment]:Returnsthemidpointofthesegment.

Note:Alsoseetool MidpointorCenter

Point

Point[Line]:Returnsapointontheline.

Point[Conic]:Returnsapointontheconicsection.

Point[Function]:Returnsapointonthefunction.

Point[Polygon]:Returnsapointonthepolygon.

Point[Vector ]:Returnsapointonthevector.

Point[Point, Vector]:Createsanewpointbyaddingthevectortothegivenpoint.

Note:Alsoseetool NewPoint

Root

Root[Polynomial]:Yieldsallrootsofthepolynomialasintersectionpointsofthe

functiongraphandthexaxis.

Root[Function, Number a]:Yieldsonerootofthefunctionusingtheinitialvalueafor

Newton'smethod.

Root[Function, Number a, Number b]:Yieldsonerootofthefunctionintheinterval

[a,b](regulafalsi).

Vertex

Vertex[Conic]:Returns(all)verticesoftheconicsection.

3.3.6. VectorCommandsCurvatureVector

CurvatureVector[Point, Function]:Yieldsthecurvaturevectorofthefunctionin

thegivenpoint.

CurvatureVector[Point, Curve]:Yieldsthecurvaturevectorofthecurveinthegiven

point.

Direction

Direction[Line]:Yieldsthedirectionvectoroftheline.

Note:Alinewithequationax+by=chasthedirectionvector(b,a).


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PerpendicularVector

PerpendicularVector[Line]:Returnstheperpendicularvectoroftheline.

Note:Alinewithequationax+by=chastheperpendicularvector(a,b).

PerpendicularVector[Vector v]:Returnstheperpendicularvectorofthegiven

vector.

Note:Avectorwithcoordinates(a,b)hastheperpendicularvector(b,a).

UnitPerpendicularVector

UnitPerpendicularVector[Line]:Returnstheperpendicularvectorwithlength1of

thegivenline.

UnitPerpendicularVector[Vector]:Returnstheperpendicularvectorwithlength1

ofthegivenvector.

UnitVector

UnitVector[Line]:Yieldsthedirectionvectorwithlength1ofthegivenline.

UnitVector[Vector]:Yieldsavectorwithlength1,whichhasthesamedirectionandorientationasthegivenvector.

Vector

Vector[Point A, Point B]:CreatesavectorfrompointAtopointB.

Vector[Point]:Returnsthepositionvectorofthegivenpoint.

Note:Alsoseetool VectorbetweenTwoPoints

3.3.7. SegmentCommandSegment

Segment[Point A, Point B]:CreatesasegmentbetweentwopointsAandB.

Segment[Point A, Number a]:CreatesasegmentwithlengthaandstartingpointA.

Note:Theendpointofthesegmentiscreatedaswell.

Note:Alsoseetools SegmentbetweenTwoPoints

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