[PDF] [PDF] Official GeoGebra Manual

[PDF] Prise en main de GeoGebra 3D - Mathématiques et sciences

3 mar 2013 · une activité sur l'étude de l'aire de la section d'une pyramide (tirée d'un ancien problème du brevet des collèges) une activité portant sur le 

[PDF] An Introduction to GeoGebra - University of Utah Math Department

Clicking on the small button in the upper right corner of the Graphics View will display an Object Properties Bar where you can change how selected objects in the 

[PDF] Official GeoGebra Manual

15 déc 2011 · GeoGebra Manual The official manual of GeoGebra the 'c' term for the line a x + b y + c = 0 (also: z-coordinate, ready for a 3D View)

[PDF] Geogebra handbook pdf - Weebly

Tutorials GeoGebra 3D Graphing App and GeoGebra 3D Graphing Tutorials GeoGebra Classic User Interface The following information is about our GeoGebra 

[PDF] GeoGebra pour le prof de maths

13 oct 2010 · avec Adobe Reader, la majorité des autres lecteurs de Pdf n'ont pas GeoGebra est un logiciel de mathématiques réunissant géométrie, algèbre et calcul : Pour la 3D, il y a des astuces, mais ce n'est pas utilisable par des 

[PDF] Using GeoGebra to Enhance Mathematics Instruction

What's GeoGebra Good for? Compatible with Mac, PC, iPad, and Construction Protocol 2D/3D Graphics View Online manual: http://wiki geogebra

[PDF] GÉOMÉTRIE DANS LESPACE ET GeoGebra

Dans une fenêtre graphique 3D, afficher le plan de base (dans GeoGebra ce plan est désigné par : PlanxOy) Construire une pyramide ABCDS à base carrée

[PDF] Handling of the Software The interface of the program is displayed

GeoGebra 3D with Algebra View, Graphics View and Graphics View 3D textbook authors and also teaching tutorials use videos This format is based on

[PDF] Lutilisation dun logiciel de géométrie dynamique, un - Eduscol

GeoGebra 3D dans plusieurs domaines de la physique 33 http://www geogebra org/manual/en/Manual pour la dernière version en anglais et https:// static

[PDF] GeoGebra Functions - Project Maths

Index Activity Topic Page 1 Introduction GeoGebra Functions 3 2 To draw the function f(x)=x2 4 3 To change the colour, etc of a graph of a function 5 4

[PDF] geogebra 3d vector

[PDF] geogebra 3d vector field

[PDF] geographics atlanta

[PDF] geographics definition

[PDF] geographics examples

[PDF] geographics meaning

[PDF] geographics paper

[PDF] geom gcn

[PDF] geometric sequences and exponential functions worksheet

[PDF] geometric shapes

[PDF] geometric shapes worksheets for 2nd grade

[PDF] geometry 8 3 skills practice answers

[PDF] geometry 8.4 trigonometry answers

[PDF] geometry chapter 11 test form 2c answer key

[PDF] geometry chapter 12 test answers

PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information.PDF generated at: Thu, 15 Dec 2011 01:15:41 UTCGeoGebra ManualThe official manual of GeoGebra.

ContentsArticlesIntroduction1Compatibility3Installation Guide4Objects6Free, Dependent and Auxiliary Objects6Geometric Objects6Points and Vectors7Lines and Axes8Conic sections8Functions9Curves10Inequalities10Intervals11General Objects11Numbers and Angles12Texts13Boolean values14Complex Numbers15Lists15Matrices17Action Objects18Selecting objects19Change Values19Naming Objects20Animation21Tracing22Object Properties22Labels and Captions23Advanced Features25Object Position25Conditional Visibility25Dynamic Colors26LaTeX27

Layers28Scripting28Tooltips30Tools31Tools31Movement Tools32Move Tool32Record to Spreadsheet Tool32Rotate around Point Tool33Point Tools33New Point Tool33Attach / Detach Point Tool34Complex Number Tool34Point on Object Tool34Intersect Two Objects Tool35Midpoint or Center Tool35Line Tools35Vector from Point Tool36Ray through Two Points Tool36Segment with Given Length from Point Tool36Line through Two Points Tool36Segment between Two Points Tool37Vector between Two Points Tool37Special Line Tools37Best Fit Line Tool38Parallel Line Tool38Angle Bisector Tool38Perpendicular Line Tool39Tangents Tool39Polar or Diameter Line Tool39Perpendicular Bisector Tool40Locus Tool40Polygon Tools41Rigid Polygon Tool41PolyLine Tool41Regular Polygon Tool42Polygon Tool42

Circle & Arc Tools42Circle with Center and Radius Tool43Circle through Three Points Tool43Circle with Center through Point Tool43Circumcircular Arc through Three Points Tool43Circumcircular Sector through Three Points Tool44Compass Tool44Circular Sector with Center between Two Points Tool44Semicircle through Two Points Tool45Circular Arc with Center between Two Points Tool45Conic Section Tools45Ellipse Tool46Hyperbola Tool46Conic through Five Points Tool46Parabola Tool46Measurement Tools47Distance or Length Tool47Angle Tool47Slope Tool48Area Tool48Angle with Given Size Tool48Transformation Tools48Translate Object by Vector Tool49Reflect Object about Line Tool49Reflect Object about Point Tool49Rotate Object around Point by Angle Tool49Reflect Object about Circle Tool50Dilate Object from Point by Factor Tool50Special Object Tools50Insert Image Tool51Probability Calculator Tool52Pen Tool52Slider Tool53Relation between Two Objects Tool53Function Inspector Tool54Insert Text Tool54Action Object Tools55Check Box to Show / Hide Objects Tool55

Insert Input Box Tool55Insert Button Tool56General Tools56Custom Tools56Show / Hide Label Tool57Zoom Out Tool57Zoom In Tool58Delete Object Tool58Move Graphics View Tool58Show / Hide Object Tool58Copy Visual Style Tool59Commands60Commands60Geometry Commands60AffineRatio Command61Angle Command61AngleBisector Command62Arc Command62Area Command63Centroid Command63CircularArc Command63CircularSector Command64CircumcircularArc Command64CircumcircularSector Command64Circumference Command65ClosestPoint Command65CrossRatio Command65Direction Command65Distance Command66Intersect Command66IntersectRegion Command67Length Command67Line Command68PerpendicularBisector Command69Locus Command69Midpoint Command70PerpendicularLine Command70

Perimeter Command71Point Command71PointIn Command71PolyLine Command72Polygon Command72Radius Command73Ray Command73RigidPolygon Command73Sector Command73Segment Command74Slope Command74Tangent Command74Vertex Command75Algebra Commands75Div Command76Expand Command77Factor Command77GCD Command78LCM Command79Max Command80Min Command81Mod Command82PrimeFactors Command83Simplify Command83Text Commands84FractionText Command84FormulaText Command84LetterToUnicode Command85Ordinal Command85RotateText Command85TableText Command85Text Command86TextToUnicode Command87UnicodeToLetter Command87UnicodeToText Command87VerticalText Command88Logic Commands88CountIf Command88

IsDefined Command89If Command89IsInRegion Command90IsInteger Command90KeepIf Command90Relation Command90Functions & Calculus Commands91Asymptote Command92Coefficients Command92CompleteSquare Command92ComplexRoot Command93Curvature Command93CurvatureVector Command93Curve Command94Degree Command94Denominator Command95Derivative Command95Extremum Command96Factors Command97Function Command98ImplicitCurve Command98Integral Command98IntegralBetween Command99Intersect Command100Iteration Command101IterationList Command101LeftSum Command101Limit Command102LimitAbove Command102LimitBelow Command103LowerSum Command103Numerator Command104OsculatingCircle Command104PartialFractions Command105PathParameter Command105Polynomial Command106RectangleSum Command106Root Command106

RootList Command107Roots Command107SolveODE Command107TaylorPolynomial Command108TrapezoidalSum Command109InflectionPoint Command109UpperSum Command109Conic Commands109Asymptote Command110Axes Command110Center Command110Circle Command110Conic Command111ConjugateDiameter Command111Directrix Command111Eccentricity Command112Ellipse Command112LinearEccentricity Command112MajorAxis Command113SemiMajorAxisLength Command113Focus Command113Hyperbola Command113Incircle Command114Parabola Command114Parameter Command114Polar Command114MinorAxis Command115SemiMinorAxisLength Command115Semicircle Command115List Commands115Append Command116Classes Command116Element Command116First Command117Frequency Command118IndexOf Command119Insert Command119Intersect Command120

Intersection Command121IterationList Command121Join Command121Last Command122OrdinalRank Command123PointList Command123Product Command123RandomElement Command124RemoveUndefined Command124Reverse Command125RootList Command125SelectedElement Command125SelectedIndex Command125Sequence Command126Sort Command127Take Command127TiedRank Command128Union Command128Unique Command128Zip Command129Vector & Matrix Commands129ApplyMatrix Command130CurvatureVector Command130Determinant Command130Identity Command131Invert Command131PerpendicularVector Command132ReducedRowEchelonForm Command132Transpose Command133UnitPerpendicularVector Command133UnitVector Command134Vector Command135Transformation Commands135Dilate Command135Reflect Command136Rotate Command136Shear Command137Stretch Command137

Translate Command137Chart Commands138BarChart Command138BoxPlot Command139DotPlot Command139FrequencyPolygon Command139Histogram Command140HistogramRight Command141NormalQuantilePlot Command141ResidualPlot Command141StemPlot Command142Statistics Commands142ANOVA Command143Classes Command144Covariance Command144Fit Command145FitExp Command145FitGrowth Command146FitLineX Command146FitLine Command146FitLog Command146FitLogistic Command147FitPoly Command147FitPow Command147FitSin Command148Frequency Command148FrequencyTable Command149GeometricMean Command150HarmonicMean Command150Mean Command150MeanX Command151MeanY Command151Median Command151Mode Command152CorrelationCoefficient Command152Percentile Command152Q1 Command153Q3 Command153

RSquare Command153RootMeanSquare Command153SD Command154SDX Command154SDY Command154Sxx Command155Sxy Command155Syy Command155Sample Command155SampleSD Command156SampleSDX Command157SampleSDY Command157SampleVariance Command157Shuffle Command158SigmaXX Command158SigmaXY Command159SigmaYY Command159Spearman Command159Sum Command160SumSquaredErrors Command161TMean2Estimate Command161TMeanEstimate Command162TTest Command162TTest2 Command163TTestPaired Command163Variance Command164Probability Commands164Bernoulli Command165BinomialCoefficient Command165BinomialDist Command166Cauchy Command167ChiSquared Command168Erlang Command168Exponential Command169FDistribution Command170Gamma Command170HyperGeometric Command171InverseBinomial Command172

InverseCauchy Command172InverseChiSquared Command172InverseExponential Command173InverseFDistribution Command173InverseGamma Command173InverseHyperGeometric Command174InverseNormal Command174InversePascal Command174InversePoisson Command175InverseTDistribution Command175InverseWeibull Command175InverseZipf Command175LogNormal Command176Logistic Command176Normal Command177Pascal Command177Poisson Command178RandomBetween Command179RandomBinomial Command179RandomNormal Command180RandomPoisson Command180RandomUniform Command181TDistribution Command181Triangular Command182Uniform Command182Weibull Command183Zipf Command183Spreadsheet Commands184Cell Command184CellRange Command185Column Command185ColumnName Command185FillCells Command185FillColumn Command186FillRow Command186Row Command186Scripting Commands186Button Command187

Checkbox Command188CopyFreeObject Command188Delete Command188Execute Command189GetTime Command189HideLayer Command189Pan Command190ParseToFunction Command190ParseToNumber Command190PlaySound Command190Rename Command192SelectObjects Command192SetActiveView Command192SetAxesRatio Command192SetBackgroundColor Command193SetCaption Command194SetColor Command194SetConditionToShowObject Command195SetCoords Command195SetDynamicColor Command195SetFilling Command196SetFixed Command196SetLabelMode Command196SetLayer Command197SetLineStyle Command197SetLineThickness Command197SetPointSize Command198SetPointStyle Command198SetTooltipMode Command199SetValue Command199SetVisibleInView Command200ShowLabel Command200ShowLayer Command200Slider Command200StartAnimation Command201InputBox Command201UpdateConstruction Command202ZoomIn Command202

ZoomOut Command203Discrete Math Commands203ConvexHull Command203DelaunayTriangulation Command204Hull Command204MinimumSpanningTree Command204ShortestDistance Command205TravelingSalesman Command205Voronoi Command205GeoGebra Commands205AxisStepX Command206AxisStepY Command206ClosestPoint Command206ConstructionStep Command206Corner Command207DynamicCoordinates Command207Name Command208Object Command208SlowPlot Command208ToolImage Command209Optimization Commands209Maximize Command209Minimize Command209CAS Specific Commands210CFactor Command213CSolutions Command213CSolve Command214CommonDenominator Command214Cross Command215Decimal Command215Dimension Command215Division Command216Divisors Command216DivisorsList Command216DivisorsSum Command217Dot Command217FractionalPart Command217Imaginary Command218

ImplicitDerivative Command218IntegerPart Command218IsPrime Command219LeftSide Command219MatrixRank Command220MixedNumber Command220NIntegral Command220NRoot Command221NSolutions Command221NSolve Command222NextPrime Command223Numeric Command223PreviousPrime Command224RandomPolynomial Command224Rationalize Command225Real Command225RightSide Command225Solutions Command226Solve Command227Substitute Command227ToComplex Command228ToExponential Command228ToPoint Command228ToPolar Command229nPr Command229Predefined Functions and Operators230User interface232Views232Graphics View233Customizing the Graphics View234Algebra View235Spreadsheet View236CAS View237Construction Protocol238Input Bar239Menubar240Toolbar240

Navigation Bar241File Menu241Edit Menu243View Menu245Perspectives246Options Menu247Tools Menu248Window Menu249Help Menu249Context Menu250Customize the Settings250Export Graphics Dialog251Export Worksheet Dialog252Properties Dialog252Redefine Dialog253Tool Creation Dialog254Keyboard Shortcuts255Options Dialog258Virtual Keyboard259Tool Manager Dialog259Accessibility260GeoGebraPrim260Publishing261Creating Pictures of the Graphics View261Upload to GeoGebraTube262Export as html Webpage262Embedding to CMS, VLE (Moodle) and Wiki264Export to LaTeX (PGF, PSTricks) and Asymptote265Printing Options266ReferencesArticle Sources and Contributors267Image Sources, Licenses and Contributors278Article LicensesLicense281

Introduction

1

IntroductionWhat is GeoGebra

GeoGebra

[1] is open source dynamic mathematics software for learning and teaching at all levels. This manual covers the commands and tools of

GeoGebra 4.0

Create dynamic constructions

Constructions in GeoGebra consist of mathematical objects of several types which can be created using

tools or commands . The tutorials may guide you through your first constructions.

ObjectsαGeometric ObjectsαGeneral ObjectsαAction ObjectsαObject PropertiesαNaming ObjectsαLabels and CaptionsαSelecting objectsαChange ValuesαAnimationαTracingαAdvanced FeaturesαScriptingToolsαAbout toolsαList of toolsCommandsαAbout commandsαList of commandsExpressionsαPredefined Functions and Operators

Introduction

2

Get to grips with GeoGebra's user interface

The main window is divided to

views . By default

Algebra View

is displayed on the left side and

Graphics View

on the right. Above these views there is a

Menubar

and

Toolbar

, underneath

Navigation Bar

can be placed. Many features of GeoGebra can be accessed via

Keyboard Shortcuts

. GeoGebra also includes accessibility features such as

Virtual Keyboard

Main componentsαMenubarαToolbarαContext MenuαNavigation BarαVirtual KeyboardαInput BarMenusαFile MenuαEdit MenuαView MenuαOptions MenuαTools MenuαWindow MenuαHelp MenuViewsαAlgebra ViewαCAS ViewαGraphics ViewαSpreadsheet ViewDialogsαProperties DialogαConstruction ProtocolαTool Creation DialogαTool Manager DialogαRedefine DialogαOptions DialogαExport Graphics DialogαExport Worksheet DialogαPrint Preview Dialog

Introduction

3

Publish your workαShare your dynamic worksheets online at GeoGebraTube [2]αPrint your construction, possibly together with the Construction ProtocolαSave image files in various formatsTroubleshootingαThe Installation Guide helps you with installation questions on different platformsαThe Compatibility page explains small differences between GeoGebra versionsαVisit our GeoGebra User Forum [3] if you have any questions or suggestionsReferences[1]http:/ / www. geogebra. org[2]http:/ / www. geogebratube. org[3]http:/ / www. geogebra. org/ forumCompatibilityGeoGebra is backward compatible in sense that every file created with older version should open flawlessly in thecurrent one. There are however several things which behave differently in 3.2 and 4.0:

αlists of angles, integrals, barcharts, histograms etc. are now visibleαlists {Segment[A,B], Segment[B,C] } are now draggableαcircle with given radius (e.g. Circle[(1,1),2]) draggableαDistance[ Point, Segment ] gives distance to the Segment (was to the extrapolated line in 3.2)αAngle[A,B,C] now resizes if B is too close to A or CαIntegral[function f,function g,a,b] is now transcribed to IntegralBetween[function f,function g,a,b].αObjects that are a translation by a free vector are now draggable, eg Translate[A, Vector[(1,1)]]LaTeX issues

The LaTeX rendering is now nicer, but some errors in LaTeX syntax which were ignored in 3.2 will cause missing texts in 4.0. αMake sure that each \left\{ has corresponding \right..

αThe array environment needs specification of columns (although it may be empty). Please use $\begin{array}{} a & b \\ c & d \\ \end{array}$ for left aligned columns or $\begin{array}{rr} a & b \\ c & d \\ \end{array}$ for right-aligned ones. Old syntax $\begin{array} a & b \\ c & d \\ \end{array}$ wouldn't work any more.

Installation Guide

4

Installation GuideWebstartWebstart reinstallation on Windows XPαStart Menu, Run..., type "javaws -viewer" into the open field and press enterαRight-click on GeoGebra -> DeleteαRe-run GeoGebra WebstartWebstart reinstallation on Windows 7αIn the Start Menu type "javaws -viewer" into the search field and press enterαRight-click on GeoGebra -> DeleteαRe-run GeoGebra WebstartWebstart reinstallation on Windows VistaαDisable UACαRestart computerαIn the Start Menu type "javaws -viewer" into the search field and press enterαRight-click on GeoGebra -> DeleteαRe-run GeoGebra WebstartαTurn UAC back onWebstart reinstallation on a MacαDelete the GeoGebra.app from my Applications folder.

αGo into the Java Preferences -> Network -> View Cache Files and delete the GeoGebra.app file that is there andre-run GeoGebra Webstart

Webstart reinstallation on LinuxαOpen a terminalαjavaws -viewerαRight-click on geogebra.jnlp -> DeleteαRe-run GeoGebra WebstartApplet ProblemsFirst, check Java is working on your computer: http:/ / www. java. com/ en/ download/ help/ testvm. xmlαThen Java Control Panel -> General -> Temporary Internet Files -> Settings -> Delete files...αTo get the Java Control Panel in Windows 7, open Control Panel then type "Java" in the search box (top right).Associating .ggb files with Webstart (Windows)

αStart Menu -> Run -> javaws -verbose -import -shortcut -association http:/ / www. geogebra. org/ webstart/geogebra. jnlp

Installation Guide

5 Problems with the offline installer (Windows) removing an old version of

GeoGebraαStart Menu -> Run -> explorer C:\Program Files\Zero G RegistryαEdit this file in Notepad: .com.zerog.registry.xml and remove the GeoGebra related bitsNB C:\Program Files\Zero G Registry is a hidden folder, so normally won't appear in C:\Program FilesNetwork install (Windows)αInstall on a standalone machineαCopy the files from C:\Program Files\GeoGebra to the networkαAssociate .ggb and .ggt files with GeoGebra.exeOther error messages

Error message "Installer User Interface Not Supported" This is a problem when your Windows username contains

unusual characters, eg !, # Solution: Create another user eg Test and install using that http:/ www. hauser-wenz. de/ s9y/ index. php?/ archives/ html

6ObjectsFree, Dependent and Auxiliary ObjectsThere are two types of objects in GeoGebra: free and dependent. Some of them can be defined to be auxiliary.Free objects

are objects whose position or value doesn't depend on any other objects. They are created by direct input or e.g.

New Point Tool. They can be moved, unless they are fixed.Dependent objectsare objects that depend on some other objects. They are created using tools and commands.Auxiliary objects

are either objects which are defined to be auxiliary by user, or objects which were created by specific tools,

e.g. Regular Polygon Tool. Spreadsheet cells are also considered to be auxiliary. They have their separate place in

Algebra View

.Geometric ObjectsGeoGebra works with many types of geometric objectsαPoints and VectorsαLines and AxesαConic sections and ArcsαFunctionsαCurvesαInequalitiesαIntervalsPaths

Some of the above mentioned objects (lines, conic sections, arcs, polygons, functions, single variable inequalities,

intervals, lists of points) are referred to as paths . One can define a point to belong to a path using the Point

Command

. Each point on a path has a path parameter, which is a number ranging from 0 to 1. To get this parameter,

you can use the

PathParameter Command

Note: Lists of other paths are also paths.

Geometric Objects

7

Regions

You can also restrict points to a

region (polygon, conic, arc, two variable inequality) using the

PointIn Command

or

Point on Object Tool

Note:

See also

Attach / Detach Point Tool

.Points and Vectors

Points and vectors may be entered via

Input Bar

in Cartesian or polar coordinates (see

Numbers and Angles

). Points can also be created using Point tools, Vector from Point Tool, Vector between Two Points Tool and a variety of commands

.Note: Upper case labels denote points, whereas lower case labels refer to vectors. This convention is not mandatory.Example:

To enter a point P or a vector v in Cartesian coordinates you may use P = (1, 0) or v = (0, 5). In order to use polar

coordinates type in P = (1; 0π) or v = (5; 90π).Note: You need to use a semicolon to separate polar coordinates. If you donαt type the degree symbol, GeoGebra willtreat the angle as if entered in radians.

CalculationsIn GeoGebra, you can also do calculations with points and vectors.Example:

You can create the midpoint M of two points A and B by entering M = (A + B) / 2 into the Input Bar. You may

calculate the length of a vector v using length = sqrt(v * v)If A = (a, b), then A + 1 returns (a + 1, b + 1). If A is a

Complex Numberscomplex number a+bλ, then A+1 returns a + 1 + bλ.

Vector Product

For two points or vectors

(a, b) (c, d) returns the z-coordinate of vector product (a, b, 0) (c, d, 0) as single number. Similar syntax is valid for lists, but the result in such case is a list. Example:{1, 2} π {4, 5} returns {0, 0, -3}{1, 2, 3} π {4, 5, 6} returns {3, 6, -3}.

Lines and Axes

8

Lines and AxesLines

You can enter a line as a linear equation in x and y or in parametric form into the Input Bar. In both cases previously

defined variables (e. g. numbers, points, vectors) can be used within the equation. Note:

You can enter a line

s name at the beginning of the input followed by a colon.Example:

Type in g: 3x + 4y = 2 to enter line g as a linear equation. You can enter a line in parametric form thus: g: X = (-5, 5)

+ t (4, -3)Define the parameters m = 2 and b = -1. Then, you can enter the equationh: y = m*x + b to get a line h in

y-intercept-form. AxesThe two coordinate axes are available in commands using the names xAxis and yAxis.

Example:

The command

PerpendicularLine

[A, xAxis] constructs the perpendicular line to the x-axis through a given point A.

Conic sectionsYou may enter a conic section as a quadratic equation in x and y. Prior defined variables (e. g., numbers, points,vectors) can be used within the conicαs equation.

Note:

The conic section

s name can be entered at the beginning of the input, followed by a colon.ExamplesConic section Input Ellipse ellell: 9 x^2 + 16 y^2 = 144Hyperbola hyphyp: 9 x^2 λ 16 y^2 = 144Parabola parpar: y^2 = 4 xCircle c1c1: x^2 + y^2 = 25Circle c2c2: (x λ 5)^2 + (y + 2)^2 = 25

Note:

If you define two parameters

a = 4 and b = 3 in advance, you may enter for example an ellipse as ell: b^2 x^2 + a^2 y^2 = a^2 b^2

Functions

9

FunctionsTo enter a function you can use previously defined variables (e. g. numbers, points, vectors) as well as otherfunctions.

Example:αFunction f: f(x) = 3 x^3 λ x^2αFunction g: g(x) = tan(f(x))αNameless function: sin(3 x) + tan(x)

Note: All available predefined functions (e. g. sin, cos, tan) are described in section

Predefined Functions and

Operators

.In GeoGebra you can also use commands to get for example, the integral and derivative of a function. You can use IfCommand to get Conditional Functions.Note: You can also use the commands f'(x) or f''(x), ² in order to get the derivatives of a previously definedfunction f(x).Example: Define function f as f(x) = 3 x^3 λ x^2. Then, you can type in g(x) = cos(f' (x + 2)) inorder to get function g.

Furthermore, functions can be translated by a vector (see

Translate Command

) and a free function can be moved with the mouse by using the Move Tool. Other Transformation Commands can be also applied to functions, but

in most cases the result is not a function but a curve.Limit Function to IntervalIn order to limit a function to an interval [a, b], you can use the Function Command or If Command.

Example:

If[x 3 x

5,x^2]

and

Function[x^2,3,5]

are both definitions of function x 2 restricted to interval [3,5]

Curves

10 CurvesThere are two types of curves in GeoGebra.Parametric curves Parametric curves of the form a(t)=(f(t),g(t)) where t is real parameter within certain range can be created using the

Curve Command

. They can be used in

Tangent Command

and

Point Command

Note:

Parametric curves can be used with pre-defined functions and arithmetic operations. For example, input c(3) returns

the point at parameter position 3 on curve c. Using the mouse you can also place a point on a curve using tool New

Point ToolNew Point or command Point CommandPoint. Since the endpoints a and b are dynamic you can use slider

variables as well (see tool Slider ToolSlider). Creating parametric curve going through given points is not possible. You can however try e.g.

FitPoly Command

to get a function going through these points.

Implicit curvesImplicit curves are polynomials in variables x and y. The can be entered directly into Input Bar.Example: x^4+y^3=2x*yInequalitiesGeoGebra supports inequalities in one or two variables. There are no limitations for inequalities to appear in AlgebraView, but only specific inequalities can be drawn in Graphics View:

αpolynomial inequalities in one variable, e.g. x^3 > x + 1 or y^2>y,αquadratic inequalities in two variables, e.g. x^2 + y^2 + x*y π 4,αinequalities linear in one variable, e.g. 2x > sin(y) or y < sqrt(x).For inequality sign you can use <, >, β, ³. The Symbols <= and => also valid.

Inequalities are similar to functions, you can test whether x and y satisfy inequality a by typing a(x,y) in the Input Bar , also when A is a point, syntax a(A) is valid. A point can be restricted to the region given by inequality using

PointIn Command

. For inequality b in one variable, e.g. in x , Point[b] yields a point restricted to the part of x-axis which satisfies inequality b Conjunction and disjunction are also supported for inequalities, e.g. (x y) (x+y 3) can be drawn.

Intervals

11

Intervals

An interval is a set of numbers between upper and lower bound. To create an interval, type e.g.

2 < x < 3

in

Input Bar

. Interval in previous example is open. You can also define closed ( 2 x 3 ) and semi-closed ( 2 x < 3 ) intervals. Note:

See also

Boolean values

To determine whether number

c belongs to interval r type r(c) into the Input Bar, the result will be a Boolean value. Generalization of intervals are

Inequalities

Commands for intervals

αMin, Max, Midpoint for an interval with lower bound a and upper bound b return numbers a, b and\(\frac{a+b}2\) respectively. The result doesn't depend on whether the interval is open, closed or semi-closed.

αPoint returns a moveable point whose x-coordinate belongs to the interval and y-coordinate is 0.

αPointIn returns a moveable point whose x-coordinate belongs to the interval and y-coordinate may be changedarbitrarily.

General ObjectsBesides Geometric Objects GeoGebra can also handleαNumbers and AnglesαComplex NumbersαBoolean valuesαListsαMatricesαTexts

Numbers and Angles

12

Numbers and AnglesNumbers

You can create numbers by using the Input Bar. If you only type in a number (e. g., 3), GeoGebra assigns a lower

case letter as the name of the number. If you want to give your number a specific name, you can type in the name

followed by an equal sign and the number (e. g., create a decimal r by typing in r = 5.32). Note: In GeoGebra, numbers and angles use a period (.) as a decimal point.

You can also use the constant ² and the Euler constant e for expressions and calculations by selecting them from the

drop down list next to the Input Bar or by using

Keyboard Shortcuts

.Note: If the variable e is not used as a name of an existing object yet, GeoGebra will recognize it as the Eulerconstant if you use it in new expressions.

Angles

Angles are entered in degree (π) or radians (rad). The constant ² is useful for radian values and can also be entered as

pi. Note:

You can enter a degree symbol (π) or the pi symbol (²) by using the following keyboard shortcuts:(Mac OS: ) for the degree symbol π (Mac OS: ) for the pi symbol ²Example: You can enter an angle ³ in degree (e. g., ³ = 60π) or in radians (e. g.,³ = pi/3).

Note:

GeoGebra does all internal calculations in radians. The degree symbol (π) is nothing but the constant ²/180

used to convert degree into radians.

Example:

If a = 30 is a number, then ³ = aπ converts number a to an angle ³ = 30π, without changing its value. If you type in b

= ³ / π, the angle ³ is converted back to the number b = 30, without changing its value.Note: For dependent angles you can specify whether they may become reflex or not on tab Basic of the PropertiesDialog.

Free Numbers and Angles

Free numbers and angles can be displayed as sliders in the Graphics View (see

Slider Tool). Using the arrow

keys, you may change the value of numbers and angles in the

Algebra View

too (see

Manual Animation

section).Limit Value to Interval

Free numbers and angles may be limited to an interval [min, max] by using tab Slider of the Properties Dialog (see

also

Slider Tool).

Texts 13 Texts

Text objects can be easily created using

Text Command

or

Insert Text Tool, or dragging an object from the

Algebra View to the Graphics View. Another way

for advanced users (described below) is typing into

Input Bar

directly.

Static textdoes not depend on any mathematical objects and is usually not affected by changes of the construction.Dynamic textcontains values of objects that automatically adapt to changes made to these objects.Mixed text

is a combination of static and dynamic text. In order to create a mixed text you may enter the static part of the

text using the keyboard (e. g., Point A =). Then, click on the object whose value you want to display in the

dynamic part of the text. Note:

GeoGebra automatically adds the syntax ("Point A = " + A ) necessary to create your mixed text: quotation

marks around the static part of the text and a plus (+) symbol to connect the different parts of the text.Input Description This is static textStatic textADynamic text (if point A exists)"Point A = " + ATwo-part mixed text using the value of point A"a = " + a + "cm"Three-part mixed text using the value of number a

Note: If an object with the name xx already exists and you want to create a static text using the object s name, you

need to enter it with quotation marks ("xx"). Otherwise, GeoGebra will automatically create a dynamic text that

gives you the value of object xx instead of its name. However, you can type any text that doesn t match any existing object s name without the quotation marks. Note:

Within a mixed text, the static part needs to be in between a pair of quotation marks. Different parts of a text

(e. g., static and dynamic parts) can be connected using plus (+) symbols. Since 4.0, the + symbols are not

mandatory.

Text objects can also use

LaTeX for typesetting math.

Boolean values

14

Boolean valuesYou can use the Boolean variables true and false in GeoGebra. Just type, for example, a = true or b =false into the Input Bar and press the Enter-key.

Check Box and Arrow Keys

Free Boolean variables can be displayed as check boxes in the

Graphics View

(see tool

Check Box to

Show/Hide Objects Tool

). By using the arrow keys of your keyboard you may also change Boolean variables in the

Algebra View (see section

Manual Animation

Note:

You may also use Boolean variables like numbers (value 0 or 1). This allows you to use a checkbox as the

dynamic speed of an animated slider allowing you to start and stop the animation. In this case, the animation button

is only shown in the Graphics View if there is also an animated slider with static (i. e. non-dynamic) speed.

Operations

You can use the following operations for Boolean variables and conditions in GeoGebra by either selecting them

from the list next to the Input Bar or by entering them using the keyboard:Operation List Keyboard Example Object types EqualΦ ==a Φ b or a == bnumbers, points, lines, conics a, bUnequalσ !=a σ b or a != bnumbers, points, lines, conics a, bLess than<>a > bnumbers a, bLess or equal thanβ <=a β b or a <= bnumbers a, bGreater or equal than³ >=a ³ b or a >= bnumbers a, bAnd¹ &&a ¹ b or a && bBooleans a, bOrμ ||a μ b or a || bBooleans a, bNot¹!¹a or !aBoolean aParallelί a ί blines a, bPerpendicular‹ a ‹ blines a, bBelongs toŒ a Œ list1number a, list of numbers list1

Complex Numbers

15

Complex NumbersGeoGebra does not support complex numbers directly, but you may use points to simulate operations with complexnumbers.Example: If you enter the complex number 3 + 4i into the Input Bar, you get the point (3, 4) in the Graphics View.This pointαs coordinates are shown as 3 + 4i in the Algebra View.Note: You can display any point as a complex number in the Algebra View. Open the Properties Dialog for the pointand select Complex Number from the list of Coordinates formats on tab Algebra.

If the variable i has not already been defined, it is recognized as the ordered pair i = (0, 1) or the complex number 0 +

1i. This also means, that you can use this variable i in order to type complex numbers into the Input Bar (e. g., q = 3

+ 4i).

Example:

Addition and subtraction:(2 + 1i) + (1 λ 2i) gives you the complex number 3 λ 1i. (2 + 1i) - (1 λ 2i) gives you the complex number 1 + 3i.Example: Multiplication and division:(2 + 1i) * (1 λ 2i) gives you the complex number 4 λ 3i. (2 + 1i) / (1 λ 2i) gives you the complex number 0 + 1i.Note: The usual multiplication (2, 1)*(1, -2) gives you the scalar product of the two vectors.GeoGebra also recognizes expressions involving real and complex numbers.Example:

3 + (4 + 5i) gives you the complex number 7 + 5i. 3 - (4 + 5i) gives you the complex number -1 - 5i. 3 / (0 + 1i)

gives you the complex number 0 - 3i. 3 * (1 + 2i) gives you the complex number 3 + 6i.

ListsUsing curly braces you can create a list of several objects (e. g. points, segments, circles).Example:

L = {A, B, C} gives you a list consisting of three prior defined points A, B, and C. L = {(0, 0), (1, 1), (2, 2)}

produces a list that consists of the entered points, as well as these nameless points. Note: By default, the elements of this list are not shown in the

Graphics View

To access particular elements of the list you can use

Element Command

. Lists can be used as arguments in list operations (mentioned further in this article) or

List Commands

Compare Lists of ObjectsYou can compare two lists of objects by using the following syntax:αList1 == List2: Checks if the two lists are equal and gives you true or false as a result.αList1 != List2: Checks if the two lists are not equal and gives you true or false as a result.List Operations