Home

GeoGebra Help Official Manual 3.2

Contents

1. 35 Lower Sum command E 43 Lowest Common Multiple commande 43 M Major Axis Commande 50 Manage Tools Tools Menu 76 Manual anima risa 79 Matrices A trates Caiarcs aetna erased 38 Apply arithmetic operations ccsssseeseeeceeeeeeeees 38 COMMINAS A ess 38 Direct IN put roo 38 Et e eg e WEE 38 Operation LE 38 SUD CHO ou 38 Max command WEE 43 60 Maximum Commande 43 60 Mean CONTAINING ME 65 MeanX command e 65 MeanY command ii 65 Median COMMAND 65 Menu Edito EE 71 el 69 aE e EE A oem ween 77 OPIO S tias A aimee 74 A E 76 MOIEN ee EE Ee 73 WY NIG EE 77 Menu tes lada 69 Midpoint Commande e ci coc E E EEE 47 Midpoint or Center tool 19 ne leg noe KEE 43 60 Minimum Commande 43 60 Minor AXIS COMIMAN e BEE 50 WIC A feet Senne ene iene ek 27 Mod command EE 43 Mode COMMA dina iii 66 Modulo Function Commande 43 Move 96 Move Drawing Pad tool 17 Ke ie A 17 Multiplication direct Input 35 N Name COM aa 56 Conic Section dd 34 ear gl ett EE 31 DM ER OF CONIC Section DEE 30 of function se ato da 30 A 30 Re el 30 IR E EE 30 A 33 EE ER Naming Obi 30 KSE get RE 11 VIEW Mesias 74 New Point TOO ura 19 New Window Mn dE 69 Window MERU ica datar nando 77 New Mens ias 69 Normal command vusciii 66 Number DECI UE et lo ba aes 32 E o A 33 A eect ieee eee ee ee 32 Object Algebraic nota caia 10 Animation Oia deta 10 Auxiliary Oe idos 7 A dasi A sovessssaredadtansiavassa
2. 59 Keeplf KeepIf Condition List Creates a new list that only contains those elements of the initial list that fulfill the condition Example KeepIf x lt 3 11 2 3 4 1 5 6 returns the new list 1 2 1 Last Last List Returns the last element of the list Last List Number n of Elements Returns a list containing just the last n elements of the list Length Length List Yields the length of the list which is the number of list elements Min Min List Returns the minimal element of the list Max Max List Returns the maximal element of the list Product Product List of Numbers Calculates the product of all numbers in the list RemoveUndefined RemoveUndefined List Removes undefined objects from a list Example RemoveUndefined Sequence 1 i i 3 1 0 5 removes the second and fourth element of the sequence which have a non integer exponent and therefore are undefined Reverse Reverse List Reverses the order of a list Sequence Sequence Expression Variable i Number a Number b Yields a list of objects created using the given expression and the index that ranges from number a to number b Example L Sequence 2 i i 1 5 creates a list of points whose y coordinates range from 1 to 5 L 2 1 2 2 2 3 2 4 2 5 Sequence Expression Variable i Number a Number b Increment Yields a list of objects created using the given expression and
3. Static tex A Dynamic text if point A exists Two part mixed text using the value of point A Three part mixed text using the value of E VW a cm 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 need to be connected using plus symbols LaTeX Formulas In GeoGebra you can write formulas as well To do so check the box LaTeX formula in the dialog window of the Insert Text tool and enter your formula in LaTeX syntax Note In order to create text that contains a LaTeX formula as well as static text you may enter the static part of the text and then add the LaTeX formula in between a set of dollar symbols Example The length of the diagonal is sqrt 2 S You can select the syntax for common formula symbols from the drop down menu next to the LaTeX checkbox This inserts the corresponding LaTeX code into the text field and places the cursor in between a set of curly brackets If you would like to create dynamic text
4. Creates a bar chart over the given interval a b that calculates the bars heights using the expression whose variable k runs from number c to number d using step width s BarChart List of Raw Data Width of Bars Creates a bar chart using the given raw data whose bars have the given width Example par Chart ll ld ll Laa ar el BarChart List of Data List of Frequencies Creates a bar chart using the list of data with corresponding frequencies Note The List of data must be a list where the numbers go up by a constant amount Examples e BarChart 10 11 12 13 14 5 8 12 0 1 e BarCharclia O Ur Sy Bh ddy Us Lear 43y ek 63 e BaerChart 10 30 Usg Usos Us 1L2 337 13 4 BarChart List of Data List of Frequencies Width of Bars w Creates a bar chart using the list of data and corresponding frequencies whose bars are of width w Note The List of data must be a list where the numbers go up by a constant amount Examples e BarChart 10 11 12 13 14 5 8 12 0 1 0 5 leaves gaps between bars e BarChart 10 11 12 13 14 5 8 12 0 1 0 produces a line graph BoxPlot BoxPlot yOffset yScale List of Raw Data Creates a box plot using the given raw data and whose vertical position in the coordinate system is controlled by variable yOffset and whose height is influenced by factor yScale Example BoxPlot 0 1 2 2 3 4 5 5 6 7 7 8 8 8 9 BoxPlot yOffset yScale Start Value a Q1 Median 03 End Value b Create
5. The resolution of an EPS graphic is always 72dpi This value is only used to calculate the true size of an image in centimeters and has no effect on the image s quality Note The transparency effect with filled polygons or conic sections is not possible with EPS 70 e PDF Portable Document Format see EPS format above Note In SVG and PDF export you have the option to export text as editable text or shapes This stores the text either as text this lets you edit the text in e g InkScape or as B zier curves this guarantees that the text looks the same even if the correct font is not installed e SVG Scaleable Vector Graphic see EPS format above e EMF Enhanced Meta Format see EPS format above Export Graphics View to Clipboard Keyboard shortcut Ctrl Shift C Mac OS Cmd Shift C This menu item copies the Graphics View to your computer s clipboard Afterwards you can easily paste this picture into other documents e g word processing document Export Graphics View as PSTricks Keyboard shortcut Ctrl Shift T Mac OS Cmd Shift T This menu item allows you to save the Graphics View as a PSTricks picture file which is a LaTeX picture format Export Graphics View as PGF Tikz This menu item allows you to save the Graphics View as a PGF TikZ picture file which is a LaTeX picture format Ka Close Keyboard shortcut A t F4 Mac OS Cmd W This menu item closes the GeoGebra window If you didn t sa
6. derivative of the function Note You can use f x instead of Derivative f as well as x instead of Derivative f 2 and so on Expand Expand Function Multiplies out the brackets of the expression Example Expand x 3 x 4 gives you f x x x 12 Factor UK English Factorise Factor Polynomial Factors the polynomial Example Factor x 2 x 6 gives you f x x 2 x 3 52 Function Function Function Number a Number b Yields a function graph that is equal to fon the interval a b and not defined outside of a b Note This command should be used only in order to display functions in a certain interval Example f x Function x 2 1 1 gives you the graph of function x in the interval 1 1 If you then type in g x 2 f x you will get the function g x 2 x but this function is not restricted to the interval 1 1 Integral Integral Function Yields the indefinite integral for the given function Note Also see command for Definite integral Polynomial Polynomial Function Yields the expanded polynomial function Example Polynomial x 3 2 yields x 6x 9 Polynomial List of n points Creates the interpolation polynomial of degree n 1 through the given n points Simplify Simplify Function Simplifies the terms of the given function if possible Examples e Simplify x x x gives you a function f x 3x e Simplify sin x cos x gives you a
7. layers are drawn on top of lower numbered layers Using the Advanced tab of the Properties Dialog you can change the layer for a certain object layers from O to 9 available Once you change the layer number for at least one object to be different from layer 0 e g to layer 3 all new objects will be drawn on the layer with the highest number that is used for any object Note After selecting any object you can select all objects in the same layer by selecting item Select Current Layer keyboard shortcut Ctrl L from the Edit menu This menu item is only available if all selected objects lie on the same layer 87 Further use of layers e For SVG export objects are grouped by layer e Layers can be controlled using the JavaScript Interface for GeoGebra applets 5 9 Redefine Redefining objects is a very versatile tool to change a construction Please note that this may also change the order of the construction steps in the Construction Protocol In GeoGebra an object may be redefined in different ways e Select the Move tool and double click on any object in the Algebra View o For free objects an editing field is opened allowing you to directly change the algebraic representation of the object Hit the Enter key in order to apply these changes o For dependent objects the Redefine dialog is opened allowing you to redefine the object e Select the Move tool and double click on any object in the Graphics View This open
8. 5 1x 4 161 creates a table as a text object with three rows and two columns All items of the table are left aligned e TableText Seguence i 2 i 1 10 creates a table as a text object with one row All items of the table are left aligned TableText List 1 List 2 List 3 Alignment of text Creates a text that contains a table of the list objects The optional text Alignment of text controls the orientation and alignment of the table text Note Possible values are vl vc vr v h hl hc hr Default is hl e v vertical i e lists are columns e h horizontal i e lists are rows left aligned 56 e r right aligned e c centered Examples e TableText 1 2 3 4 1 4 9 16 v creates a text with two columns and four rows whose elements are left aligned o TableText Ll 2 3 4rr1lp4 9 L0 5 Hn creates a text with two rows and four columns whose elements are left aligned TableTextE till 2 1234173242529 234 0 yr creates a text with one column whose elements are right aligned Text Text Object Returns the formula for the object as a text object Note By default values are substituted for variables Example If a 2 and c a then Text c returns the text 4 Text Object Boolean Returns the formula for the object as a text object The Boolean variable determines if values are substituted for variables true or if variable names are shown in the
9. 79 DECO ME aria aia 79 CCA pl tad 79 NC Man 80 Manta o O 79 Oslo 79 EE ee 79 Plus MiNUS Key Sisi 79 A A en meet ee 79 EE 79 Antihyperbolic Antihyperbolic cosine direct Input 35 Antihyperbolic sine direct Input 35 Antihyperbolic tangent direct Input 36 Append COMMANG WE 58 Arc Arc cosine direct Input 35 Arc sine direct Input 35 Arc tangent direct Input 35 Circular Arc with Center between Two Points tool 23 Circular Arc Commande 54 Circumcircular Arc through Three Points tool 23 Circumcircular Arc Commande 55 90 comman O A 54 Area Area between two functons 42 command A A SA 41 Definite integral ara 41 42 EGOO EE 24 Arithmetic Operations EE 35 POW E 33 ANIMATO EE 79 Asymptote Commande 49 Automatic animation id 79 Auxiliary OD CUS eer a E atan 7 8 Vew MENU e 73 Axes COMAN eea a EA aie 49 E Ern 9 Ps lee E E 34 A ENEE 9 A EEA 9 A O et MR PROS a 17 SCIE ee 9 ne E 9 Show hide individually ocooooooonocnncnncnnnnnnnoss 9 TEMAS o Eco 9 te EE 9 View TE E EE 73 KAMIS ANCA AMIS 2icscveseuiucioesveenieaunwunomynncmetnd 34 Axis Step COMMMANG sia an 41 B BacCkerOund Image rennuna a a 29 Bar Chart ell Ee MEN 63 Best Fit Line tool EE 20 Binomial Coefficient commande 41 Bisector Angle Bisector Commande 49 Angle Bisector tool 20 Perpendicular Bisector Commande 50 Perpendicular Bisector tool 21 Boolean Boolean Variable tool 25 Ri ege 36 ShowW hi
10. B and point A e Drag point A along the x axis to see point B moving along its locus line 2 2 13 Geometric Transformation Tools The following geometric transformations work for points lines conic sections polygons and images e Dilate Object from Point by Factor UK English Enlarge Object from Point by Factor Select the object to be dilated Then click on a point to specify the dilation center and enter the dilation factor into the text field of the appearing dialog window also see commands Dilate US and Enlarge UK 25 LC N Reflect Object about Line UK English Reflect Object in Line Select the object you want to reflect Then click on a line to specify the mirror line of reflection also see command Reflect Reflect Object about Point UK English Reflect Object in Point Select the object you want to reflect Then click on a point to specify the mirror point of reflection also see command Reflect kV f Reflect Point about Circle UK English Reflect Point in Circle This tool allows you to invert a point in a circle Select the point you want to invert Then click on a circle to specify the mirror circle of inversion also see command Reflect Rotate Object around Point by Angle Select the object you want to rotate Then click on a point to specify the center of rotation and enter the rotation angle into the text field of the appearing dialog window also see command Rota
11. FitSin List of Points Calculates the regression curve in the form a b sin cx d Note The list should have at least 4 points preferably more The list should cover at least two extremal points The first two local extremal points should not be too different from the absolute extremal points of the curve Histogram Histogram List of Class Boundaries List of Heights Creates a histogram with bars of the given heights The class boundaries determine the width and position of each bar of the histogram Example Histogram 0 1 2 3 4 5 12 6 8 3 1 createsa histogram with 5 bars of the given heights The first bar is positioned at the interval 0 1 the second bar is positioned at the interval 1 2 and so on Histogram List of Class Boundaries List of Raw Data Creates a histogram using the raw data The class boundaries determine the width and position of each bar of the histogram and are used to determine how many data elements lie in each class Example Histogrammi id 2 Ce 2 IDO Lels Dal 12 Dery 2 2 2 5 4 0 creates a histogram with 3 bars with the heights 5 first bar 2 second bar and 1 third bar InverseNormal InverseNormal Mean u Standard Deviation o Probability P Calculates the function O P o u where O 7 is the inverse of the probability density function O for N 0 1 Note Returns the x coordinate with the given probability to the left under the normal distribution curve Mean com
12. In addition you can set the Scale of your printout in cm and change the Orientation of the paper used portrait or landscape Note In order to update the Print Preview after you made changes to the text or layout of the printout you need to press the Enter key Printing the Construction Protocol If you want to print the Construction Protocol you first need to open the Construction Protocol dialog window by using the View menu Then you can open the Print Preview window of the Construction Protocol from the File menu of this new window Again you may enter Title Author and a Date or change the Scale or paper Orientation before printing your Construction Protocol Note You may switch the different columns Name Definition Command Algebra and Breakpoint of the Construction Protocol on and off by using the View menu of the Construction Protocol dialog window 1 4 2 Creating Pictures of the Graphics View Saving the Graphics View as a Picture You can save the Graphics View of your constructions as a picture file on your computer Note The full Graphics View will be saved as a picture If your construction does not use all the available space in the Graphics View you might want to e use tools Y Move Drawing Pad E Zoom In and or SL Zoom Out in order to place your construction in the upper left corner of the Graphics View Afterwards you may reduce the size of the GeoGebra window by dragging one of its corners with the
13. Mac OS Ctr click on an object and select a Properties from the appearing Context Menu e Select item lA Properties from the Edit menu e Select the Move tool and double click on an object in the Graphics View In the appearing Redefine dialog window click on the button Properties In the Properties Dialog objects are organized by types e g points lines circles in the list on the left hand side which makes it easier to handle large numbers of objects You need to select one or more objects from this list in order to change its their properties Note By clicking on a heading in the list of objects e g Point you can select all objects of this type and therefore quickly change the properties for all these objects You can modify the properties of selected objects using the tabs on the right hand side e g Basic Color Style Advanced Note Depending on the selection of objects in the list a different set of tabs may be available Close the Properties Dialog when you are done with changing properties of objects 1 2 3 The Context Menu The Context Menu provides a quick way to change the behavior or advanced properties of an object Right click Mac OS Ctr click on an object in order to open its Context Menu For example it allows you to change the object s algebraic notation e g polar or Cartesian coordinates implicit or explicit equation and to directly access features like fb Rename Delete
14. Reflect COMMING anoa eae or Reflect Object about Line tool Reflect Object about Point tool Reflect Object in Line tool Reflect Object in Point tool Reflect Point about Circle tool Reflect Point in Circle tool Reflex anple inienn rE Meuse N ORAT Refresh Views View Menu Regular Polygon tool Relation Remainder Commande Remove Undefined commande Rename Context Men na ads Fast renaming option aislar Restore Defaultsetting S aes mee r ees Default Settings Options Menu occccccccnccnnnnnnnnnnos Default Toolbar REVElSE COMIMANG ME Right Angle Style Options Menu ias SA id ROO enee le EE Rotate Command todos Rotate around Point tool Rotate Object around Point by Angle tool Rounding Direct INPUT id ls tarse Options MU rad all E Row een glo ET dana S Save Save AS Mens Save Settings Options Menu A A dees Scalar product direct input oocccccccocccnnncnonononnnnananonos SD CORMAN lar aia Second axis Command WEE Second axis length Commande Sector Circular Sector with Center between Two Points Circular Sector Commande Circumcircular Sector through Three Points too Circumcircular Sector Commande A its 98 Segment COM aos 48 Segment between Two Points too l 19 Segment with Given Length from Point tool 19 Select O ane aces 16 Select All Edit Menu 72 Select Ancestors Edit Menu 72 Sele
15. be pasted into other documents e g a word processing document Note In Order to export your construction at a certain scale in cm please use the menu item E Graphics View as Picture in the File menu Export 1 4 3 Creating Interactive Webpages GeoGebra allows you to create interactive webpages so called Dynamic Worksheets from your files In the File menu you need to select item Export before you can click on item E Dynamic Worksheet as Webpage html This opens the export dialog window for Dynamic Worksheets e Atthe top of the export window you can enter the Title Author and a Date for your Dynamic Worksheet e Tab General allows you to add some text above and below the dynamic construction e g a description of the construction and some tasks You can also determine if the construction itself may be included directly into the webpage or if it can be opened by clicking on a button e Tab Advanced allows you to change the functionality of the dynamic construction e g show a reset icon double click should open the GeoGebra application window as well as to modify the user interface shown in the interactive applet e g show the Toolbar modify height and width 14 Note If the size of your applet is too big to fit on a computer screen with standard resolution 1024 x 768 you may want to resize it before the actual export as a Dynamic Worksheet Note Several files are created when you export a Dynamic Worksheet
16. by an equal sign In the example below the new point is named S Example To get the intersection point of two lines g and h you can enter S Intersect g h see command Intersect Note You can also use indices within the names of objects A is entered as A_1 while Sag is created using s AB Automatic Completion of Commands When you type a command into GeoGebra s Input Bar the software tries to automatically complete the command for you This means that after you typed in the first two letters of 39 the command into the Input Bar GeoGebra displays the first command of an alphabetically sorted list that starts with these letters e n order to accept this suggestion and place the cursor in between the brackets hit the Enter key e Ifthe suggested command is not the one you wanted to type in just keep typing GeoGebra will adapt its suggestions to the letters you enter 3 3 1 General Commands ConstructionStep ConstructionStep Returns the current Construction Protocol step as a number ConstructionStep Object Returns the Construction Protocol step for the given object as a number Delete Delete Object Deletes the object and all its dependents objects Note Also see tool 7 Delete Object Relation Relation Object a Object b Shows a message box that gives you information about the relation of object a and object b Note This command allows you to find out whether two objects are equal if a po
17. compatible matrices Multiplication examples e Matrix Number Multiplies every element of the matrix by the given number e Matrixl Matrix2 Uses matrix multiplication to calculate the resulting matrix Note The rows of the first and columns of the second matrix need to have the same number of elements Example 1 2 3 4 5 6 1 2 3 14 5 6 gives you the matrix 9 12 15 19 26 33 29 40 STI e 2x2 Matrix Point or Vector Multiplies the matrix with the given point vector and gives you a point as a result Example 11 2 3 4 3 4 gives you the point A 11 25 e 3x3 Matrix Point or Vector Multiplies the matrix with the given point vector and gives you a point as a result Example 1 2 3 14 5 6 0 O 1 1 2 gives you the point A 8 20 Note This is a special case for affine transformations where homogenous coordinates are used x y 1 for a point and x y 0 for a vector This example is therefore equivalent to o ee Siy ide Sak Ger O 1 A lee Dee A Other examples see also section Matrix Commands e Determinant Matrix Calculates the determinant for the given matrix e Invert Matrix Inverts the given matrix e Transpose Matrix Transposes the given matrix 3 2 10 Complex Numbers and Operations GeoGebra does not support complex numbers directly but you may use points to simulate operations with complex numbers 38 Example If you enter the
18. complex number 3 4i into the Input Bar you get the point 3 4 in the Graphics View This point s coordinates are shown as 3 4i in the Algebra View Note You can display any point as a complex number in the A gebra View Open the Properties Dialog for the point and 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 O 1i This also means that you can use this variable in order to type complex numbers into the Input Bar e g q 3 41 Addition and subtraction examples e 2 11 1 21 gives you the complex number 3 1i e 2 1i 1 21 gives you the complex number 1 3i Multiplication and division examples e 2 11 1 2i gives you the complex number 4 3i e 2 11 1 21 gives you the complex number O 1i Note The usual multiplication 2 1 1 2 gives you the scalar product of the two vectors Other examples GeoGebra also recognizes expressions involving real and complex numbers e 3 4 51 gives you the complex number 7 5i e 3 4 51 gives you the complex number 1 5i e 3 0 11 gives you the complex number 0 3i e 3 1 21 gives you the complex number 3 6i 3 3 Commands Using commands you can produce new and modify existing objects Note A command s result may be named by entering a label followed
19. degree symbol e Alt P Mac OS Ctrl P for the pi symbol 7t Example You can enter an angle a in degree e g a 60 or in radians e g a 1 3 Note GeoGebra does all internal calculations in radians The degree symbol is nothing but the constant 71 180 used to convert degree into radians Examples e lfa 30isanumber thena a converts number a to an angle a 30 without changing its value e lfyoutypeinb a the angle ais converted back to the number b 30 without changing its value 32 Sliders and Arrow Keys Free numbers and angles can be displayed as sliders in the Graphics View see tool 2 Slider Using the arrow keys you may change the value of numbers and angles in the Algebra View too see section Manual Animation 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 tool Slider Note For dependent angles you can specify whether they may become reflex or not on tab Basic of the Properties Dialog 3 2 2 Points and Vectors Points and vectors may be entered in Cartesian or polar coordinates see section Numbers and Angles Note Upper case labels denote points whereas lower case labels refer to vectors Examples e To enter a point P or a vector v in Cartesian coordinates you may use P 1 0 Orv 0 5 e n order to use polar coordinates type inP 1 0 orv 5 90 Note You
20. e HTML file e g circle html this file includes the worksheet itself e GGB file e g circle ggb this file includes your GeoGebra construction e JAR several files these files include GeoGebra and make your worksheet interactive All these files e g circle html circle ggb and the geogebra jar files have to be in one folder directory to let the dynamic construction work The exported HTML file e g circle html can be viewed with any Internet browser e g Mozilla Internet Explorer Safari In order to let the dynamic construction work Java has to be installed on the computer You can get Java from http www java com without charge If you want to use your Dynamic Worksheet in your school s computer network ask your local network administrator to install Java on the computers Note You can edit the Dynamic Worksheet s text with many word processing systems e g FrontPage OpenOffice Writer by opening the exported HTML file You may also edit the Dynamic Worksheet s applet by opening the GGB file in GeoGebra and saving it with the same name afterwards 15 2 Geometric Input 2 1 General Notes The Graphics View shows the graphical representation of mathematical objects e g points vectors segments polygons functions curves straight lines conic sections Whenever the mouse is moved over one of these objects a description appears as a roll over text and the object is highlighted There are several
21. find tools that create different types of points in the Point Toolbox default icon and tools that allow you to apply geometric transformations in the Transformation Toolbox default icon N 1 1 2 Algebra View Using the Input Bar you can directly enter algebraic expressions in GeoGebra After hitting the Enter key your algebraic input appears in the Algebra View while its graphical representation is automatically displayed in the Graphics View Example The input f x x 2 gives you the function f in the Algebra View and its function graph in the Graphics View In the Algebra View mathematical objects are organized as free and dependent objects If you create a new object without using any other existing objects it is classified as a free object If your newly created object was created by using other existing objects it is classified as a dependent object Hint If you want to hide the algebraic representation of an object in the Algebra View you may specify the object as an auxiliary object Right click Mac OS Ctrl click on the corresponding object in the Algebra View and select Properties from the appearing Context Menu On tab Basic of the Properties Dialog you may specify the object as an Auxiliary Object By default auxiliary objects are not shown in the Algebra View but you can change this setting by selecting the item Auxiliary Objects from the View menu Note that you are able to modify objects in the Algebra View as
22. function f x tan x e Simplify 2 sin x cos x gives you a function f x sin 2 x TaylorPolynomial TaylorPolynomial Function Number a Number n Creates the power series expansion for the given function about the point x a to order n 3 3 13 Parametric Curve Command Curve Curve Expression el Expression e2 Parameter t Number a Number b Yields the Cartesian parametric curve for the given x expression e1 and y expression e2 using parameter t within the given interval a b Example Input of c Curve 2 cos t 2 sin t t 0 2 pi creates a circle with radius 2 around the origin of the coordinate system Note Parametric curves can be used with pre defined functions and arithmetic operations Example Input c 3 returns the point at parameter position 3 on curve c 53 Note Using the mouse you can also place a point on a curve using tool New Point or command Point Since the parameters a and b are dynamic you could use slider variables as well see tool Slider Commands for Parametric Curves Curvature Point Curve Calculates the curvature of the curve in the given point CurvatureVector Point Curve Yields the curvature vector of the curve in the given point Derivative Curve Returns the derivative of the parametric curve Derivative Curve Number n Returns the n derivative of the parametric curve Length Curve Number tl Number t2 Yields the length of the curve between the par
23. given point Direction Direction Line Yields the direction vector of the line Note A line with equation ax by c has the direction vector b a 47 PerpendicularVector PerpendicularVector Line Returns the perpendicular vector of the line Note A line with equation ax by c has the perpendicular vector a b PerpendicularVector Vector v Returns the perpendicular vector of the given vector Note A vector with coordinates a b has the perpendicular vector b a UnitPerpendicularVector UnitPerpendicularVector Line Returns the perpendicular vector with length 1 of the given line UnitPerpendicularVector Vector Returns the perpendicular vector with length 1 of the given vector UnitVector UnitVector Line Yields the direction vector with length 1 of the given line UnitVector Vector Yields a vector with length 1 which has the same direction and orientation as the given vector Vector Vector Point A Point B Creates a vector from point A to point B Vector Point Returns the position vector of the given point Note Also see tool Vector between Two Points 3 3 7 Segment Command Segment Segment Point A Point B Creates a segment between two points A and B Segment Point A Number a Creates a segment with length a and starting point A Note The endpoint of the segment is created as well Note Also see tools Segment between Two Points and Segment with Given Length
24. ios 25 2 2 13 Geomettie Transtormation TOO EE 25 2 2 14 HE ee EE 26 2 2 15 Mist ASS 28 3 ALGEBRAICINPUL lt a 30 3 1 General NOLES ida 30 3 2 Direct Npe EEEE EE EEA 32 32i INUMIDEES and ANEIS ns 32 322 Gelee Meier el 33 BD O II A 33 Be CONIC SECORS EE 34 AL FUNCIONO e ti riera 34 3 2 6 Pre defined Functions and Operations cccsecccosssccccosscccesccsccussessuessescsecsseccsessccssccnsassenssesses 35 3 2 Boolean Variables and gie E e e E 36 32 06 EE and Operation S ee EE Ee e EE 37 S20 MetencObjects and Opera ee deed 38 3 2 10 Complex NUMDErS and OPeratiOns ui iaa Gevdsa arca nr nadaa ria ae eS 38 3 3 L ein Eu KEE 39 33 le GENGA ele en El e Sct OE 40 Da2 BOONE MANOS Os 40 333 INUMmMbercCommanas ata oi 41 Se Angle ei e ET EE 45 3325y 72 aE GX g ay 6 Y 1 218 Sasan TRO Ee EES 45 O VECINA FEI EE 47 Deh SELMEN COMMANA a a E a E 48 338 Ray COMMER DEE 48 330 Polygon Command WEE EE 49 3 3 10 Ene Commmand EE 49 3 3 11 Conic e AESA fal ue PEE daa 51 3 3 12 PUI ELION geint te LEE 52 3 3 13 Parametric Curve Commande 53 3 3 14 Are and Sector ein len e LEE 54 3 3 15 HES A ele wie ae e LEE 55 3 3 16 Locus Command aaa de di 58 3 3 17 Be ue Sequence Commands EE 58 3 3 18 Geometric Transformation Commande 61 3 3 19 STATISTICS Command as 63 3 3 20 ele DT Ee ei ET EE AN N 67 3 3 21 Mati COMA e LEE 67 A IMENUTREMS ee Ee 69 4 1 File Mena aia E A A A A a ions 69 4 2 ECG Meni
25. length e List Number Adds the number to every element of the list e Listl List2 Subtracts the elements of the second list from corresponding elements of the first list Note The lists need to be of the same length e List Number Subtracts the number from every element of the list Multiplication and Division examples e Listl List2 Multiplies corresponding elements of two lists Note The lists need to be of the same length If the two lists are compatible matrices matrix multiplication is used e List Number Multiplies every list element with the number e Listl List2 Divides elements of the first list by corresponding elements of the second list Note The two lists need to be of the same length e List Number Divides every list element by the number e Number List Divides the number by every element of the list Examples using functions e List 2 Squares every element of the list e sin List Applies the sine function to every element of the list 37 3 2 9 Matrix Objects and Operations GeoGebra also supports matrices which are represented as a list of lists that contain the rows of the matrix 1 2 3 Example In GeoGebra 1 2 3 4 5 6 7 8 9 represents the matrix 44 5 6 7 8 9 Matrix Operations Addition and subtraction examples e Matrixl Matrix2 Adds the corresponding elements of two compatible matrices e Matrixl Matrix2 Subtracts the corresponding elements of two
26. mouse e use the selection rectangle in order to specify which part of the Graphics View should be exported and saved as a picture 13 e You may create points called Export_1 and Export_2 which will be used to define diagonally opposite corners of the export rectangle Note Points Export and Export must be within the visible area of the Graphics View In the File menu select item Export before clicking on item E Graphics View as Picture In the appearing dialog window you may specify the Format Scale in cm and the Resolution in dpi of the output picture file Note The true size of the exported image is shown at the bottom of the export window just above the buttons both in centimeters and pixel Please find more information about the different picture files available in section Export D Graphics View as Picture png eps Copying the Graphics View to Clipboard There are different ways of copying the Graphics View to your computer s clipboard e Inthe Edit menu you may select item Graphics View to Clipboard e Inthe File menu you first need to select item Export before you can click on item Graphics View to Clipboard e Inthe Export Graphics View as Picture dialog window menu File Export Graphics View as Picture png eps you may click on the button Clipboard This feature copies a screenshot of the Graphics View to your system s clipboard as a PNG see PNG format picture This picture can
27. n of the initial list Union Union List 1 List 2 Joins the two lists and removes elements that appear multiple times 3 3 18 Geometric Transformation Commands Dilate UK English Enlarge Dilate Point A Number Point S Dilates point A from point S using the given factor 61 Dilate Line Number Point S Dilates the line from point S using the given factor Dilate Conic Number Point S Dilates the conic section from point S using the given factor Dilate Polygon Number Point S Dilates the polygon from point S using the given factor Note New vertices and segments are created too Dilate Image Number Point S Dilates the image from point S using the given factor Note Also see tool e Dilate Object from Point Reflect Reflect Point A Point B Reflects point A about point B Reflect Line Point Reflects the line about the given point Reflect Conic Point Reflects the conic section about the given point Reflect Polygon Point Reflects the polygon about the given point Note New vertices and segments are created as well Reflect Image Point Reflects the image about the given point Reflect Point Line Reflects the point about the given line Reflect Line g Line h Reflects line g about line h Reflect Conic Line Reflects the conic section about the line Reflect Polygon Line Reflects the polygon about the line Note New vertices and segments are created as well Reflect Ima
28. need to use a semicolon to separate the two coordinates If you don t type in the degree symbol GeoGebra will treat the angle as if entered in radians In GeoGebra you can also do calculations with points and vectors Examples e You can create the midpoint M of two points A and B by entering M A B 2 into the Input Bar e You may calculate the length of a vector v using length sqrt v v 3 2 3 Lines and Axes Lines You can enter a line as a linear equation in x and y or in parametric form into the nput 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 Examples e Typeing 3x 4y 2 to enter line g as a linear equation e Define a parameter t e g t 3 before entering line g in parametric form using g X 5 5 t 4 3 33 e Define the parametersm 2andb 1 Then you can enter the equation h y m x btoget a line h in y intercept form Axes The two coordinate axes are available in commands using the names xAxis and yAxis Example The command Perpendicular A xAxis constructs the perpendicular line to the x axis through a given point A 3 2 4 Conic Sections You 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 conic s equation Note The conic sect
29. text false Example If a 2 and c a then Text c true returns the text 4 Text c false returns the text a Text Object Point Returns the formula for the object as a text object at the position of the given point Example Text hello 2 3 draws the text at the position 2 3 Text Object Point Boolean Returns the formula for the object as a text object at the position of the given point The Boolean variable determines if values are substituted for variables true or if variable names are shown in the text false TextToUnicode TextToUnicode Text Turns the text into a list of Unicode numbers one for each character Examples e TextToUnicode Some text gives you the list of Unicode numbers 183 111 109 101 32 116 101 120 116 e If text1is hello then TextToUnicode text1 gives you the list of Unicode numbers 104 101 108 108 111 UnicodeToLetter UnicodeToLetter Integer Converts the integer Unicode number back into a letter which is displayed as a text object in the Graphics View Example UnicodeToLetter 97 gives you the text a 57 UnicodeToText UnicodeToText List of Integers Converts the integer Unicode numbers back into text Example UnicodeToText 104 101 108 108 111 gives you the text hello 3 3 16 Locus Command Locus Locus Point Q Point P Returns the locus line of point Q which depends on point P Note Point P has to be a point on
30. the appearing dialog you can specify the output and input objects of your tool and choose names for the Toolbar icon and corresponding command Note Your tool can be used both with the mouse and as a command in the Input Bar All tools are automatically saved in your GGB construction file a Manage Tools Using the Manage tools dialog you can delete a tool or modify its name and icon You can also save selected tools to a GeoGebra Tools File GGT This file can be used later on File menu Open to load the tools into another construction Note Opening a GGT file doesn t change your current construction but opening a GGB file does Customize Toolbar You can customize the tools in GeoGebra s Toolbar by selecting Customize Toolbar in the menu Tools This is especially useful for Dynamic Worksheets where you want to restrict the available tools in the Toolbar 76 Note The current Toolbar setting is saved with your construction in a GGB file 4 6 Window Menu I New Window Keyboard shortcut Ctrl N Mac OS Cmd N This menu item opens a new GeoGebra window that uses the default settings of the GeoGebra interface Note If you change and save some of these settings the new GeoGebra window will open using your customized settings List of Open GeoGebra Windows If you have more than one GeoGebra window open this menu item allows you to switch between these different windows Note This might be helpful when you ar
31. the T up and J down arrow keys of your keyboard in order to navigate through prior input step by step Note Click on the little question mark to the left of the Input Bar in order to display the help feature for the Input Bar Insert Name Value or Definition of an Object into the Input Bar Insert the name of an object Activate tool Move and select the object whose name you want to insert into the Input Bar Then press the F5 key on your keyboard Note The name of the object is appended to any expression you typed into the Input Bar before pressing the F5 key Insert the value of an object There are two ways of inserting an object s value e g 1 3 3x 5y 12 into the Input Bar e Right click Mac OS Ctrl click on the object and select item E Copy to Input Bar from the appearing Context Menu e Activate tool Move and select the object whose value you want to insert into the Input Bar Then press the F4 key on your keyboard Note The value of the object is appended to any expression you typed into the nput Bar before pressing the F4 key Insert the definition of an object There are two ways of inserting an object s definition e g A 4 2 c Circle A B into the nput Bar e Alt click on the object to insert the object s definition and delete whatever input might have been in the Input Bar before e Activate tool Move and select the object whose definition you want to insert into the Input Bar Th
32. the index that ranges from number a to number b with given increment Example L Sequence 2 i i 1 3 0 5 creates a list of points whose 60 y coordinates range from 1 to 3 with an increment of 0 5 L 2 1 2 1 5 2 2 2 2 5 2 3 Note Since the parameters a and b are dynamic you could use slider variables as well Sort Sort List Sorts a list of numbers text objects or points Note Lists of points are sorted by x coordinates Examples e Sort 3 2 1 gives you the list 1 2 3 e Sort pears apples figs gives you the list elements in alphabetical order e Sort 3 2 2 5 4 1 gives you 2 5 3 2 4 1 Sum Sum List Calculates the sum of all list elements Note This command works for numbers points vectors text and functions Examples e Sum 1 2 3 gives you a number a 6 e Sum x 2 x 3 gives you f x X x e Sum Sequence i i 1 100 gives you a number a 5050 EN 2 2 3 gives you a point A 3 5 EN 2 3 gives you point B 4 2 e Sum a b c gives you the text abc Sum List Number n of Elements Calculates the sum of the first n list elements Note This command works for numbers points vectors text and functions Example Sum 1 2 3 4 5 6 4 gives you the number a 10 eo Sum l e Sum l Take Take List Start Position m End Position n Returns a list containing the elements from positions m to
33. tools modes to tell GeoGebra how it should react to mouse input in the Graphics View see section Construction Tools For example clicking on the drawing pad can create a new point see tool New Point intersect two objects see tool gt T Intersect Two Objects or create a circle see Circle tools 2 2 Construction Tools The following construction tools or modes can be activated by clicking on the buttons of the Toolbar You can click on the small arrow in the lower right corner of an icon to open a Toolbox with similar other tools Note With most construction tools you can easily create new points by clicking on empty Spaces on the drawing pad Selecting Objects To select an object means to click on it with the mouse after selecting the Move tool If you want to select several objects at the same time you could draw a selection rectangle Select the Move tool and click on the position of the first corner of your desired selection rectangle Hold the left mouse key pressed down and move the pointer to the position of the diagonally opposite corner of your desired selection rectangle After releasing the mouse button all objects within the selection rectangle are selected Note By holding the Ctrl key Mac OS Cmd key while clicking on different objects you can select several objects at the same time Fast Renaming of Objects To quickly rename a selected or newly created object just start typing to open the Rena
34. within the formula you need to click on an object causing GeoGebra to insert its name as well as the syntax for mixed text 27 Some important LaTeX commands are explained in following table Please have a look at any LaTeX documentation for further information LaTeX input Result a dedorb L b sin alpha b ER 2 2 15 Image Tool d if 2 Insert Image This tool allows you to insert an image into the Graphics View First specify the location of the image in one of the following two ways e Click in the Graphics View to specify the position of the image s lower left corner e Click on a point to specify this point as the lower left corner of the image Then a file open dialog appears that allows you to select the image file from the files saved on your computer Note After selecting the tool 2 Insert Image you can use the keyboard shortcut A t click in order to paste an image directly from your computer s clipboard into the Graphics View Properties of Images The position of an image may be absolute on screen or relative to the coordinate system You can specify this on tab Basic of the Properties Dialog of the image You may specify up to three corner points of the image on tab Position of the Properties Dialog This gives you the flexibility to scale rotate and even distort images also see command Corner e Corner 1 position of the lower left corner of the image 28 e Corner 2 position of the lo
35. 0 speed multiplier Go up in Construction Protocol 85 Key iid ioe eg g ie Mac OS Cmd Cmd Shift Mac OS Ctrl Increase selected number angle x0 1 speed multiplier press Shift only x10 speed multiplier Right arrow NN e MIONE PERN x100 speed multiplier gt point to the right Go up in Construction Protocol Decrease selected number angle move selected O meri Left arrow lt Ge SE speed j x100 speed multiplier point to the left W press Shift multiplier only Go down in Construction Protocol Decrease selected number angle Move selected point down x0 1 speed Down arrow multiplier speed NY Go to newer entry wi press Shift multiplier in Input Bar only history x10 x100 speed multiplier Go down in Construction Protocol Construction Protocol Construction Protocol Additional keyboard commands e Alt Shift Mac OS Ctrl Shift Upper case Greek letters e Spreadsheet View Ctrl Alt C copies values not the formulae Note The degree symbol A t O Mac OS Ctrl O and the symbol x for pi A t P Mac OS Ctrl P can also be used in the slider dialog window for interval min max and the increment 5 7 Labels and Captions Show and Hide Labels You can show or hide the labels of objects in the Graphics View in different ways e Select the tool Show Hide Label and click on the object whose label you would like to show or hide e Open the Context Menu for the desired object
36. ALAN CC ta iio 64 Ee EE 41 KENE ee ed 42 54 Reef e LEE 47 54 CUVE a 53 BIE nnn BURGE ERENT Rn ARR Re OG Re mE E 40 91 EE 52 54 RN ln ET E 67 Dita EEN 61 Diretti Mina 47 lee eg EE 49 DISTIN aa 42 DIW DIVISION Lars 42 Element tio 58 lie ON 51 Ela 61 EE 52 NU le EE 46 EE eeh 52 age 52 le EE 58 FItEX D EE 64 EIERE 64 A EE 64 FILOS es een ne ROE ees eer core anon 64 o Eo o eu 64 ELLO E 64 EE 64 A O 64 A A 46 FORMU AVE XC cc rara ias 55 Fraction LOK ices ais di 56 EISEN EE EE 53 E RE 42 Greatest COMMON DiVISOF oocccocccccnoncncnoncncncnanononos 42 ele EE 42 Highest COMMON Factor oooocccncccccnnnnnnncnnnnnnnnonnnnnnnnos 42 Histogram BEE 65 HDD Tras 51 If Boolean COMMANG leia 40 let gl Lait GEN 46 A aaa 59 leng Sie E Een D 42 UE 42 53 ge EE 46 o A 59 InverseNormMal nvidia 65 TA e SN CAPT OR ane cot Amn DY OSE OREO RDM TON Set SD 68 En e BEE 40 Ehek EE 41 gg e EEN 42 A O e A 59 LEE 59 Reeg 60 A el et a 60 O O 43 Least COMMON Multiple o oooooooccccnnnnnnnnnnnnnnno 43 E A O 43 54 60 LetterToUnicO rain ida 56 IDEA 49 LinearEccentricity sucia das 43 LOW CRS UNI ay cle cn 43 Lowest Common Multiple cccccccccceeeessseeeeeees 43 Tei EE 50 MaX re Eeer 43 60 E le DEE 43 60 Mea EE 65 METE ais 65 lh GE 65 92 MEN dd 65 MIO Mirada ras 47 A a O Ee 43 60 MINIMUM astas 43 60 MINOS a 50 o EE 43 MOE osa 66 Mod l FUNCION ais 43 Nimes dt o ae e 56 leid TEE 66 ODJECE eier
37. EE 11 Navigate ant ee Tee a o e 11 A II AA 13 View En EEN 74 Construction step Construction Protocol 11 Navigation Batan dai is 11 Construction Step Command EE 40 Construction Tools 6 16 Activate Aia laa 16 Context EE EE 10 Animation DA o e ad oe 10 CODY e ee e TEE 10 E O Sane Eee 10 A E eee 10 RACE NA o tol 10 88 Tracetobpreadcheert AAA 10 Continuity Options MENU occcooccccnnccnnncnncnnnnoccnnnnccnnnnnns 75 Convert Degree LE EE 32 G ele Ee BEE ER Coordinate axes E ON 9 STOW TEE 9 A ea Ae cae eed Sonera 73 Coordinate grid SA A O nent 9 SOW Ae cami 9 VIEW IMIG erger cacice cennceceeencsorstns seussacudensens Siia 73 A A 33 Cartesian coordinates 33 Coordinate Style Settings ccccccccsssssseeeeeeeeeess 12 Coordinates style Options Menu 75 POlar e ee gelt E EE 33 a UE 35 VC OOM ALE ad 35 Copy Copy to Input Bar Context Men 10 Copy Visual Style COOL vecindad 17 Corner COMMON iia 46 Correlation Coefficient Commande 64 COSINE direct INPUT Ac root arica 35 Count If commande 58 Covariance Commande 64 Create New Tool Tools Menu 76 Cross Ratio command mic dns 41 C bic root direct Input acojan 35 Curvature Command aii as 42 54 Curvature Vector Commande 47 54 Curve ele ie LALO EE 53 Customize 93 EE ele EN 8 A een 12 Ree Le EN 9 76 User ntertace 8 D Decimal places Options Menu 75 Decana 32 DEME Inter nesosita A S 42 EE 42 Area
38. EE 71 EPS d E GE 70 GRIES 71 PGE AT ita 71 PING TOR Maree eda decis 70 PSTricks TO Mtra 71 ROLE 28 EE 28 SA cies ech easier 28 SOTO Mao 71 TRAYS AOTC ri lio 29 Picture export File Menu oocccccnncccccnncnoccnnnnnnnnnnonanonns 70 Plus Minus keys Animation 79 PMCC command rica 64 PNG TO Mt ea 70 Point A E Une WEE 47 97 Point Capturing Options Menu Point Style Re del EE EE KREE Polar COMMAND a Polar or Diameter Line tool Polar coordinates ssesssssssssseesssrsrrreserersessrssresrersseee Polygon Polynomial Commande Pre defined functions oooccccnnncnnncnnnnonnnnnnnnnnononanananonos Principal axis Command ase OA A Principal axis length commande Print Construction PFOLOCOM rente Gf aphics VIEW mandor anaa OPOS deeg Print Preview Product moment correlation coefficient command Product command EE Properties Dialog rastas Edit En EEN Protocol export as webpage ccccccccncnnccccncnnnnncnnnananenonons PSTricks Export FIle Ment reisene aa Se E EE Q Quartile Q1 eene Le t ele ln TL R Radians convert to degree oooooccnnnccccnnncnnncnnnnnnnnoncnnnnanos Radius command WEE Random COMMAN e DEE ii dite E EE New random numbers Random Between Commande Random Binomial commande Random Normal commande Random Poisson Commande Ray Recompute All Objects View Menu s ssosssssssesessssseen Record to Spreadsheet Context Meliana ti A EA Redo Edit Mera
39. GeoGebra Help Official Manual 3 2 Noa Markus Hohenwarter and Judith Hohenwarter www geogebra org GeoGebra Help 3 2 Last modified April 22 2009 Authors Markus Hohenwarter markus geogebra org Judith Hohenwarter judith geogebra org GeoGebra Online http www geogebra org Contents 1 WHAILISGEROGEBRA Sarai dani 6 1 1 Multiple Views for Mathematical Objects ccsccscssccsccscceccsccccescescescncceccecceccecceccsccecsccscssescescs 6 Bas E eege VW EE 6 ELZ Algebra AEN EE Ee 7 TLS let EENEG 8 1 2 GeoGebra as a Tool for Teaching and Learning Mathematics ccsccscsscssccsccsccccescecceccscceccsccecens 8 ZE CUSTOMIZING tneUser INTEN ACC ER 8 E22 THE ele E Ee E T O 10 Ee HIVES EE a E EE 10 1 3 GeoGebra as a Presentation Tool BEE 11 Laie Tie Navi ato Ba nunsio a A E a A N 11 L32 Te CONSHTUCON PrOtOCO EE 11 33 Customize d En gl E 12 1 4 GeoGebra as an Authoring TOO siii ci a 13 LAL PANIN EES 13 LZ Creating Pictures Of the Graphics Vie Wii AAAA ATEAN EE 13 LAS Creating Interactive e 14 Zi GEOMETRICINPUT aaa 16 2 1 General Notes as 16 2 2 Construction TOO Sist id 16 Peak EH Re tt tc 17 DD exe NOW Kee AAA II EE 18 e NECIO TOOS rd et adoeindat EA 19 A A E Ke e EE 19 E RIV POO EE 20 220 PONER TOO Sarai 20 Ll ANA TOO Sd ada 20 2 2 8 CONIC ECHO TOO taa aba 21 22 9 A ee Be ee EE 22 2 2 10 Number and Angle Too Susini ii 23 2 2 11 Boolean Vala GOO lua di 25 2 2 12 LOCUS TOO lua
40. Length Commande 44 le gelen Dra 58 Fit commands FItEX p e ul un En A EE 64 FitLog command ss ii 64 FIRLOBISTIC commanda rrera nra e e 64 FIEP OY Commana i n 64 SNE ele le E ne debi 64 FitSin command A O 64 Fit Line COMMING iaa 64 Fit Line X command sist iria fed 64 A a eesbuaes 36 Focus COMMON drid 46 Font Size t l cca ee es ee ees 13 Options VICK Usadas tie 75 Format Copy Visual Style cc ooooononoccncnnncnncnnonannnnnoss 17 A EEN 27 Formula Text command iii isa 55 Fraction Text Commande 56 RE esis A 7 Function Commande 53 Functions Direct IMO E 34 Exponential function 35 Limit to interval cccooocccnncnnnnccnnncnnnnacannncnnnncnnananono 35 No cd 30 Pre defined functions ooccccnccccncnnnnccnnnnccnnnccnnacnnnoss 35 G Gamma function direct Input 35 GED command WEE E 42 GeoGebra Forum Help Menu 78 GeoGebraWiki Help Men 78 Geometric Transformations o A A Se E 25 SPADES VIEW snc ed 6 Create DICLUIES eer geg 13 CUSTOM E EE 8 EXportas pituto 13 EXPO AL Cala 14 Export de d e Meel Ee WEE 14 Export to Clipboard Edit Menu 72 Geometric INOUE tato 16 MOVE ODIECES iria 7 PU esse oe o pases eat es 13 Update Print Drevlew 13 LA EE 8 Greatest Common Divisor Commande 42 Greek letter upper Case occccccccccnnocccccnnnnnnonnnnanoconannnnnnnos 86 Grid CO eeh 9 CUSTOMIZE tasa 9 Distance of INES aia 9 A EE 9 O 9 Ee A A 9 MeWMen Und ida 73 H FICE COMMING BEE 42 Help Comm
41. N ALS AXES its 73 Coordinate Grid oococccnncccccnccnnonnnnnnnnnnnnononanonnnnnnoss 73 AP O 73 Horizontal SHINE E 73 A O 73 Navigation E 74 Recompute All Objects occccccccnnconncccnnnnnnncnnnnonnnos 74 Refresh VIEWS dicci n 74 Spreadsheet View 73 Views Algebra Vie W erte EA E 7 Ee el lte 6 Spreadsheet View 8 Visibility Conditional ariadna 80 Properties Diario 10 Vistiall StVle CONV 17 W Window MENU sssssessssssssssseeerssssssssrersresssssssrreeresssssseeene 77 NGQW AV TIO Wester ds 77 www geogebra org Help Men 77 e e EE 8 CENTER o ZOO ii aia 8 Zoom n LOO cial mune 18 Zoom Out tool ich nisi iaa 18 ZOOM rectang lEs rc dais 9
42. Note This command also draws the area between the function graph of f and the x axis Integral Function f Function g Number a Number b Yields the definite integral of the difference f x g x in the interval a b Note This command also draws the area between the function graphs of f and g Note Also see command for ndefinite Integral Iteration Iteration Function Number x0 Number n Iterates the function n times using the given start value xO Example After defining f x x 2 the command Iteration f 3 2 gives you the result 3 81 42 LCM LCM Number a Number b Calculates the least common multiple of two numbers a and b UK English LCM lowest common multiple LCM List of numbers Calculates the least common multiple of the elements of the list UK English LCM lowest common multiple Length Length Vector Yields the length of the vector Length Point A Yields the length of the position vector of the given point Length Function Number xl Number x2 Yields the length of the function graph in the interval x1 x2 Length Function Point A Point B Yields the length of the function graph between the two points A and B Note If the given points do not lie on the function graph their x coordinates are used to determine the interval Length Curve Number tl Number t2 Yields the length of the curve between the parameter values t1 and t2 Length Curve c Point A Point B Yields t
43. OMMAND aos 52 PON OM Aroa 53 Exponential function direct Input 35 Exponentiation direct Input sessesesssssssereerssssssseesee 35 Export Construction Protocol as webpnage 12 Dynamic Worksheet coccccconcnnccccccnncnnncnnanoccncnonnnonons 14 Files created WEE 15 GGB Mita 15 Height width 14 HTM Efile cias ta 15 JAR Med bei 15 Show reset ICON ia 14 SHOW Toolbar ina ada dida 14 E nen DEE 14 Tab Cnel nin caera 14 Dynamic Worksheet as Webpage File Menu 70 File WIG EE 70 Graphics View as PGF TikZ Eileen 71 AAA nO en ae eT cE 13 as Picture File Menu 70 as PSTricks File Menu 71 SUE 14 e elei te RE 14 to Clipboard Edit Menu 72 to Clipboard File Menu dd Interactive webpage cccccccccncnnococcnnnonononononccnnnnnnnnnnns 14 Interactive workeheet 14 EXDOMM RECTAN CIE ia tia 14 Extremum Commande 46 F Factor COMIMANG EE 52 PACtOFIAl Greet npt cio NE 35 Factorise Commande 52 Fl Menorca ei 69 EE O oy AE Oe TE 71 Export Dynamic Worksheet as Webpage html 70 Graphics View as PGF TikZ reee 71 Graphics View as Picture png eps 70 Graphics View as PSTFICKS ccccoooncccnnncnocnnnnnos 71 Graphics View to Clipboard 71 NOW aan 69 NeWwWiNdOW sra 69 A ot E nice E EE E a 69 94 Print PTE EW e E C tahoe 70 SEN 69 A A A RREA 70 Filling Properties Dialog sessssnesesssssssseererssssssserrerree 10 First axis COMMAND ri keene 50 First Axis
44. Object in Une 26 99 Reflect Object in Point 26 Reflect Point about Circle 26 Reflect Point in Circle 26 Regular POIV RON nino 20 A cousin en t 17 Rotate around Point ccccsseccessscccenssscesenscseeeeses 18 Rotate Object around Point by Angle 26 Segment between TWO POINTS occccccccocccnnncnnnoonnss 19 Segment with Given Length from Point 19 SENTI e 23 Show Hide Label 18 SHOW HIDE OD ei 18 A A maaan 24 O 25 BEL 21 Translate Object by Vector eessssessssseseessseerressesee 26 Vector between TWO Points 19 Vector from POIN str E e 19 A O 18 Zoom EE 18 TOONS EE 6 CUSTOMIZE EE 9 76 e VE e EE 6 Restore default E 9 WO OND OK AAA 7 O q eee eee rrr ee eer eee ee 16 Kee EE 16 A A A renee os 16 Ancla 23 Angle TOO oia ais 23 A aa aeeaiei Baiat 22 Boolean Variable Tool 25 CONIC Section Tools 21 General e ole EEN 17 Geometric Transformation Tools 25 ee a Tank 28 Insert Text TOO dia ita 26 NE TOO EEN 20 E EE 25 WIAA SS TOUS ai E E T 76 Number Tools iiscctsamsatentessssaieaatwemasetiaseaceeyteinens 23 PODES 18 Polygon Lee dee 20 Ray e EE 20 Sector TO Sa ii 22 Segmento aaa 19 Transformation TOO Susini aa 25 User defined E e e CES 76 81 Vector TOD Sit iaa 19 Kail A O ene A 76 Create NEW TOO Is 76 Customize TODD olas 76 Manage e EE 76 JEE Eege 88 RI EE 74 Trace On Context Menu 10 88 Trace to Spreadsheet Context Menu 10 Transformation d
45. Points Only and Automatic Note The setting Automatic shows the labels of newly created objects if the A gebra View is shown LA Font Size This menu item determines the font size for labels and text in points pt Note If you are using GeoGebra as a presentation tool increasing the font size makes it easier for your audience to read text labels and algebraic input you are using 75 Language GeoGebra is multilingual and allows you to change the current language setting This affects all input including command names and all output Note No matter which language was selected the globe icon O will lead you back to the language menu All language names are always displayed in English a Drawing Pad This menu item opens a dialog where properties of the Drawing Pad e g coordinate grid and axes background color can be set Note You can also open this dialog window by right clicking Mac OS Ctr click on the drawing pad Save Settings GeoGebra remembers your favorite settings e g settings in the Options menu current Toolbar and Graphics View settings if you select Save settings in the Options menu Restore Default Settings You can restore the default settings of GeoGebra using this menu item 4 5 Tools Menu ER a Create New Tool Based on an existing construction you can create your own tools in GeoGebra After preparing the construction of your tool choose Create new tool in the Tools menu In
46. Trace On Animation On or El Copy to Input Bar Note If you open the Context Menu for a point in the Graphics View it gives you the option Trace to Spreadsheet only if the Spreadsheet View is active Once selected this feature allows you to record the coordinates of the point in the Spreadsheet View if it is moved Note Selecting a Properties in the Context Menu opens the Properties Dialog where you can change the properties of all objects used 10 1 3 GeoGebra as a Presentation Tool 1 3 1 The Navigation Bar GeoGebra offers a Navigation Bar that allows you to navigate through the construction steps of a prepared GeoGebra file Select item Navigation Bar for Construction Steps in the View menu in order to display the Navigation Bar at the bottom of the Graphics View The Navigation Bar provides a set of navigation buttons and displays the number of construction steps e g 2 7 means that currently the second step of a total of 7 construction steps is displayed e button go back to step 1 e button go back step by step e button go forward step by step e EM button go to the last step e gt Play automatically play the construction step by step Note You may change the speed of this automatic play feature using the text box to the right of the gt Play button e Ul Pause pause the automatic play feature Note This button only appears after you click on the Play button e button This button opens the Constructi
47. Vector Translates the line by the vector Translate Conic Vector Translates the conic by the vector Translate Function Vector Translates the function by the vector Translate Polygon Vector Translates the polygon by the vector Note New vertices and segments are created as well Translate Image Vector Translates the image by the vector Translate Vector Point Translates the vector v to point Note Also see tool Translate Object by Vector 3 3 19 Statistics Commands BarChart BarChart Start Value End Value List of Heights Creates a bar chart over the given interval where the number of bars is determined by the length of the list whose elements are the heights of the bars Example BarChart 10 20 1 2 3 4 5 gives you a bar chart with five bars of specified height in the interval 10 20 BarChart Start Value a End Value b Expression Variable k From Number c To Number d Creates a bar chart over the given interval a b that calculates the bars heights using the expression whose variable k runs from number cto number d Example If p 0 1 q 0 9 and n 10 are numbers then Barcnartl D tm 0 BinomialCoefficient n k p k q n k k 0 n gives you a bar chart in the interval 0 5 n 0 5 The heights of the bars depend on the probabilities calculated using the given expression BarChart Start Value a End Value b Expression Variable k From Number c To Number d Step Width s
48. Webpage html Mac OS Cmd Close Mac OS only AAA COMES IEEE Teen E O O O E E PIO E O O E E ES E Exponent CO Exponent ES E LL E E pl LL E E LL Exponent EQ OKA A AA ooo po E E O EE Zoom Out minus or plus number angle Increase selected Zoom In plus or minus number angle Increase selected Zoom In not equal to a PARES lr II ess than or equal to lt Geet lr II ess than or equal to lt greater than or equal to gt reater than or equal P e q a 2 Start editing selected object Algebra View Insert definition of selected object Input Bar Insert value of selected object Input Bar 84 Ctrl Shift Alt SR Mac OS Cmd Shift Mac OS Cmd Mac OS Ctrl Insert name of F5 selected object Input Bar Update random numbers Toggle focus Enter between Graphics Left click Open Context Menu on object Properties Dialog of Graphics View on background Right click Mac OS Ctrl click in Graphcis View Click and drag Fast Drag Mode on object Zoom rectangle on background Zoom in out Accelerated Zoom in Scroll Wheel Zoom in out Applet Zoom out Delete current Delete selection Delete current Backspace selection Increase selected number angle Move selected point up x0 1 speed multiplier Up arrow speed E d T Go to prior entries j ip press Shift multiplier in Input Bar only history x10 x10
49. a A A a A 56 OSC UALS Cire Erase a 52 54 Parabola o 52 Parameter orena A S 44 ger EEN 44 PerpendicularBisector sessssssseseeressssssrerrersesss 50 Perpendicular me 50 Perpendicularector 48 A A eee 47 el eN AE EET E EE EE E E E 50 PO ON nda 49 PON ISI 53 ProdU GE 60 Si A O oer are 66 RER ee EE 66 A A Oe an rs 44 REMAN EEN 44 RandombpetWee Ian cose 44 RandomBinomMial ccccncccnccnnnnncnnnccnnnaccnnacnnnanananinos 44 RandomNormal rasa 44 RINOOMPO Ee ra ria 44 Dee 48 A EE 62 Rea do 40 REMAIN e e o 43 Removellndefnegd 60 A E PO Cn E E ones 60 A A A OME SP EAT CONE OOOO 47 ROCA EE 62 e 67 SD standard deviation ssesessesseseresessserrerreee 66 EE Kate dss 55 EE ee ee 48 A A A 55 SermiMalor Avislengthb 44 SemiMinorAvislength 44 SA os ads 60 E ES 66 MA e Da es 66 A er eee me eee eee ae 66 A A detonators heen anes 53 ele 44 8701 EE 61 A E E 61 A le 66 CL E ent ea aie as 66 CA E 66 SA A 56 EC 61 REN eer eee eee ery ey Pree eee re 50 54 FA Ke ee e enn TEE 53 HR EE 57 TEXTO UNICO Eo 57 Translate ii io 63 TADOS usa 68 eeler stands 44 Trapezo dal SUM 44 Be Ulaf Der 46 Unicode Tole cd iaa 57 Uiicode TO Tetas nia 58 UNO RE 61 UnitPerpendicularVector ssssssssessssssesserresssssssseee 48 UNItVECTO 1 48 UNES 6 EE 45 ET 67 Ve o cum 48 EE 47 COMME 7 39 Accept suggestion in 40 Angle Coma 45 ALC COMMBANGS rial nas 54 Automatic completion oocccnccccnnncnnoccnnnnnnnon
50. a will use the first two empty columns of the Spreadsheet View to record the values of the selected objects 2 a b Relation Select two objects to get information about their relation in a pop up window also see command Relation 17 o Rotate around Point Select the center point of the rotation first Then you may rotate free objects around this point by dragging them with the mouse also see command Rotate ES Show Hide Label Click on an object to show or hide its label a Show Hide Object Select the object you want to show or hide after activating this tool Then switch to another tool in order to apply the visibility changes to this object Note When you activate this tool all objects that should be hidden are displayed in the Graphics View highlighted In this way you can easily show hidden objects again by deselecting them before switching to another tool e Zoom In Click on any place on the drawing pad to zoom in also see section Customizing the Graphics View Note The position of your click determines the center of zoom a Zoom Out Click on any place on the drawing pad to zoom out also see section Customizing the Graphics View Note The position of your click determines the center of zoom 2 2 2 Point Tools xX Intersect Two Objects Intersection points of two objects can be created in two ways also see command ntersect e Selecting two objects creates all intersection points if pos
51. ameter values t1 and t2 Length Curve c Point A Point B Yields the length of curve c between two points A and B that lie on the curve OsculatingCircle Point Curve Yields the osculating circle of the curve in the given point Tangent Point Curve Creates the tangent to the curve in the given point 3 3 14 Arc and Sector Commands Note The algebraic value of an arc is its length and the value of a sector is its area Arc Arc Conic Point A Point B Returns a conic section arc between two points A and B on the conic section c Note This only works for a circle or ellipse Arc Conic Number t1 Number t2 Returns a conic section arc between two parameter values t1 and t2 on the conic section Note Internally the following parameter forms are used e Circle r cos t r sin t where r is the circle s radius e Ellipse a cos t b sin t where a and b are the lengths of the semimajor and semiminor axis CircularArc CircularArc Point M Point A Point B Creates a circular arc with midpoint M between points A and B Note Point B does not have to lie on the arc Note Also see tool Circular Arc with Center between Two Points CircularSector CircularSector Point M Point A Point B Creates a circular sector with midpoint M between two points A and B Note Point B does not have to lie on the arc of the sector 54 Note Also see tool 4 Circular Sector with Center between Two Points CircumcircularArc C
52. an object e g line segment circle Note Also see tool Locus 3 3 17 List and Sequence Commands Append Append List Object Appends the object to the list Example Append 1 2 3 4 gives you 1 2 3 4 Append Object List Appends the list to the object Example Append 4 1 2 3 gives you 4 1 2 3 Countlf CountIf Condition List Counts the number of elements in the list satisfying the condition Examples e Countlf x lt 3 1 2 3 4 5 gives you the number 2 e CountlIf x lt 3 A1 A10 where A1 A10 is a range of cells in the spreadsheet counts all cells whose values are less than 3 Element Element List Number n Yields the o element of the list Note The list can contain only elements of one object type e g only numbers or only points First First List Returns the first element of the list First List Number n of elements Returns a new list that contains just the first n elements of the list 58 Insert Insert Object List Position Inserts the object in the list at the given position Example Insert Tee ly Le 3 Ay D gt 34 places x at the third position and gives you the list 1 2 x 3 4 5 Note If the position is a negative number then the position is counted from the right Example Insert x 2 1 2 3 4 5 1 places x at the end of the list and gives you the list 1 2 3 4 5 x Insert List 1 List 2 Position Inserts all elements of l
53. and Mt 7 tel Dal aiei ce eee a 7 31 Ke aile E 6 Help E A 77 About Ucense aian 78 GeoGebra FO EE 78 GeoGebra Wik nenni onra tcs 78 e e EE 78 WWW SeOogebra org 77 Highest Common Factor commande 42 Histogram command nds 65 Horizontal Splitting View Menu 73 Hyperbola COM lau ee 51 ee 22 Hyperbolic Hyperbolic cosine direct Input 35 Hyperbolic sine direct Input 35 Hyperbolic tangent direct Input 35 If Command WE 40 52 Conditional FUNCTION anar do 52 Image E ya aaa gek 46 DIST sisi is 28 Insertimage tolueno ata 28 Properties Dialog ai 28 RO rea e 28 SO iu cos 28 SDECHY COMES nun daa data 28 Imaginary Un escandalos 39 Increment Manual animation ccccncccnncccnnnncnnccnnnancno 80 NGG EE 30 39 Inflection Point commande 46 INP ado 7 30 DIFE PU dios 32 a TE e EE 7 31 e ei E 31 SIO WN PU Eat 31 View En E EE 73 Insert E A 59 Definition in Input Bar 31 INSERT Image Tola aia 28 INSEE Text e e EEN 26 Name in Input EE 31 Value Input EE 31 meert Image ee EE 28 Insert TENG LOO EE 26 Integer Division Commande 42 Integral Command ld 42 53 95 DEFINITE inter 42 Indefinite Integral 53 Interactive Webpage Exvport 14 Interactive worksheet Export 14 Interface E ei E 8 Intersect All intersection points 18 Allow outlying intersection points 18 COMM E EEN 46 Intersect Two Objects tool 18 One intersection point cccoononccnononocononononnononnnns 18 Intersection Comman
54. and select Show Label e Open the Properties Dialog for the desired object and check or un check the checkbox Show Label on tab Basic Name and Value In GeoGebra every object has a unique name that can be used to label the object in the Graphics View In addition an object can also be labeled using its value or its name and value You can change this label setting in the Properties Dialog on tab Basic by selecting the corresponding option Name Value or Name amp Value from the drop down menu next to the checkbox Show Label Note The value of a point is his coordinates while the value of a function is its equation Caption However sometimes you might want to give several objects the same label for example to label the four edges of a square a In this case GeoGebra offers captions for all objects in addition to the three labeling options mentioned above You can set the caption of an object on tab Basic of the Properties Dialog by entering the desired caption into the text field called Caption Afterwards you can select the labeling option Caption from the drop down menu next to the checkbox Show Label 5 8 Layers Note In GeoGebra layers are used to determine which object to select or drag when the user clicks on multiple objects at the same time By default all objects are drawn on layer 0 which is basically the background layer of the Graphics View A total of 10 layers are available numbers O to 9 and higher numbered
55. ard Deviation Generates a random number from a normal distribution with given mean and standard deviation RandomPoisson Mean Generates a random number from a Poisson distribution with given mean SemiMajorAxisLength SemiMajorAxisLength Conic Returns the length of the semimajor axis half of the major axis of the conic section SemiMinorAxisLength SemiMinorAxisLength Conic Returns the length of the semiminor axis half of the minor axis of the conic section Slope Slope Line Returns the slope of the given line Note This command also draws the slope triangle whose size may be changed on tab Style of the Properties Dialog Note Also see tool 4 Slope TrapezoidalSum UK English TrapeziumSum TrapezoidalSum Function Number a Number b Number n Calculates the trapezoidal sum of the function in the interval a b using n trapezoids Note This command draws the trapezoids of the trapezoidal sum as well 44 UpperSum UpperSum Function Number a Number b Number n Calculates the upper sum of the function on the interval a b using n rectangles Note This command draws the rectangles of the upper sum as well 3 3 4 Angle Command Angle Angle Vector vl Vector v2 Returns the angle between two vectors v1 and v2 between 0 and 360 Angle Line g Line h Returns the angle between the direction vectors of two lines g and h between 0 and 360 Angle Point A Point B Point C Returns the angl
56. as Webpage html Keyboard shortcut Ctrl Shift W Mac OS Cmd Shift W This menu item allows you to export your current construction as a webpage HTML in order to create a so called Dynamic Worksheet Please see section Creating Interactive Webpagesfor more information Keyboard shortcut Ctr Shift P Mac OS Cmd Shift P This menu item allows you to save GeoGebra s Graphics View as a picture file on your computer In the appearing dialog window you can select the picture file Format and change the Scale in cm and Resolution in dpi of the picture When exporting the Graphics View as a picture you can choose out of the following formats e PNG Portable Network Graphics This is a pixel graphics format The higher the resolution dpi the better the quality 300dpi will usually suffice PNG graphics should not be scaled subsequently to avoid a loss of quality PNG graphic files are well suited for the use on web pages HTML and in word processing documents Note Whenever you insert a PNG graphic file into a word processing document menu Insert Image from file make sure that the size is set to 100 Otherwise the given scale in cm would be changed e EPS Encapsulated Postscript This is a vector graphics format EPS pictures may be scaled without loss of quality EPS graphic files are well suited for the use with vector graphics programs e g Corel Draw and professional text processing systems e g LaTex
57. ations To create numbers coordinates or equations see section Direct Input you may also use the following pre defined functions and operations Note The pre defined functions need to be entered using parentheses You must not put a space between the function name and the parentheses Operation Function Input EES ee E 35 Operation Function put Antihyperbolic tangent Greatest integer less than or equal floor Least integer greater than or equal 3 2 7 Boolean Variables and Operations You can use the Boolean variables true and false in GeoGebra Just type for example a trueorb 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 3 Check Box to Show Hide objects 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 O 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 the
58. b Axes you can for example change the line style and units of the coordinate axes and set the distance of the tickmarks to a certain value Note that you can customize both axes individually by clicking on tabs xAxis or yAxis Furthermore you can also change the ratio between the axes and hide or show the axes individually e On tab Grid you can for example change the color and line style of the coordinate grid and set the distance for grid lines to a certain value In addition you may also set the grid to be sometric Note Scaling the axes is possible in every mode by pressing and holding the Shift key PC also Ctrl key while dragging the axis Note The Properties Dialog of the Graphics View is different from the Properties Dialog for objects Customizing the Toolbar The Toolbar can be customized by selecting Customize Toolbar from the Tools menu Select the too or toolbox you want to remove from the Toolbar in the list on the left hand side of the appearing dialog window and click button Remove gt in order to remove the tool toolbox from the Toolbar Note You can restore the default Toolbar by clicking on the button Restore Default Toolbar in the left lower corner of the dialog window 1 2 2 The Properties Dialog The Properties Dialog allows you to modify properties of objects e g size color filling line style line thickness visibility You can open the Properties Dialog in several ways e Right click
59. between two FUNCTIONS cccccoccnccnnncnncnononnnnnoss 42 Definition insert in Input Bar 31 Degree Convert e lee EEN 32 SIMIO dicta oa calida 24 32 Delete Command MEN 40 CON id 10 Delete Object tool 17 Eo 72 A eet oe ee dO Ne re 74 Dependent object nn 7 Derivative HEUSDEN Ee 52 54 Determinant commande 67 Dilate COMUN rra edad 61 Dilate Object from Point by Factor too l 25 Direct input EE li 32 Ee PEO SE A A AN 34 CONIC SECTIONS sctesuctt A 34 FUNCION EE 34 tt ER NEES 32 POINTS ee ER A SEEeEe ER Direction command EE 47 Directrix command cda 49 Distance Reegel Le EE 42 Distance or Length tool 24 Div command EE 42 Division direct INPUT EE 35 Drawing Pad Options Menu 76 Dynami col E aA 82 Dynamic text EEN 27 Dynamic Worksheet Edit Applets oia an 15 elt ad ENEE 15 Gi EE 14 Export File Menu ccccccocooccccncnoncnonononaconcnnnonononnnnnnns 70 E VE 15 EE Mer aid 71 DA o 72 Export Graphics View to Clipboard o 72 Properties Dialog nessssssssseersssssssrereresressssrreeres 73 REOOO Ee 71 Selecta ll ds 72 EE 72 Select Current Layer ooooocccnncccccnccnoconnnnnnnonnnnananononnnos 72 Selec EDESA Auca 72 ite EE 71 ee ele osos 31 Element command iiser tere Eege 58 Ellipse Command aros 51 Ee e BEE 22 EMF ett E EE 71 Enlarge COM o oi 61 Enlarge Object from Point by Factor tool 25 EPSON bus 70 EUler CON baaa 32 Expand C
60. corresponding objects in the construction Note The exported HTML file can be viewed with any Internet browser e g Firefox Internet Explorer and edited with many text processing systems e g OpenOffice Writer 1 3 3 Customize the Settings GeoGebra allows you to change and save settings using the Options menu For example you may change the Angle Unit from Degree to Radians or change the Point Style Checkbox Size and Right Angle Style In addition you may change how Coordinates are displayed on screen and which objects are labeled Labeling Please see the section about the Options menu for more information You can save your customized settings by selecting item Save Settings from the Options menu After doing so GeoGebra will remember your customized settings and use them for every new GeoGebra file you create Note You may restore the default settings by selecting Restore Default Settings from the Options menu 12 Note If you use GeoGebra as a presentation tool you might want to increase the Font Size Options menu so your audience can easily read text and labels of objects 1 4 GeoGebra as an Authoring Tool 1 4 1 Printing Options Printing the Graphics View GeoGebra allows you to print the Graphics View of your constructions You can find the corresponding item Print Preview in the File menu In the appearing Print Preview dialog window you can specify the Title Author and a Date for the construction
61. cs View as well Thereby the name of the object matches the name of the spreadsheet cell used to initially create it e g A5 C1 Note By default soreadsheet objects are classified as auxiliary objects in the Algebra View You can show or hide these auxiliary objects by selecting Auxiliary Objects from the View menu 1 2 GeoGebra as a Tool for Teaching and Learning Mathematics 1 2 1 Customizing the User Interface The user interface of GeoGebra can be customized by using the View menu For example you can hide different parts of the interface e g the Algebra View Spreadsheet View or Input Bar by checking or un checking the corresponding menu item in the View menu Showing and Hiding Objects You may show or hide objects in the Graphics View in different ways e You may use tool Show Hide Object to show or hide objects e Open the Context Menu and select item Show Object to change the visibility status of the selected object e Inthe Algebra View the icon to the left of every object shows its current visibility state shown or hidden You may directly click on the little marble icon in order to change the visibility status of an object e You can also use the tool Check Box to Show Hide Objects in order to show or hide one or several objects Customizing the Graphics View In order to adjust the visible part of the drawing pad in the Graphics View you can drag the drawing pad by using tool Move Drawin
62. ct Current Layer Edit Menu 72 Select Descendants Edit Menu 72 Several Objects ssssssnsessssssseerrsesssssssrererssssseeseere 16 Select All Properties Dialog 10 Selection rectangle ccccccoononocnoncnnnnnnnnnnnnancnnnonnnnnnnnnos 16 Semicircle comman eee e os 55 Ee EE 23 Semimajor Axis Length commande 44 Semiminor Axis Length Commande 44 Sequence COMMING tai 60 Settings Ancla UD ee ege 12 A a ian tedvaderonduanes 12 Coordinate Style E 12 EE en EEN 12 Increase Font Size nee 13 E a S 12 PONI SV EE 12 Restore Default Setting 12 Right Angle Stvle REENEN 12 GIE gege 12 Show Hide t 8 Show Hide Label tool 18 Show Hide Object tool 18 Sigma SIZMAXX Commande 66 SISA Vo command ME 66 SIGMA YY COMMAING EE 66 SIM CUPS CUMIN cscs ee 35 Significant Figures Options Menu 75 Simplify COMINO Cia EA 53 el re Let EE 53 Sine direct let ata is aes 35 Size Properties Dialog sicarios mecano 10 e TEE 33 Re ie BEEN 24 Slope COMMANA cet iuitaeel a 44 O AA ARUN te PPT ame NS SUC A IN 25 S rt command DEE 61 iS A cca easveraens 8 Colin PUE 8 Colmena ee de 8 VIEW MENU atan 73 Square root direct Input 35 Standard deviation Commande 66 SOU as 27 Subtraction direct ul EE 35 SUM ele gl E Le MEET 61 A A 71 SXX COMMAaNd Ao dd 66 SKY COMME 66 A oreren E E NEA 66 T Table Text COMMON 56 EI COMMMMANG WEE 61 Tangent Command KEEN 50 54 Ee 21 Tangent direct Input 35 Taylor Polynomial c
63. d hare two lines and you would like a text to be shown whenever these lines are parallel then you could use g has a conditional statement for the text 80 5 3 User Defined Tools GeoGebra allows you to create your own construction tools based on an existing construction Once created your custom tool can be used both with the mouse and as a command in the Input Bar All tools are automatically saved in your GeoGebra file Creating a User Defined Tool First create the construction your tool should be able to create later on In the Tools menu click on B Create New Tool in order to open the corresponding dialog box Now you need to fill in the three tabs Output Objects Input Objects and Name and Icon in order to create your custom tool Example Create a Square tool that creates a square whenever you click on two existing points or on two empty spots in the Graphics View e Construct a square starting with two points A and B Construct the other vertices and connect them with the tool Polygon to get the square poly1 e Select B Create New Tool in the Tools menu e Specify the Output Objects Click on the square or select it from the drop down menu Also specify the edges of the square as Output Objects e Specify the Input Objects GeoGebra automatically specifies the Input Objects for you here points A and B You can also modify the selection of input objects using the drop down menu or by clicking on them in your cons
64. de 59 Inverse Normal commande 65 Invert command ie 68 Is Defined command ua 40 Ela e dun EE 41 Iteration I eina le EE 42 Iteration List commande 59 J JavaScript VALI acera 82 Join Command EE 59 K ITA A O RA 60 Keyboard SHONMGCUTS veia a a 83 L Labeling le ei EE EE 75 EE ege was 12 E O 87 Caption A eee eae S 87 Name and value 87 Showy Md aca iaa di 87 Language Options Men 76 Last COMIMANG A a ae e aaas 60 LaTeX A A 27 LM Sitios 87 e Be eil Te RE 43 Least Common Multiple commande 43 Length Command E 43 54 60 Distance or Length tool 24 Letter To Unicode commande 56 License Help Menu 78 Limit FUNCTION TO INTEIVAl NEE 35 Valle en TE ER Value Of NUMDE F onere E A 33 Line CORI ME 49 DEE HEEN ee deele 33 Line style Properties Dialog 10 Line thickness Properties Dialog 10 Line through Two Points tool 20 A A atin da Moe oats 30 33 Perpendicular Line Commande 50 Perpendicular Line tool 21 Linear Eccentricity Commande 43 List Direct lee dd 37 EI O aeene eons 37 Lists AA das 37 Apply arithmetic operations snesesesssseeeeesssssseee 37 Apply pre defined function 37 COMPAR Geese eset sted oa 37 DIVISION cect ec eee fete cece E EETA A ES 37 Multiple 37 Pre defined FUNCTIONS cccoooocncccnnnnncnnncnnanocnnanonononnns 37 NA E 37 Rate EE 88 89 Command eaea ia 58 FOO BEE 25 Logarithm base 10 direct Input 35 base ele eg le E EE 35 natural direct Ipput
65. de ODJECU oriceon a a 36 Manabi a e ds 36 Box Plot COMMON diia ia 64 Break Mba ERN 12 C Caption E 87 A A A E 87 Name 87 Names Vale sa 87 Cartesian coordinates ara 33 Ceiling directinputii is 36 Cell Cell Range commande 67 fei EE 8 KI 8 Center COMMAND 45 Centre command aaa 45 Centr oid commandi uni 45 Checkbox Checkbox to Show Hide Objects too 25 Size Options leede eegEg deeg 75 SIZE Settings ls 12 Circle Circle through Three Points tool 22 Circle with Center and Radius Too l 21 Circle with Center through Point too l 22 COMME 51 Osculating Circle Commande 52 Circumference Commande 41 Close File MEM 71 Color DINAMICA e 82 Properties Dialog tdci 10 Column Column Name Commande 67 Command EE 67 Command ANER NO ondas 41 A A ea ete tune ee 45 ANBleBIS de EE 49 A HE 58 A 54 LEE 41 ASVMPTOTE eege 49 PS a E a eT 49 PONS SUC EE 41 A A A 63 BinomialCoefficient cooncccnccnnnnccnnnocinancnnnananananos 41 Ge EE 64 A ee 67 A oee are EEE EEE E ES 45 CENE a a E E MORENO TAS 45 CONTO O esas nen 45 Girele E EE 51 Gircular EE 54 EE 54 CIFCUMCTCUS TA Cassina da 55 Bild feele el Ee TEE 55 Circumference EE 41 E 67 GOAN Na ME ad al 67 L e el 51 ConjugateDiameter ocnconccccnonconoconannnononcnnonncnononanos 49 Construction Ste 40 COMPE a 46 CorrelationCoefficient sssesssesrseserrreerrresrresserreesee 64 Eet EE 58 COV
66. e Lines circles and conic sections These objects can be named by typing in the name and a colon in front of their equations or commands Examples g y x 3 c x 1 2 y 2 2 4 hyp XUL y Z e Functions You can name functions by typing for example f x org x in front of the function s equation or commands Examples h x 2 x 4 q x x 2 trig x sin x Note e lf you don t manually assign a name to an object GeoGebra assigns the names of new objects in alphabetical order e You can create indices within the names of objects by using an underscore For example A is entered as A 1 and Sagis entered as s AB Change Values There are two ways of manipulating a free object s value e Change the value of the object by entering its name and the new value in the nput Bar see section Direct Input 30 Example If you want to change the value of an existing number a 3 type a 5 nto the Input Bar and press the Enter key e Edit the algebraic representation Activate tool Move and double click on the object in the Algebra View This opens a text box where you can edit the object s value Press the Enter key to apply your changes Note While free objects values can be changed directly the values of dependent objects can only be influenced by changing their parent objects or by redefining the dependent object Display Input Bar History After placing the cursor in the Input Bar you can use
67. e Properties Dialog a A Angle with Given Size Select two points A and B and type the angle s size into the text field of the appearing window also see command Angle Note This tool creates a point C and an angle a where a is the angle ABC em Area This tool gives you the area of a polygon circle or ellipse as a number and shows a dynamic text in the Graphics View also see command Area cm e Distance or Length This tool gives you the distance between two points two lines or a point and a line as a number and shows a dynamic text in the Graphics View It can also give you the length of a segment the circumference of a circle or the perimeter of a polygon also see commands Distance and Length a 2 Slider Click on any free place in the Graphics View to create a slider for a number or an angle The appearing dialog window allows you to specify the Name Interval min max and Increment of the number or angle as well as the Alignment and Width of the slider in pixel Note In the Slider dialog window you can enter a degree symbol or pi 7t for the interval and increment by using the following keyboard shortcuts e Alt O Mac OS Ctrl O for the degree symbol e Alt P Mac OS Ctrl P for the pi symbol 7t The position of a slider may be absolute in the Graphics View this means that the slider is not affected by zooming but always remains in the visible part of the Graphics View or relative to
68. e corner of the text n 1 2 3 4 Note The numbering of the corners is counter clockwise and starts at the lower left corner Extremum UK English TurningPoint Extremum Polynomial Yields all local extrema of the polynomial function as points on the function graph Focus Focus Conic Yields all foci of the conic section InflectionPoint InflectionPoint Polynomial Yields all inflection points of the polynomial as points on the function graph Intersect Intersect Line g Line hl Yields the intersection point of lines g and h Intersect Line Conic Yields all intersection points of the line and conic section max 2 Intersect Line Conic Number nl Yields the n intersection point of the line and the conic section Intersect Conic cl Conic c2 Yields all intersection points of conic sections c1 and c2 max 4 Intersect Conic cl Conic c2 Number n Yields the ol intersection point of conic sections c1 and c2 Intersect Polynomial f1 Polynomial 2 Yields all intersection points of polynomials f1 and f2 Intersect Polynomial f1 Polynomial 2 Number n Yields the n intersection point of polynomials f1 and f2 Intersect Polynomial Line Yields all intersection points of the polynomial and the line Intersect Polynomial Line Number n Yields the n intersection point of the polynomial and the line Intersect Function f Function g Point A Calculates the intersection point of func
69. e e Sana is 25 Translate COMMAND i 63 Translate Object by Vector tool 26 Transparency Of picture 29 Transpose Commande 68 Trapezium SUM Commande 44 Trapezoidal Sum Commande 44 Trigonometric FUNCTIONS ccccccnncnnncccnnnnnnnnnnnnnnnnnnnnnnnnnnns 34 Antihyperbolic cosine ccoooococccnnnnnnnnnnononannnnnnnss 35 Antihyperbollc TE 35 Antihyperbolic tLangent 36 A EE 35 Le E 35 ALETA a AAA E a i aTa 35 COSINE EG 35 Hyperbolic COSING ccccccccccononnocncnnnnnnnncnnnaancnnnnnnnonons 35 Hyperbolic SICA Na 35 Hyperbolic tangent seeeessessssereersssssssersrersssse 35 le EE 35 Eet E 35 t rning Point command eeren ergeet eege 46 U Undo Edit MERU rnit oi a Ei 71 Unicode Unicode To Letter commande 57 Unicode To Text Commande 58 Union command ocios 61 Unit Perpendicular Vector Commande 48 Unit Vector command Rca 48 LSD SES UM commando ales 45 User defined Ee e a5 tics saciid acid 76 81 EE vices A av a ese 81 ON 81 Se one ae RE eI POR ORE EE 81 User interface Custonmlze 8 V Value CANT 30 Insertin Input Ba e 31 Variance command eiii is 67 Vector COM MIL ia 48 BY a e ra IA PU na n a dodo cid 33 NaMe e ea E ESS 30 33 Perpendicular Vector commande 48 Vector between Two Points tool 19 Vector from Point TOO soii ds 19 Vertex CON Manda dd 47 SA Eu EE 73 100 ASCO VIEWS rias 73 AUN OD sn 73 A A A 73 Command aaron 74 Construction Protocol oeren ee rs dek ee 74 COOKI
70. e enclosed by BA and BC between 0 and 360 where point B is the apex Angle Point A Point B Angle a Returns the angle of size a drawn from point A with apex B Note The point Rotate A a B is created as well Angle Conic Returns the angle of twist of a conic section s major axis see command Axes Angle Vector Returns the angle between the x axis and given vector Angle Point Returns the angle between the x axis and the position vector of the given point Angle Number Converts the number into an angle result between O and 2pi Angle Polygon Creates all angles of a polygon in mathematically positive orientation i e counter clockwise Note If the polygon was created in counter clockwise orientation you get the interior angles If the polygon was created in clockwise orientation you get the exterior angles Note Also see tools lt Angle and lt lt Angle with Given Size 3 3 5 Point Commands Center UK English Centre Center Conic Returns the center of a circle ellipse or hyperbola Note Also see tool Midpoint or Center Centroid Centroid Polygon Returns the centroid of the polygon 45 Corner Corner Number n of Corner Creates a point at the corner of the Graphics View n 1 2 3 4 which is never visible on screen Corner Image Number n of Corner Creates a point at the corner of the image n 1 2 3 4 Corner Text Number n of Corner Creates a point at th
71. e using GeoGebra as a presentation tool and want to have several GeoGebra files open at the same time as well as to toggle between them 4 7 Help Menu Help This menu item gives you access to the HTML version of the GeoGebra Help document Depending on how you installed GeoGebra on your computer you may need Internet access for this feature e If you downloaded and installed GeoGebra using the installer file from the webpage you don t need Internet access in order to use the help feature The HTML version of the GeoGebra Help document is locally saved on your computer when you install GeoGebra e lf you used GeoGebraWebstart to install GeoGebra on your computer you need to have Internet access in order to use this help feature If you don t have Internet access you will get an error message Note The HTML version of this GeoGebra Help document is available online at http www geogebra org help GI www geogebra org If you have Internet access this menu item opens the GeoGebra webpage in your default web browser http www geogebra org 77 B GeoGebra Forum If you have Internet access this menu item opens the GeoGebra online User Forum in your default web browser http www geogebra org forum Note You can post and answer GeoGebra related questions and problems in the GeoGebra User Forum GeoGebraWiki If you have Internet access this menu item opens the GeoGebraWiki webpage in your default web browser h
72. ection through these points also see command Conic Note If four of these five points lie on a line the conic section is not defined ge E Ellipse Select the two foci of the ellipse Then specify a third point that lies on the ellipse also see command Ellipse N Da Hyperbola Select the two foci of the hyperbola Then specify a third point that lies on the hyperbola also see command Hyperbola Parabola Select a point and the directrix of the parabola also see command Parabola 2 2 9 Arc and Sector Tools Note In GeoGebra the algebraic value of an arc is its length The value of a sector is its area 22 bw 4 Circular Arc with Center between Two Points First select the center point M of the circular arc Then select the starting point A of the arc before you select a point B that specifies the length of the arc also see command CircularArc Note While point A always lies on the circular arc point B does not have to lie on it A Circular Sector with Center between Two Points First select the center point M of the circular sector Then select the starting point A of the sector s arc before you select a point B that specifies the length of the sector s arc also see command CircularSector Note While point A always lies on the sector s arc point B does not have to lie on it C J Circumcircular Arc through Three Points Selecting three points A B and C creates a circular arc through th
73. elect two points A and B in order to create a segment between A and B also see command Segment Note In the A gebra View the segment s length is displayed a e Segment with Given Length from Point Click on a point A that should be the starting point of the segment Specify the desired length a of the segment in the appearing window also see command Segment Note This tool creates a segment with length a and endpoint B which may be rotated around the starting point A by using tool Move 19 2 2 5 Ray Tool j Ray through Two Points Selecting two points A and B creates a ray starting at A through B also see command Ray Note In the A gebra View the equation of the corresponding line is displayed 2 2 6 Polygon Tools D gt Polygon Successively select at least three points which will be the vertices of the polygon Then click the first point again in order to close the polygon also see command Polygon Note In the Algebra View the polygon s area is displayed Regular Polygon Select two points A and B and specify the number n of vertices in the text field of the appearing dialog window This gives you a regular polygon with n vertices including points A and B also see command Polygon 2 2 7 Line Tools e Angle Bisector Angle bisectors can be defined in two ways also see command AngleBisector e Selecting three points A B and C produces the angle bisector of the enclosed angle where point B is the ape
74. en press the F3 key on your keyboard Note The definition of the object replaces any expression you typed into the Input Bar before pressing the F3 key 31 3 2 Direct Input GeoGebra can work with numbers angles points vectors segments lines conic sections functions and parametric curves You can enter these objects into the Input Bar by using their coordinates or equations and pressing the Enter key 3 2 1 Numbers and Angles Numbers 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 typinginr 5 32 Note In GeoGebra numbers and angles use a period as a decimal point You can also use the constant rand the Euler constant e for expressions and calculations by selecting them from the drop down menu 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 Euler constant if you use it in new expressions Angles Angles are entered in degree or radians rad The constant rt is useful for radian values and can also be entered as pi Note You can enter a degree symbol or the pi symbol 77 by using the following keyboard shortcuts e Alt O Mac OS Ctrl O for the
75. es a hyperbola with focal points F and G and semimajor axis length a Note Condition O lt 2a lt Distance F G Hyperbola Point F Point G Segment Creates a hyperbola with focal points F and G where the length of the semimajor axis equals the length of segment s Hyperbola Point F Point G Point A Creates a hyperbola with foci F and G passing through point A Note Also see tool Hyperbola 51 OsculatingCircle OsculatingCircle Point Function Yields the osculating circle of the function in the given point OsculatingCircle Point Curve Yields the osculating circle of the curve in the given point Parabola Parabola Point F Line g Returns a parabola with focal point F and directrix g Note Also see tool e Parabola 3 3 12 Function Commands Conditional Functions You can use the Boolean command f in order to create a conditional function Note You can use derivatives and integrals of such functions and intersect conditional functions like normal functions Examples e f x If x lt 3 sin x x 2 gives you a function that equals sin x for x lt 3 and x for x gt 3 e a 3 A b 2 0 tests whether a equals 3 and b is greater than or equal to 0 Note Symbols for conditional statements e g A 2 can be found in the list next to the right of the nput Bar Derivative Derivative Function Returns the derivative of the function Derivative Function Number n Returns the n
76. ese points Thereby point A is the starting point of the arc point B lies on the arc and point C is the endpoint of the arc also see command CircumcircularArc W Circumcircular Sector through Three Points Selecting three points A B and C creates a circular sector through these points Thereby point A is the starting point of the sector s arc point B lies on the arc and point C is the endpoint of the sector s arc also see command CircumcircularSector o Semicircle Select two points A and B to create a semicircle above the segment AB also see command Semicircle 2 2 10 Number and Angle Tools Ae Angle With this tool you can create angles in different ways also see command Angle e Click on three points to create an angle between these points The second point selected is the vertex of the angle e Click on two segments to create the angle between them e Click on two lines to create the angle between them e Click on two vectors to create the angle between them 23 e Click on a polygon to create all angles of this polygon Note If the polygon was created by selecting its vertices in counter clockwise orientation the Angle tool gives you the interior angles of the polygon Note Angles are created in counter clockwise orientation Therefore the order of selecting these objects is relevant for the Angle tool If you want to limit the maximum size of an angle to 180 un check Allow Reflex Angle on tab Basic of th
77. ficient BinomialCoefficient Number n Number r Calculates the binomial coefficient n choose r Circumference Circumference Conic Returns the circumference of a circle or ellipse CrossRatio CrossRatio Point A Point B Point C Point D Calculates the cross ratio of four collinear points A B C and D where A AffineRatio B C D AffineRatio A C D 41 Curvature Curvature Point Function Calculates the curvature of the function in the given point Curvature Point Curve Calculates the curvature of the curve in the given point Distance Distance Point A Point B Yields the distance of two points A and B Distance Point Line Yields the distance of the point and the line Distance Line g Line h Yields the distance of the parallel lines g and h Note The distance of intersecting lines is O Thus this command is only interesting for parallel lines Note Also see tool Distance or Length GCD UK English HCF GCD Number a Number b Calculates the greatest common divisor of numbers a and b UK English HCF highest common factor GCD List of Numbers Calculates the greatest common divisor of the list of numbers UK English HCF highest common factor Integer Division Div Number a Number b Calculates the integer quotient for division of number a by number b Integral Integral Function Number a Number b Returns the definite integral of the function in the interval fa b
78. from Point 3 3 8 Ray Command Ray Ray Point A Point B Creates a ray starting at point A through point B Ray Point Vector v Creates a ray starting at the given point which has the direction vector v Note Also see tool Ray through Two Points 48 3 3 9 Polygon Command Polygon Polygon Point A Point B Point C Returns a polygon defined by the given points A B C Polygon Point A Point B Number n Creates a regular polygon with n vertices including points A and B Note Also see tools Polygon and Regular Polygon 3 3 10 Line Commands AngleBisector AngleBisector Point A Point B Point C Returns the angle bisector of the angle defined by points A B and C Note Point B is apex of this angle AngleBisector Line g Line h Returns both angle bisectors of the lines Note Also see tool lt Angle Bisector Asymptote Asymptote Hyperbola Yields both asymptotes of the hyperbola Axes Axes Conic Returns the major and minor axis of a conic section ConjugateDiameter ConjugateDiameter Line Conic Returns the conjugate diameter of the diameter that is parallel to the line relative to the conic section ConjugateDiameter Vector Conic Returns the conjugate diameter of the diameter that is parallel to the vector relative to the conic section Directrix Directrix Parabola Yields the directrix of the parabola Line Line Point A Point B Creates a line through t
79. g Pad and use the following ways of zooming e You may use the tools E Zoom In and E Zoom Out in order to zoom in the Graphics View Note The position of your click determines the center of zoom e You may use the scroll wheel of your mouse in order to zoom in the Graphics View e You may use keyboard shortcuts to zoom in Ctrl and to zoom out Ctrl e After right clicking Mac OS Ctri click on an empty spot on the drawing pad a Context Menu appears which allows you to Zoom e You may specify a zoom rectangle by right clicking Mac OS Cmd click on an empty spot in the Graphics View and dragging the mouse to the opposite corner of your desired zoom rectangle Release the mouse button in order to finish the zoom rectangle which will then automatically adjust to fill all the space in the Graphics View You can also show or hide the coordinate axes and a coordinate grid in the Graphics View by using the View menu Note Another way of showing or hiding the axes and the grid is by right clicking Mac OS Ctri click on the drawing pad and selecting the corresponding items L Axes or Grid from the appearing Context Menu Customizing Coordinate Axes and Grid The coordinate axes and grid can be customized using the Properties Dialog of the Graphics View After right clicking Mac OS Ctri click on the drawing pad you can open this dialog window by selecting Properties from the appearing Context Menu of the Graphics View e On ta
80. ge Line Reflects the image about the line Reflect Point Circle Inverts the point in the circle Note Also see tools d Reflect Object about Point A Reflect Object about Line Reflect Point about Circle Rotate Rotate Point Angle Rotates the point by the angle around the axis origin Rotate Vector Angle Rotates the vector by the angle around the starting point of the vector Rotate Line Angle Rotates the line by the angle around the axis origin Rotate Conic Angle Rotates the conic section by the angle around the axis origin Rotate Polygon Angle Rotates the polygon by the angle around the axis origin Note New vertices and segments are created as well Rotate Image Angle Rotates the image by the angle around the axis origin Rotate Point A Angle Point B Rotates point A by the angle around point B Rotate Line Angle Point Rotates the line by the angle around the point Rotate Vector Angle Point Rotates the vector by the angle around the point Rotate Conic Angle Point Rotates the conic section by the angle around the point Rotate Polygon Angle Point Rotates the polygon by the angle around point B Note New vertices and segments are created as well Rotate Image Angle Point Rotates the image by the angle around the point 62 Note Also see tool Rotate Object around Point by Angle Translate Translate Point Vector Translates the point by the vector Translate Line
81. he length of curve c between two points A and B that lie on the curve Length List Yields the length of the list which is the number of elements in the list Note Also see tool Distance or Length LinearEccentricity LinearEccentricity Conic Calculates the linear eccentricity of the conic section Note The linear eccentricity is the distance between a conic s center and its focus or one of its two foci LowerSum LowerSum Function Number a Number b Number n Yields the lower sum of the given function on the interval a b with n rectangles Note This command draws the rectangles for the lower sum as well Minimum and Maximum Min Number a Number b Yields the minimum of the given numbers a and b Max Number a Number b Yields the maximum of the given numbers a and b Modulo Function Mod Integer a Integer b Yields the remainder when integer a is divided by integer b 43 Parameter Parameter Parabola Returns the parameter of the parabola which is the distance of directrix and focus Perimeter Perimeter Polygon Returns the perimeter of the polygon Radius Radius Circle Returns the radius of the circle Random commands RandomBetween Min Integer Max Integer Generates a random integer between min and max inclusive RandomBinomial Number n of Trials Probability pol Generates a random number from a binomial distribution with n trials and probability p RandomNormal Mean Stand
82. iders into the text boxes for the color components e Close the Properties Dialog and change the values of the sliders in order to find out how each color component influences the resulting color of the polygon Note You could also animate the sliders with different speeds in order to see the color of the polygon change automatically 5 5 JavaScript Interface Note GeoGebra s JavaScript interface is interesting for users who have some experience in HTML editing In order to enhance your Dynamic Worksheets and increase their interactivity GeoGebra applets provide a JavaScript interface For example you could create a button to randomly generate new configurations of a dynamic construction 82 Please see the document GeoGebra Applets and JavaScript http www geogebra org in Help for examples and information about using JavaScript with GeoGebra applets 5 6 Keyboard Shortcuts Ctrl Ctrl Shift Alt M ney Mac OS Cmd mies Mac OS Ctrl Show hide beta betap A a Export Spreadsheet Graphics View View to Clipboard po eg i Refreshviews phip a Go gammy ___ current Ce E 0 Open JL degree symbol Export Graphics Print preview View as pi TU Picture png eps Select Select GE GE es rar hide Save Spreadsheet sigma O Spreadsheet View 83 Ctrl Ctrl Shift Alt Mac OS Cmd Shift Mac OS Ctrl Export Dynamic Worksheet omega w as
83. ies from the Edit menu The Transparency of an image can be changed in order to see objects or axes that lie behind the image You can set the transparency of an image by specifying a Filling value between O and 100 on tab Style of the Properties Dialog 29 3 Algebraic Input 3 1 General Notes The algebraic representations of mathematical objects e g values coordinates equations are shown in the A gebra View You can create and modify objects by using the Input Bar at the bottom of the GeoGebra window see sections Direct Inputand and Commands Note Always press the Enter key after typing algebraic input into the Input Bar Note Pressing the Enter key at any time toggles the focus between the Input Bar and the Graphics View This allows you to enter expressions and commands into the Input Bar without having to click on it with the mouse first Naming Objects You can assign a certain name to an object when you create it using the nput Bar e Points In GeoGebra points are always named using upper case letters Just type in the name e g A P and an equal sign in front of the coordinates or commands Examples C 2 4 P 1 180 Complex 2 i e Vectors In order to distinguish between points and vectors vectors need to have a lower case name in GeoGebra Again type in the name e g v u and an equal sign in front of the coordinates or commands Examples v 1 3 u 3 90 complex 1 2i
84. indow either vertically or horizontally into the different views Input Bar This menu item allows you to show or hide the Input Bar at the bottom of the GeoGebra window 73 Command List This menu item allows you to show or hide the list of commands next to the Input Bar at the bottom of the GeoGebra window E Construction Protocol This menu item allows you to open the Construction Protocol in a new window Navigation Bar for Construction Steps This menu item allows you to show or hide the Navigation Bar at the bottom of the Graphics View Refresh Views Keyboard shortcut Ctrl F Mac OS Cmd F This menu item allows you to repaint all views on screen Note You can use this menu item to delete any traces of points or lines in the Graphics View Recompute All Objects Keyboard shortcut F9 This menu item recomputes all objects used in your GeoGebra file Note You can use this menu item to create new random numbers if you used any in your GeoGebra file 4 4 Options Menu Global options may be changed in the menu Options Note To change object settings please use the Context Menu and Properties Dialog amp Point Capturing This menu item determines if the point capturing is On or Off or if points are captured by the grid item On Grid Note Option Automatic turns the point capturing On when the grid or the coordinate system are shown and turns it Off if they are hidden ei Angle Unit This menu ite
85. int lies on a line or conic or if a line is tangent or a passing line to a conic Note Also see tool Relation 3 3 2 Boolean Commands If If Condition Object Yields a copy of the object if the condition evaluates to true and an undefined object if it evaluates to false If Condition Object a Object b Yields a copy of object a if the condition evaluates to true and a copy of object b if it evaluates to false IsDefined IsDefined Object Returns true or false depending on whether the object is defined or not 40 Isinteger IsInteger Number Returns true or false depending whether the number is an integer or not 3 3 3 Number Commands AffineRatio AffineRatio Point A Point B Point C Returns the affine ratio A of three collinear points A B and C where C A A AB Area Area Point A Point B Point C Calculates the area of the polygon defined by the given points A B C Area Conic c Calculates the area of a conic section c circle or ellipse Note e n order to calculate the area between two function graphs you need to use the command Integral e Also see tool M4 Area AxisStep AxisStepX Returns the current step width for the x axis AxisStepY Returns the current step width for the y axis Note Together with the Corner and Sequence commands the AxisStep commands allow you to create custom axes also see section Customizing Coordinate Axes and Grid BinomialCoef
86. ion s name can be entered at the beginning of the input followed by a colon Examples e Ellipse ell ell 9 x 2 16 y 2 144 e Hyperbola hyp hypt 9 x 2 16 y 2 144 e Parabola par bars V2 4 x e Circle ci Cle 2 2 25 e Circle c2 CZs X DL F ty 2 Zo Note If you define two parameters a 4andb 3 in advance you may enter for example an ellipse as ell b 2 x 2 a 2 y 2 a 2 bi 3 2 5 Functions of x To enter a function you can use previously defined variables e g numbers points vectors as well as other functions Examples e Function f Fiz 3 o x 2 e Function g g x tan f x e Nameless function sin 3 x tan x Note All available pre defined functions e g sin cos tan are described in section Pre defined Functions and Operations In GeoGebra you can also use commands to get for example the Integral and Derivative of a function Note You can also use the commands f x or x in order to get the derivatives of a previously defined function f x Example Define function fas f x 3 x 3 x 2 Then you can type in g x cos f x 2 in order to get function g 34 Furthermore functions can be translated by a vector see command Translate and a free function can be moved with the mouse by using tool Move Limit Function to Interval In order to limit a function to an interval a b you can use the command Function 3 2 6 Pre defined Functions and Oper
87. ircumcircularArc Point A Point B Point C Creates a circular arc through three points A B and C where A is the starting point and C is the endpoint of the circumcircular arc Note Also see tool 3 Circumcircular Arc through Three Points CircumcircularSector CircumcircularSector Point A Point B Point C Creates a circular sector whose arc runs through the three points A B and C Point A is the starting point and point C is the endpoint of the arc Note Also see tool Y Circumcircular Sector through Three Points Sector Sector Conic Point A Point B Yields a conic section sector between two points A and B on the conic section Note This works only for a circle or ellipse Sector Conic Number tl Number t2 Yields a conic section sector between two parameter values t1 and t2 on the conic section Note Internally the following parameter forms are used e Circle r cos t r sin t where r is the circle s radius e Ellipse a cos t b sin t where a and b are the lengths of the semimajor and semiminor axis Semicircle Semicircle Point A Point B Creates a semicircle above the segment AB Note Also see tool Semicircle 3 3 15 Text Commands FormulaText FormulaText Object Returns the formula for the object as a LaTeX text Example If a 2 and f x a x then FormulaText f returns 2 x as a LaTeX text FormulaText Object Boolean Returns the formula for the object as LaTeX text The Boo
88. ist1 in list2 at the given position Example Insert 11 12 11 2 3 4 5 3 places the elements of list1 at the third and following position s of ist2 and gives you the list 1 2 11 12 3 4 5 Note If the position is a negative number then the position is counted from the right Example Insert 11 12 1 2 3 4 5 2 places the elements of list1 at the end of list2 before its last element and gives you 1 2 3 4 11 12 5 Intersection Intersection List 1 List 2 Gives you a new list containing all elements that are part of both lists IterationList Lteraticnilase Funckron Number 0 Number ni Gives you a list of length n 1 whose elements are iterations of the function starting with the value x0 Example After defining function f x x 2 the command L IterationList f 3 2 gives you the list L 3 9 81 Join Join List 1 List 2 Joins the two or more lists Note The new list contains all elements of the initial lists even if they are the same The elements of the new list are not re ordered Example Join 5 4 3 1 2 3 creates the list 5 4 3 1 2 3 Join List of lists Joins the sub lists into one longer list Note The new list contains all elements of the initial lists even if they are the same The elements of the new list are not re ordered Examples e Join 1 2 creates the list 1 2 e Join 1 2 3 3 4 8 7 creates the list 1 2 3 3 4 8 7
89. le alternates between Decreasing and Increasing e gt Increasing The slider value is always increasing After reaching the maximum value of the slider it jumps back to the minimum value and continues the animation e lt Decreasing The slider value is always decreasing After reaching the minimum value of the slider it jumps back to the maximum value and continues the animation Note While an automatic animation is activated GeoGebra remains fully functional This allows you to make changes to your construction while the animation is playing 5 1 2 Manual Animation To manually change a number or angle continuously select the Move tool Then click on a free number or angle and press either the or key or the arrow keys on you keyboard Keeping one of these keys pressed allows you to produce manual animations 79 Example If a point s coordinates depend on a number k like in P 2 k k the point will move along a straight line when k is changed continuously Note You can adjust the increment of the slider on tab Slider of the Properties Dialog of this object Keyboard Shortcuts e Shift arrow key gives you a step width of 0 1 units e Ctrl arrow key gives you a step width of 10 units Alt arrow key gives you a step width of 100 units Note A point on a line can also be moved along its line using the or key 5 2 Conditional Visibility Apart from just showing or hiding certain objects you can als
90. lean variable determines if values are substituted for variables true or if variable names are shown in the text false Examples If a 2 and f x a x then 55 FormulaText f true returns 2x as a LaTeX text FormulaText f false returns ox as a LaTeX text FractionText FractionText Number Converts the number to a fraction which is displayed as a LaTeX text object in the Graphics View Example Ifa y 1 5 x 2 is a line then FractionText Slope a gives you the fraction 3 2 as a text LetterToUnicode LetterToUnicode Letter Turns a single letter into its Unicode number Note The letter needs to be in between a set of quotation marks Example LetterToUnicode a returns the number 97 Name Name Object Returns the name of an object as a text in the Graphics View Note Use this command in dynamic text for objects that might be renamed The Name command is the opposite of the Object command Object Object Name of Object as Text Returns the object for a given name The result is always a dependent object Note The Object command is the opposite of the Name command Example If points A1 A2 A20 exist and slider n 2 then Object A n gives you a copy of point A2 TableText TableText List 1 List 2 List 3 Creates a text that contains a table of the list objects Note By default every list is displayed in a new column of the table Examples TableTextlix 2 4 x53
91. m determines whether angles are displayed in Degree or Radians rad Note Input is always possible in both ways degree and radian 74 Rounding This menu item allows you to set the number of decimal places or significant figures displayed on screen Continuity GeoGebra allows you to turn the continuity heuristic On or Off in the Options menu The software uses a near to heuristic to keep moving intersection points e g line conic conic conic close to their old positions and avoid jumping intersection points Note By default this heuristic is turned off For user defined tools continuity is always turned off as well Point Style This menu item determines whether by default points are displayed as e dots o circles or x crosses lw Checkbox Size This menu item allows you to set the size of checkboxes to Regular or Large Note If you are using GeoGebra as a presentation tool or are working with an interactive whiteboard setting the checkbox size to Large might make it easier for you to use checkboxes Jp Right Angle Style Determines whether right angles are displayed as a rectangle o a dot e or like all other angles Off Coordinates This menu item determines whether coordinates of points are displayed as A x y or A x y 4 Labeling You can specify whether the label of a newly created object should be shown or not You can choose between the settings All New Objects No New Objects New
92. m using the keyboard Keyboard Eemer Object types numbers points Equal b ora b lines conics a b numbers points Unequal a zb or a lines conics a b E l Greater than Less or equal bora lt numbers a b than Greater or bora gt numbers a b Cem a A bora amp amp b Booleans a b AN a V bora Booleans a b wo oa Boolean a Parallel cr ai insae Perpendicular 4 amp 4 alb ines b 36 3 2 8 List Objects and Operations Using curly braces you can create a list of several objects e g points segments circles Examples e L A B C gives you a list consisting of three prior defined points A B and C e 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 Compare Lists of Objects You can compare two lists of objects by using the following syntax e Listl List2 Checks if the two lists are equal and gives you true or false as a result e Listl List2 Checks if the two lists are not equal and gives you true or false as a result Apply Pre defined Operations and Functions to Lists Note If you apply operations and pre defined functions to lists you will always get a new list as a result Addition and Subtraction examples e Listl List2 Adds corresponding elements of two lists Note The two lists need to be of the same
93. mands Mean List of Numbers Calculates the mean of the list elements MeanX List of Points Calculates the mean of the x coordinates of the points in the list MeanY List of Points Calculates the mean of the y coordinates of the points in the list Median Median List of Numbers Determines the median of the list elements 65 Mode Mode List of Numbers Determines the mode s of the list elements Examples e Mode 1 2 3 4 returns an empty list e Mode 1 1 1 2 3 4 returns the list 1 e Mode 1 1 2 2 3 3 4 returns the list 11 2 3 Normal Normal Mean u Standard Deviation o Variable Value x Calculates the function M x u o where is the probability density function for N 0 1 Note Returns the probability for a given x coordinate value or area under the normal distribution curve to the left of the given x coordinate Quartile commands Ql List of Numbers Determines the lower quartile of the list elements Q3 List of Numbers Determines the upper quartile of the list elements SD SD List of Numbers Calculates the standard deviation of the numbers in the list Sigma commands SigmaXX List of Numbers Calculates the sum of squares of the given numbers Example In order to work out the variance of a list you may use SigmaXX list Length list Mean list 2 SigmaXX List of Points Calculates the sum of squares of the x coordinates of the given points SigmaXY List of x Coo
94. me dialog for this object Then type in the new name of the selected object and click on the OK button 16 2 2 1 General Tools Copy Visual Style This tool allows you to copy visual properties e g color size line style from one object to one or more other objects To do so first select the object whose properties you want to copy Then click on all other objects that should adopt these properties Delete Object Click on any object you want to delete also see command Delete Note You can use the Undo button if you accidentally deleted the wrong object Drag and drop free objects with the mouse If you select an object by clicking on it in Move mode you may e delete the object by pressing the Delete key e move the object by using the arrow keys see section Manual Animation Note You can quickly activate the Move tool by pressing the Esc key of your keyboard Move te Drag and drop the drawing pad in the Graphics View to change its visible area Note e You can also move the drawing pad by pressing the Shift key MS Windows also Ctrl key and dragging it with the mouse in any mode e Inthis mode you can also scale each of the axes by dragging it with the mouse Move Drawing Pad 56 ZA KR 12 Record to Spreadsheet This tool allows you to move an object and to record a sequence of its values in the Spreadsheet View This tool works for numbers points and vectors Note GeoGebr
95. nding spreadsheet cells Al 1 A2 4 A3 9 The command CellRange Al A3 then gives you the list 1 4 9 Column Column Spreadsheet Cell Returns the column of the cell as a number starting at 1 Example Column B3 gives you number a 2 since column B is the second column of the spreadsheet ColumnName ColumnName Spreadsheet Cell Returns the column name of the cell as a text Example ColumnName A1 gives you a text A in the Graphics View Row Row Spreadsheet Cell Returns the row number of a spreadsheet cell starting at 1 Example Row B3 gives you number a 3 3 3 21 Matrix Commands Determinant Determinant Matrix Returns the determinant of the matrix Example Determinant l 2 3 4 gives you the number a 2 67 Invert Invert Matrix Inverts the given matrix Example Invert 1l 2 13 4 gives you the inverse matrix 2 1 1 5 0 5 Transpose Transpose Matrix Transposes the matrix Example Transpose 1 2 13 4 gives you the matrix 1 3 2 4 68 d Menu Items 4 1 File Menu _ New Window Keyboard shortcut Ctrl N Mac OS Cmd N This menu item opens a new GeoGebra window that uses the default settings of the GeoGebra user interface Note If you change and save some of these settings the new GeoGebra window will open using your customized settings New This menu item opens a new and empty user interface in the same GeoGebra wi
96. ndow You are asked if you would like to save the existing construction before opening the new user interface Note The new user interface adopts the settings used for the prior construction For example if the coordinate axes were hidden before selecting the menu item New the axes will be hidden in the new user interface as well Open Keyboard shortcut Ctrl O Mac OS Cmd O This menu item allows you to open a GeoGebra file file name extension GGB that is saved on your computer Note In order to open a GeoGebra file you can also drag it with the mouse to the GeoGebra window and drop it there Save Keyboard shortcut Ctr S Mac OS Cmd S This menu item allows you to save your current construction as a GeoGebra file file name extension GGB on your computer Note If the file was saved before this menu item overwrites the old file by using the same file name 69 Save As This menu item allows you to save your current construction as a GeoGebra file file name extension GGB You will be asked to enter a new name for your GeoGebra file before it is saved on your computer Eb Print Preview Keyboard shortcut Ctr P Mac OS Cmd P This menu item opens the Print Preview window for the Graphics View You may specify Title Author Date and the Scale of your printout in cm Note Press the Enter key after you made a change in order to update the preview of your printout Export Fa Dynamic Worksheet
97. nnnnnanos 39 Boolean Commande 40 Change Sure ON ina 40 Command list View Menu c oocccnnccnnnccnnnacinncinnnanana 74 Command syntax Kelt ugueeedru ergeet dE eege 7 Conic Section Commande 51 Curve Commande 53 Function COMMINOS ri 52 General COMMONS iria 40 Geometric Transformation Commande 61 Help tel dein on El EE 7 Line COMIMANG Sn 49 List COMMand Sarria 58 LISEOF COMMMAN GS ee geess 7 Locus Commande 58 ausge eigene LEE 67 Number Commandes 41 Parametric Curve Commande 53 POMtCOMMANIOS ii 45 Polygon comman eisen e 49 E ne BE 48 Sector Re unn e ld 54 Segment Commande 48 Sequence COMMANAS ccccccccnnncccccnnnnnnnnnnnnnnccnnnnnnnnnnns 58 Spreadsheet Commande 67 Statistics COMMANDOS ido ias 63 Text Commands ia ados 55 Transformation Commande 61 VECLOr COMMANC Sissi sitar 47 Compass TOO iii E 22 Compasses e EE 22 Complex number AdditiON WEEN 39 Direct MEI eeh 38 A A O 39 Operations ltda sn 38 Subtraction traia 39 Conditional Tunctons 52 Conditional visite 80 EXIStING ODIECES tada 80 New objects anidan 80 Conic section Conic through Five Points tool 22 COMIC SCOMMAN dera 51 DESEM ie 34 NM rates ateo dos Eee do e 30 34 Conjugate Diameter Commande 49 Construction Protocol titan aia diia 11 Break DOMO 12 CHANGE A EE 12 Seil rv 12 13 Export as webpage ooccccccncccnnonononcnonnnnnnononanncnnnnnnnonnns 12 Export colorful Construction Protocol 12 Insert NeW ia 12 MOOI
98. nter the command Locus into the Input field Note The point whose locus you would like to create must depend on another point s movement which is restricted to move along an object e g line segment circle Example e Create a segment a between the points A 1 1 and B 1 1 e Place a point C on the segment so it is restricted to move along segment a e Create a point P that depends on point C e g P x C x C 72 e Use either tool or command Locus in order to create the locus of point P in dependence on point C o Tool amp Locus Click first on point P and then on point C o Command Locus Enter Locus P C into the Input Bar and hit the Enter key e Note The locus created in this example is the graph of a parabola on the interval 1 1 89 6 Index A About License Help Menu 78 Absolute value direct Input 35 Addition direct inpUt esssssssesesressssssesereressssssserreres 35 Affine Ratio commande 41 O A O 7 Modify ODO oca 7 Show hide View Menu 73 Algebraic input Input Bar 30 Angle Allow TEREX and 24 Angle Bisector Commande 49 Angle BISeCtO tool medie 20 Angle with Given Size tool 24 Angles ipod Mii da 45 ei ue Te 45 Direct Input cios 32 ien e 24 LUNE E 33 DEE ee Ee 33 TOG EP A eee 23 Angle Unit Options Mena ts tise atmo tees 74 ST 12 ARM as 79 Animation DUO Nte 79 TREIE 79 Animation On Context Menu 10 ATOW Sri 79 Automatic animation 79 EE ee dee
99. nts 30 Colas 10 COMMING DEE 56 Copy to Mput Balcarce 10 Delete geeiert 10 Dependent eet geegent EEN ERC 7 PP A EEE AEE A EEE 31 FASE PEM MIA Baena 16 O 10 A A wanes 7 Insert Definition in Input Bar 31 Name lMiNPpuUbLB Is 31 Value In Input Blair 31 A chat uid als day gl 10 Be Ce 10 NAM oa 30 R a E 10 A EE 16 Select several objects ooooocccccononnnonnnnnancnnnnnnnnnnns 16 SHOW 7 MO od 8 E 10 Trace ON EE 10 VIOU EEN 10 Open Men ET 69 Riegel En EE 74 Amele Uta era 74 CREEK DO Claridad 75 CON EN EE 75 Coordinates lO nara recia 75 Decimal Places iia 75 Drawing TE 76 FOME SIZE ore dad 75 E GE 75 Language aae ee 76 Ge in Elei 74 PONES s 75 Restore Default Settings essssssessenessssssserresrssses 76 Right Angle Style AAA 75 ROUNOM eras oa leed 75 Saye SEM 76 Il lei 75 Osculating Circle Commande 54 P Parabola COMME 52 Ke re BE 22 Parallel Line td 21 Parameter Commande 44 Parametric Curve Commande 53 Parentheses direct Input 35 Pause animation eh Seege eege dese 79 PDF OR ee 71 Perimeter COMM aN Cie lila 44 Perpendicular Perpendicular Bisector Commande 50 Perpendicular Bisector tool 21 Perpendicular Line commande 50 Perpendicular Line tool 21 Perpendicular Vector Commande 48 PGF TikZ Export File Menu occcccccccccnonnococcnnnonnnnnnonacccnnnnnnnonons 71 EOL Dri s 71 Pi CONSI docs also 32 SM 24 32 Picture Background Min ET 29 o cle eae ea 28 EMF TOM a
100. o have their visibility status depend on a certain condition For example you would like an object to appear on screen if you check a checkbox positioned in the Graphics View or if a slider is changed to a certain value Conditionally Show or Hide Existing Objects You can use the tool Y Checkbox to Show Hide Objects in order to create a checkbox that controls the visibility of one or more existing objects on screen Alternatively you could also create a Boolean variable e g b true using the Input Bar and make it visible as a checkbox in the Graphics View by changing its visibility status e g use tool Show Hide Object or use the Context Menu In order to use this Boolean variable as a condition for the visibility of certain objects you need to follow the steps described below Changing the Visibility of Newly Created Objects In the Properties Dialog you can enter a condition for the visibility of an object on tab Advanced Note You can select the logic operators e g gt A ll from the drop down menu in order to create your conditional statements Examples e fa isa slider then the conditional statement a lt 2 means that the corresponding object is only shown in the Graphics View if the slider s value is less than 2 e If bis a Boolean variable you can use b as a conditional statement The corresponding object is shown whenever the value of b is true and is hidden when the value of b is false e fgan
101. oats Ns 71 4 3 View Menti PO e ee eeneg 73 4 4 Options WGI EE 74 4 5 TOOLS Menu A o 76 4 6 Window Ment al 77 4 7 Better 77 5 SPECIAL GEOGEBRA FEATURES cccssssscccssesseccccecseecceessseceeesseeeeeenesereeeceseeeennsseesaeensers 79 5 1 ANIMACION acota 79 Dlls GAUTOMIAUIC ANIMA EE 79 SL MINA AAA a do 79 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 5 10 Conditional VISIDINGY EE 80 User Defined TOONS ai Ee 81 Dynamic Colors ia daa 82 Javascript Intenace iia AAA A idos 82 KEV OO Al BR leie TEE 83 Labels and Capone aan 87 WAV ENS cai sv crscs inca csucaicaccewssnwes vancuse vices vaanidanuedecencuadccveces EE 87 E E srne a a A a A 88 Trace and LOCUS id arn A 88 INDEX GE 90 1 What is GeoGebra GeoGebra is dynamic mathematics software that joins geometry algebra and calculus It is developed for learning and teaching mathematics in schools by Markus Hohenwarter and an international team of programmers 1 1 Multiple Views for Mathematical Objects GeoGebra provides three different views of mathematical objects a Graphics View a numeric Algebra View and a Spreadsheet View They allow you to display mathematical objects in three different representations graphically e g points function graphs algebraically e g coordinates of points equations and in spreadsheet cells Thereby all representations of the same object are linked dynamically and adapt automatically to changes made to any
102. of the representations no matter how they were initially created 3 GeoGebra o File Edit View Options Tools Window Help Palla SIE 9 Le N a 2 Move E DI a Pe gt O El AN am Drag or select objects Esc d Free Objects dJ Dependent Objects Toolbar Toolbar Help Algebra View Spreadsheet View Graphics View J TI Input Input Bar gt y a Command z 1 1 1 Graphics View Using the construction tools available in the Toolbar you can do geometric constructions in the Graphics View with the mouse Select any construction tool from the Toolbar and read the Toolbar Help next to the Toolbar in order to find out how to use the selected tool Any object you create in the Graphics View also has an algebraic representation in the Algebra View Note After activating the tool Move you are able to move objects in the Graphics View by dragging them with the mouse At the same time their algebraic representations are dynamically updated in the A gebra View Every icon in the Toolbar represents a toolbox that contains a selection of similar construction tools In order to open a toolbox you need to click on the small arrow in the lower right corner of the Toolbar icon Hint Construction tools are organized by the nature of resulting objects or the functionality of the tools You will
103. ommande 53 Text COMMANG usar oa aneles 57 Dynamic d EE 27 A wis 27 Insert Text too nani 26 ESTO FOU MU ee 27 IEN 27 Ee EE 27 Text To Unicode Commande 57 Tool A A E 20 Angle with Given Size 24 A EEN 24 Best FIt LINEA reien ee rane so 20 Checkbox to Show Hide Objects ccecce 25 Circle through Three Points 22 Circle with Center and Radius 21 Circle with Center through Point 22 Circular Arc with Center between Two Points 23 Circular Sector with Center between Two Points 23 Circumcircular Arc through Three Points 23 Circumcircular Sector through Three Points 23 COMPASS cuand necia 22 COMPASSES siete ese cee eis a id 22 Conic through Five Points 22 Delete Obeid ic dees 17 Dilate Object from Point by Factor eesssssssssseesss 25 DISTANCE OR LN tia tod 24 le EG 22 Enlarge Object from Point by Factor cc 25 A ie E E 22 MISC GE TEE ee 28 Insert Ee 26 Intersect Two Oblects 18 Line through Two Points 20 Eora t EE A O 25 Midpoint or Center 19 WIOW S EE 17 Move Drawing Punset 17 IS De brad 19 PD ol 22 Para divida 21 Perpendicular Bisector sssssssssssereerssssssseerrersssse 21 Perpendicular Une 21 Polar or Diameter Lime 21 PON OM ces 20 Ray through Two Points 20 Record to Spreadsheet sessssssssseeresssssseerrerssssss 17 Reflect Object about Line sesssseeseseeesseseersseererssss 26 Reflect Object about Point 26 Reflect
104. on your customized tool will be part of the GeoGebra Toolbar Note You can remove the custom tool from the Toolbar after opening item Customize Toolbar from the Tools menu Then select your custom tool from the list of tools on the left hand side of the appearing dialog window and click button Remove gt Don t forget to save your settings after removing the custom tool e After saving your custom tool on your computer as a GGT file you can import it into a new GeoGebra window at any time Just select item E Open from the File menu and open the file of your custom tool Note Opening a GeoGebra tool file GGT in GeoGebra doesn t affect your current construction It only makes this tool part of the current GeoGebra Toolbar 5 4 Dynamic Colors In GeoGebra you can change the color of objects using tab Color of the Properties Dialog However you can also have the color of an object change dynamically Open the Properties Dialog for a certain object whose color you would like to change and click on tab Advanced There you will find a section called Dynamic Colors with text boxes for the color components Red Green and Blue Note In each of these text boxes you can enter a function with range 0 1 Example e Create three sliders a b and c with an interval from O to 1 e Create a polygon whose color should be influenced by the slider values e Open the Properties Dialog for the polygon poly1 and enter the names of the three sl
105. on Protocol 1 3 2 The Construction Protocol You can access the interactive Construction Protocol by selecting item Construction Protocol from the View menu It is a table that shows all construction steps The Construction Protocol allows you to redo a prepared construction step by step using the Navigation Bar at the bottom of the Construction Protocol dialog Navigating and Modifying the Construction Protocol You may use the keyboard to navigate in the Construction Protocol e Use the T up arrow of your keyboard to go to the previous construction step e Use the J down arrow of you keyboard to go to the next construction step e Use the Home key to go to the beginning of the Construction Protocol e Use the End key to go to the end of the Construction Protocol e Use the Delete key in order to delete the selected construction step Note This may also affect other objects that depend on the selected object construction step You may also use the mouse in order to navigate in the Construction Protocol e Double click a row in order to select a construction step e Double click the header of any column in order to go to the start of the Construction Protocol 11 e Drag and drop a row in order to move a construction step to another position in the Construction Protocol Note This is not always possible due to the dependencies between different objects e Right click a row in order to open the Context Menu for the object of this con
106. rdinates List of y Coordinates Calculates the sum of the products of the x and y coordinates SigmaXY List of Points Calculates the sum of the products of the x and y coordinates Example You can work out the covariance of a list of points using SigmaXY list Length list MeanX list MeanY list SigmaYY List of Points Calculates the sum of squares of y coordinates of the given points Commands for statistic quantities Sxx List of Numbers Calculates the statistic 2 x X x 3 x n Sxx List of Points Calculates the statistic X x 3 x 2 x n using the x coordinates of the given points Sxy List of Numbers List of Numbers Calculates the statistic 2 xy 2 x 2 y n Sxy List of Points Calculates the statistic 2 xy 2 x 2 y n 66 Syy List of Points Calculates the statistic 3 y X y 3 y n using the y coordinates of the given points Note These quantities are simply unnormalized forms of the variances and covariance of X and Y given by Sxx N var X Syy N var Y and Sxy N cov X Y Example You can work out the correlation coefficient for a list of points using Sxylilast sgrtu sxx listl Syy listl Variance Variance List of Numbers Calculates the variance of list elements 3 3 20 Spreadsheet Commands CellRange CellRange Start Cell End Cell Creates a list containing the cell values in this cell range Example Enter the following values into the correspo
107. s a box plot for the given statistical data in interval a b CorrelationCoefficient CorrelationCoefficient List of x Coordinates List of y Coordinates Calculates the product moment correlation coefficient using the given x and y coordinates CorrelationCoefficient List of Points Calculates the product moment correlation coefficient using the coordinates of the given points Covariance Covariance List 1 of Numbers List 2 of Numbers Calculates the covariance using the elements of both lists Covariance List of Points Calculates the covariance using the x and y coordinates of the points FitLine FitLine List of Points Calculates the y on x regression line of the points FitLineX List of Points Calculates the x on y regression line of the points Note Also see tool 7 Best Fit Line Other Fit Commands FitExp List of Points Calculates the exponential regression curve FitLog List of Points Calculates the logarithmic regression curve FitLogistic List of Points Calculates the regression curve in the form a 1 b SY kx 64 Note The first and last data point should be fairly close to the curve The list should have at least 3 points preferably more FitPoly List of Points Degree n of Polynomial Calculates the regression polynomial of degree n FitPow List of Points Calculates the regression curve in the form a x Note All points used need to be in the first quadrant of the coordinate system
108. s the Redefine dialog and allows you to redefine the object e Change any object by entering its name and the new definition into the nput Bar e Open the Properties Dialog and change the definition of an object on tab Basic Note Fixed objects cannot be redefined In order to redefine a fixed object you need to free it first using tab Basic of the Properties Dialog Examples e In order to place a free point A on an existing line h you first need to double click on the point A to open the Redefine dialog window Then enter the command Point h in the appearing text field and press the Enter key To remove point A from this line and make it free again you need to redefine it to some free coordinates like 1 2 e Another example is the conversion of a line h through two points A and B into a segment Open the Redefine dialog for line h and enter the command Segment A B in the appearing text field 5 10 Trace and Locus Objects can leave a trace in the Graphics View when they are moved Use the Context Menu to switch this Trace On Then modify the construction so that the object whose trace you turned on changes its position and leaves a trace Note You can turn off the trace of an object by un checking Trace On in the Context Menu The menu item LGS Refresh Views in the View menu clears all traces 88 You can also let GeoGebra automatically create the locus of a point by either using tool amp Locus with the mouse or e
109. section to get the polar line e Selecta line or a vector and a conic section to get the diameter line ye Tangents Tangents to a conic section can be produced in several ways also see command Tangent e Selecting a point A and a conic c produces all tangents through A to c e Selecting a line g and a conic c produces all tangents to c that are parallel to line g e Selecting a point A and a function f produces the tangent line to f in x x A Note x A represents the x coordinate of point A If point A lies on the function graph the tangent runs through point A 2 2 8 Conic Section Tools Circle with Center and Radius Select the center point M and enter the radius in the text field of the appearing dialog window also see command Circle 21 d Circle with Center through Point a Selecting a point M and a point P defines a circle with center M through P also see command Circle Note The circle s radius is the distance MP d a Circle through Three Points Selecting three points A B and C defines a circle through these points also see command Circle Note If the three points lie on one straight line the circle degenerates to this line VE Compass UK English Compasses Select a segment or two points to specify the radius Then click on a point that should be the center of the new circle a e Conic through Five Points Selecting five points produces a conic s
110. sible e Directly clicking on an intersection of the two objects creates only this single intersection point Note For segments rays or arcs you may specify whether you want to Allow outlying intersections on tab Basic of the Properties Dialog This can be used to get intersection points that lie on the extension of an object For example the extension of a segment or a ray is a Straight line 18 Ki Midpoint or Center You may click on either two points or one segment to get its midpoint You can also click on a conic section circle or ellipse in order to create its center point also see commands Center and Midpoint A D New Point Click on the drawing pad in the Graphics View in order to create a new point The coordinates of the point are fixed when the mouse button is released Note e By clicking on a segment straight line polygon conic section function or curve you can create a point on this object also see command Point e Clicking on the intersection of two objects creates this intersection point also see tool Intersect Two Objects and command ntersect 2 2 3 Vector Tools A Select the starting point and then the end point of the vector also see command Vector Vector between Two Points Ga Vector from Point Select a point A and a vector v to create the new point B A v as well as the vector from A to B also see command Vector 2 2 4 Segment Tools A Segment between Two Points S
111. stors of the selected object meaning all objects the selected one depends on Note You need to select the dependent object prior to using this menu item Graphics View to Clipboard Keyboard shortcut Ctrl Shift C Mac OS Cmd Shift C This menu item copies the Graphics View to your computer s clipboard Afterwards you can easily paste this picture into other documents e g word processing document 72 a Properties Keyboard shortcut Ctrl E Mac OS Cmd E This menu item opens the Properties Dialog which allows you to modify the properties of all objects used in the GeoGebra file 4 3 View Menu Jl Axes This menu item allows you to show or hide the coordinate axes in the Graphics View Note You can use the Properties Dialog for Graphics View in order to customize the coordinate axes H Grid This menu item allows you to show or hide the coordinate grid in the Graphics View Note You can use the Properties Dialog for Graphics View in order to customize the coordinate grid Algebra View Keyboard shortcut Ctrl Shift A Mac OS Cmd Shift A This menu item allows you to show or hide the Algebra View Spreadsheet View Keyboard shortcut Ctrl Shift S Mac OS Cmd Shift S This menu item allows you to show or hide the Spreadsheet View Auxiliary Objects This menu item allows you to show or hide auxiliary objects in the A gebra View Horizontal Splitting This menu item allows you to split the GeoGebra w
112. struction step Note You can insert construction steps at any position Select the construction step below you would like to insert a new construction step Leave the Construction Protocol window open while you create a new object This new construction step is immediately inserted into the selected position of the Construction Protocol Using the column Breakpoint in the View menu of the Construction Protocol window you are able to define certain construction steps as Breakpoints This allows you to group several objects together When navigating through your construction using the Navigation Bar groups of objects are shown at the same time Note You may switch the different columns of the Construction Protocol on and off by using the View menu of the Construction Protocol window Exporting the Construction Protocol as a Webpage GeoGebra allows you to export the Construction Protocol as a webpage First you need to open the Construction Protocol using the View menu Then you can open the File menu of the appearing Construction Protocol window and select item Export as Webpage In the export window of the Construction Protocol you can enter Title Author and a Date for the construction and choose whether or not you want to include a picture of the Graphics View and the Algebra View In addition you can also choose to export a Colorful Construction Protocol This means that objects in the Construction Protocol will match the color of the
113. te Translate Object by Vector Select the object you want to translate Then click on the translation vector also see command Translate 2 2 14 Text Tool Insert Text With this tool you can create static and dynamic text or LaTeX formulas in the Graphics View also see section Text Commands At first you need to specify the location of the text in one of the following ways e Click in the Graphics View to create a new text at this location e Click on a point to create a new text that is attached to this point Then a dialog appears where you may enter your text 26 Note You may specify the position of a text as absolute on screen or relative to the coordinate system on tab Basic of the Properties Dialog Static text does not depend on any mathematical objects and is usually not affected by changes of the construction Dynamic text contains 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 1 Description _
114. the coordinate system see Properties Dialog of the corresponding number or angle Note In GeoGebra a slider is the graphical representation of a free number or free angle You can easily create a slider for any existing free number or angle by showing this object in the Graphics View see Context Menu see tool Show Hide Object 24 A Slope This tool gives you the slope of a line and shows a slope triangle in the Graphics View also see command Slope 2 2 11 Boolean Variable Tool e Check Box to Show Hide Objects Clicking in the Graphics View creates a check box see section Boolean Variables and Operations that allows you to show and hide one or more objects In the appearing dialog window you can specify which objects should be affected by the check box Note You may select these objects from the list provided in the dialog window or select them with the mouse in any view 2 2 12 Locus Tool vax Locus Select a point B that depends on another point A and whose locus should be drawn Then click on point A to create the locus of point B also see command Locus Note Point A has to be a point on an object e g line segment circle Example e Type f x x 2 2 x 1 into the Input Bar and press the Enter key e Place a new point A on the x axis see tool New Point see command Point e Create pointB x A f x A that depends on point A e Select tool amp Locus and successively click on point
115. the x coordinate of point A Tangent Point Curve Creates the tangent to the curve in the given point Note Also see tool Tangents 50 3 3 11 Conic Section Commands Circle Circle Point M Number r Yields a circle with midpoint M and radius r Circle Point M Segment Yields a circle with midpoint M whose radius is equal to the length of the given segment Circle Point M Point A Yields a circle with midpoint M through point A Circle Point A Point B Point C Yields a circle through the given points A B and C Note Also see tools Q Compass Circle with Center through Point Circle with Center and Radius and Circle through Three Points Conic Conic Point A Point B Point C Point D Point E Returns a conic section through the five given points A B C D and E Note If four of the points lie on one line the conic section is not defined Note Also see tool 2 Conic through Five Points Ellipse Ellipse Point F Point G Number a Creates an ellipse with focal points F and G and semimajor axis length a Note Condition 2a gt Distance F G Ellipse Point F Point G Segment Creates an ellipse with focal points F and G where the length of the semimajor axis equals the length of the given segment Ellipse Point F Point G Point A Creates an ellipse with foci F and G passing through point A F Note Also see tool Ellipse Hyperbola Hyperbola Point F Point G Number a Creat
116. tions f and g by using Newton s method with initial point A Intersect Function Line Point A Calculates the intersection point of the function and the line by using Newton s method with initial point A 46 Note Also see tool Intersect two Objects Midpoint Midpoint Point A Point B Returns the midpoint of points A and B Midpoint Segment Returns the midpoint of the segment Note Also see tool Midpoint or Center Point Line Returns a point on the line Conic Returns a point on the conic section Function Returns a point on the function Polygon Returns a point on the polygon Vector Returns a point on the vector Point Vector Creates a new point by adding the vector to the given point Point Pont Point Point POLE Point FI FI FI F F Note Also see tool New Point Root Root Polynomial Yields all roots of the polynomial as intersection points of the function graph and the x axis Root Function Number a Yields one root of the function using the initial value a for Newton s method Root function Number a Number b Yields one root of the function in the interval a b regula falsi Vertex Vertex Conic Returns all vertices of the conic section 3 3 6 Vector Commands CurvatureVector CurvatureVector Point Function Yields the curvature vector of the function in the given point CurvatureVector Point Curve Yields the curvature vector of the curve in the
117. truction e Specify the Tool Name and Command Name for your new tool Note The Tool Name will appear in GeoGebra s Toolbar while the Command Name can be used in GeoGebra s Input Bar e You may also enter text to be shown in the Toolbar Help e You can also choose an image from you computer for the Toolbar icon GeoGebra resizes your image automatically to fit on a Toolbar button Saving a User Defined Tool You can save your custom tools so you can reuse them in other GeoGebra constructions In the Tools menu select a Manage Tools Then select the custom tool you want to save from the appearing list Click on button Save As in order to save your custom tool on your computer Note User defined tools are saved as files with the file name extension GGT so you can distinguish custom tool files from usual GeoGebra files GGB Accessing a User Defined Tool If you open a new GeoGebra interface using item New from the File menu after you created a custom tool it will still be part of the GeoGebra Toolbar However if you open a new GeoGebra window item _ New Window from the File menu or open GeoGebra on another day your custom tools won t be part of the Toolbar any more 81 There are different ways of making sure that your user defined tools are displayed in the Toolbar of a new GeoGebra window e After creating a new user defined tool you can save your settings using item Save Settings from the Options menu From now
118. ttp www geogebra org wiki Note The GeoGebra Wiki is a pool of free instructional materials created with GeoGebra by users from all around the world i About License This menu item opens a dialog window that gives you information about the license of GeoGebra and gives credit to people who support the GeoGebra project by contributing in many different ways e g programming translations 78 5 Special GeoGebra Features 5 1 Animation 5 1 1 Automatic Animation GeoGebra allows you to animate one or several free numbers and or angles at the same time if they are shown as sliders in the Graphics View If you want to animate a free number or angle in GeoGebra you need to right click Mac OS Ctri click on the number or angle and select Animation On from the appearing Context Menu In order to stop the animation you need to un check Animation On in the same Context Menu Note After animating a free number or a free angle an animation button appears in the lower left corner of the Graphics View It allows you to either pause or continue an animation In the Properties Dialog on tab Slider you can change the behavior of the animation On the one hand you may control the Speed of the animation Note A speed of 1 means that the animation takes about 10 seconds to run once through the interval of the slider On the other hand you may change how the animation cycle is repeated e lt gt Oscillating The animation cyc
119. ve your construction prior to selecting Close you are asked if you would like to do so 4 2 Edit Menu Zen Undo Keyboard shortcut Ctrl Z Mac OS Cmd Z This menu item allows you to undo your activities step by step Note You can also use the Undo button to the right of the Toolbar C Redo Keyboard shortcut Ctrl Y Mac OS Cmd Shift Z 71 This menu item allows you to redo your activities step by step Note You can also use the Redo button to the right of the Toolbar 4 Delete Keyboard shortcut Delete key This menu items allows you to delete selected objects and their dependent objects Note You need to select the objects you want to delete first e g use a selection rectangle Select All Keyboard shortcut Ctrl A Mac OS Cmd A This menu item allows you to select all objects used in your construction Select Current Layer Keyboard shortcut Ctr L Mac OS Cmd L This menu item allows you to select all objects that are on the same layer as a selected object Note You need to select one object that lies on the desired layer prior to using this menu item Select Descendants Keyboard shortcut Ctrl Shift Q Mac OS Cmd Shift Q This menu item allows you to select all objects that depend on the selected object Note You need to select the parent object prior to using this menu item Select Ancestors Keyboard shortcut Ctr Q This menu item allows you to select all objects that are ance
120. well Make sure that you activate the Move tool before you double click on a free object in the Algebra View In the appearing text box you can directly edit the algebraic representation of the object After hitting the Enter key the graphical representation of the object will automatically adapt to your changes If you double click on a dependent object in the Algebra View a dialog window appears allowing you to Redefine the object GeoGebra also offers a wide range of commands that can be entered into the Input Bar You can open the list of commands in the right corner of the Input Bar by clicking on the button Command After selecting a command from this list or typing its name directly into the Input Bar you can press the F1 key to get information about the syntax and arguments required to apply the corresponding command 1 1 3 Spreadsheet View In GeoGebra s Spreadsheet View every cell has a specific name that allows you to directly address each cell For example the cell in column A and row 1 is named A1 Note These cell names can be used in expressions and commands in order to address the content of the corresponding cell In the spreadsheet cells you can enter not only numbers but all types of mathematical objects that are supported by GeoGebra e g coordinates of points functions commands If possible GeoGebra immediately displays the graphical representation of the object you entered in a spreadsheet cell in the Graphi
121. wer right corner of the image Note This corner may only be set if Corner 1 was set before It controls the width of the image e Corner 4 position of the upper left corner of the image Note This corner may only be set if Corner 1 was set before It controls the height of the image Example Create three points A B and C to explore the effects of the corner points e Set point A as the first and point B as the second corner of your image By dragging points A and Bin Move mode you can explore their influence e Now remove point B as the second corner of the image Set point A as the first and point C as the fourth corner and explore how dragging the points now influences the image e Finally you may set all three corner points and see how dragging the points distorts your image Example You already saw how to influence the position and size of your image If you want to attach your image to a point A and set its width to 3 and its height to 4 units you could do the following e Set Corner 1toA e SetCorner2toA 3 0 e SetCorner4toA 0 4 Note If you now drag point A in Move mode the size of your image does not change You may specify an image as a Background Image on tab Basic of the Properties Dialog A background image lies behind the coordinate axes and cannot be selected with the mouse any more Note In order to change the background setting of an image you may open the Properties Dialog by selecting a Propert
122. wo points A and B Line Point Parallel Line Creates a line through the given point parallel to the given line Line Point Direction Vector v Creates a line through the given point with direction vector v 49 Note Also see tool Line through Two Points MajorAxis MajorAxis Conic Returns the major axis of the conic section MinorAxis MinorAxis Conic Returns the minor axis of the conic section PerpendicularLine PerpendicularLine Point Line Creates a line through the point perpendicular to the given line PerpendicularLine Point Vector Creates a line through the point perpendicular to the given vector Note Also see tool Perpendicular Line PerpendicularBisector PerpendicularBisector Point A Point B Yields the perpendicular bisector of the line segment AB PerpendicularBisector Segment Yields the perpendicular bisector of the segment Note Also see tool Perpendicular Bisector Polar Polar Point Conic Creates the polar line of the given point relative to the conic section Note Also see tool D Polar or Diameter Line Tangent Tangent Point Conic Creates all tangents through the point to the conic section Tangent Line Conic Creates all tangents to the conic section that are parallel to the given line Tangent Number a Function Creates the tangent to the function at x a Tangent Point A Function Creates the tangent to the function at x x A Note x A is
123. x e Selecting two lines produces their two angle bisectors Note The direction vectors of all angle bisectors have length 1 Ba Best Fit Line Create the best fit line for a set of points in the following ways also see command FitLine e Create a selection rectangle that contains all points e Selecta list of points to create their corresponding best fit line Line through Two Points Selecting two points A and B creates a straight line through A and B also see command Line Note The line s direction vector is B A 20 n Parallel Line Selecting a line g and a point A defines a straight line through A parallel to g also see command Line Note The line s direction is the direction of line g x Perpendicular Bisector Click on either a segment s or two points A and B in order to create a perpendicular bisector also see command PerpendicularBisector Note The bisector s direction is equivalent to the perpendicular vector of segment s or AB also see command PerpendicularVector p nn e e Perpendicular Line Selecting a line g and a point A creates a straight line through A perpendicular to line g also see command PerpendicularLine Note The line s direction is equivalent to the perpendicular vector of g also see command PerpendicularVector Polar or Diameter Line This tool creates the polar or diameter line of a conic section also see command Polar e Selecta point and a conic

Download Pdf Manuals

Related Search

Related Contents

  Satisfait  Eureka 6997 Vacuum Cleaner User Manual  Istruzioni d`Uso  ワイヤレスパソコンアダプタC 取扱説明書  4590 TSM (v2.03) Service Manual for GPE  DP Hotels Provence&Cote d`Azur_FR    Instructions    

Copyright © All rights reserved. Failed to retrieve file