![24
• 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 the Properties Dialog.
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 α, where α is the angle ABC.
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).
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).
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 (π) for the interval
and increment by using the following keyboard shortcuts:
• Alt‐O (Mac OS: Ctrl‐O) for the degree symbol °
• Alt‐P (Mac OS: Ctrl‐P) for the pi symbol π
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 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).](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-24-320.jpg)
![28
Some important LaTeX commands are explained in following table. Please have a look at any
LaTeX documentation for further information.
LaTeX input Result
a cdot b ba ⋅
frac{a}{b}
b
a
sqrt{x} x
sqrt[n]{x} n
x
vec{v} v
r
overline{AB} AB
x^{2} 2
x
a_{1} 1a
sinalpha +
cosbeta
βα cossin +
int_{a}^{b} x dx ∫
b
a
xdx
sum_{i=1}^{n} i^2 ∑=
n
i
i1
2
2.2.15. Image Tool
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:
• Click in the Graphics View to specify the position of the image’s lower left corner.
• 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 Insert Image, you can use the keyboard shortcut Alt‐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).
• Corner 1: position of the lower left corner of the image](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-28-320.jpg)
![31
Example: If you want to change the value of an existing number a = 3, type
a = 5 into the Input Bar and press the Enter‐key.
• 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 the ↑ up and ↓ 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.
• Right click (Mac OS: Ctrl‐click) on the object and select item Copy to Input Bar
from the appearing Context Menu.
• 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 Input
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 Input Bar.
• Alt click on the object to insert the object’s definition and delete whatever input
might have been in the Input Bar before.
• Activate tool Move and select the object whose definition you want to insert into
the Input Bar. Then, 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-31-320.jpg)
![33
Sliders and Arrow Keys
Free numbers and angles can be displayed as sliders in the Graphics View (see tool
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:
• To enter a point P or a vector v in Cartesian coordinates you may use P = (1, 0)
or v = (0, 5).
• In order to use polar coordinates type in P = (1; 0°) or v = (5; 90°).
Note: You 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:
• You can create the midpoint M of two points A and B by entering
M = (A + B) / 2 into the Input Bar.
• You may calculate the length of a vector v using length = sqrt(v * v)
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 Input Bar.
In both cases previously defined variables (e. g. numbers, points, vectors) can be used within
the equation.
Note: You can enter a line’s name at the beginning of the input followed by a colon.
Examples:
• Type in g: 3x + 4y = 2 to enter line g as a linear equation.
• Define a parameter t (e. g., t = 3) before entering line g in parametric form using
g: X = (-5, 5) + t (4, -3).](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-33-320.jpg)
![34
• Define the parameters m = 2 and b = -1. Then, you can enter the equation
h: y = m*x + b to get 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 section’s name can be entered at the beginning of the input followed by a
colon.
Examples:
• Ellipse ell: ell: 9 x^2 + 16 y^2 = 144
• Hyperbola hyp: hyp: 9 x^2 – 16 y^2 = 144
• Parabola par: par: y^2 = 4 x
• Circle c1: c1: x^2 + y^2 = 25
• Circle c2: c2: (x – 5)^2 + (y + 2)^2 = 25
Note: If you define two parameters a = 4 and b = 3 in advance, you may enter for
example an ellipse as ell: b^2 x^2 + a^2 y^2 = a^2 b^2.
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:
• Function f: f(x) = 3 x^3 – x^2
• Function g: g(x) = tan(f(x))
• 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 f''(x),… in order to get the derivatives of a
previously defined function f(x).
Example: Define function f as f(x) = 3 x^3 – x^2. Then, you can type in
g(x) = cos(f' (x + 2)) in order to get function g.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-34-320.jpg)
![35
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 Operations
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
Addition +
Subtraction -
Multiplication * or Space key
Scalar product * or Space key
Division /
Exponentiation ^ or 2
Factorial !
Gamma function gamma( )
Parentheses ( )
x‐coordinate x( )
y‐coordinate y( )
Absolute value abs( )
Sign sgn( )
Square root sqrt( )
Cubic root cbrt( )
Random number between 0 and 1 random( )
Exponential function exp( ) or ℯx
Logarithm (natural, to base e) ln( ) or log( )
Logarithm to base 2 ld( )
Logarithm to base 10 lg( )
Cosine cos( )
Sine sin( )
Tangent tan( )
Arc cosine acos( )
Arc sine asin( )
Arc tangent atan( )
Hyperbolic cosine cosh( )
Hyperbolic sine sinh( )
Hyperbolic tangent tanh( )
Antihyperbolic cosine acosh( )
Antihyperbolic sine asinh( )](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-35-320.jpg)
![38
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.
Example: In GeoGebra, {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} represents the matrix
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
987
654
321
.
Matrix Operations
Addition and subtraction examples:
• Matrix1 + Matrix2: Adds the corresponding elements of two compatible
matrices.
• Matrix1 – Matrix2: Subtracts the corresponding elements of two compatible
matrices.
Multiplication examples:
• Matrix * Number: Multiplies every element of the matrix by the given number.
• Matrix1 * 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}, {4, 5, 6}} gives
you the matrix {{9, 12, 15}, {19, 26, 33}, {29, 40, 51}}.
• 2x2 Matrix * Point (or Vector): Multiplies the matrix with the given
point/vector and gives you a point as a result.
Example: {{1, 2}, {3, 4}} * (3, 4) gives you the point A = (11, 25).
• 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}, {4, 5, 6}, {0, 0, 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:
{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * {1, 2, 1}.
Other examples (see also section Matrix Commands):
• Determinant[Matrix]: Calculates the determinant for the given matrix.
• Invert[Matrix]: Inverts the given matrix
• 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-38-320.jpg)
![39
Example: If you enter the 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 Algebra 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 0 + 1i. This also means, that you can use this variable i in order to type
complex numbers into the Input Bar (e. g., q = 3 + 4i).
Addition and subtraction examples:
• (2 + 1i) + (1 – 2i) gives you the complex number 3 – 1i.
• (2 + 1i) - (1 – 2i) gives you the complex number 1 + 3i.
Multiplication and division examples:
• (2 + 1i) * (1 – 2i) gives you the complex number 4 – 3i.
• (2 + 1i) / (1 – 2i) gives you the complex number 0 + 1i.
Note: The usual multiplication (2, 1)*(1, -2) gives you the scalar product of the two
vectors.
Other examples:
GeoGebra also recognizes expressions involving real and complex numbers.
• 3 + (4 + 5i) gives you the complex number 7 + 5i.
• 3 - (4 + 5i) gives you the complex number ‐1 ‐ 5i.
• 3 / (0 + 1i) gives you the complex number 0 ‐ 3i.
• 3 * (1 + 2i) gives you the complex number 3 + 6i.
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 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: A1 is entered as A_1 while SAB 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](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-39-320.jpg)
![40
the command into the Input Bar, GeoGebra displays the first command of an alphabetically
sorted list that starts with these letters.
• In order to accept this suggestion and place the cursor in between the brackets, hit
the Enter‐key.
• If the 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 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 point
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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-40-320.jpg)
![41
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 λ of three
collinear points A, B, and C, where C = A + λ * 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:
• In order to calculate the area between two function graphs, you need to use the
command Integral.
• Also see tool 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).
BinomialCoefficient
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 λ
of four collinear points A, B, C, and D, where
λ = AffineRatio[B, C, D] / AffineRatio[A, C, D].](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-41-320.jpg)
![42
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 0. 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 [a , b].
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 Indefinite Integral
Iteration
Iteration[Function, Number x0, Number n]: Iterates the function n times using
the given start value x0.
Example: After defining f(x) = x^2 the command Iteration[f, 3, 2] gives
you the result (32
)2
= 81.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-42-320.jpg)
![43
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 x1, 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 t1, Number t2]: Yields the length of the curve between the
parameter 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.
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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-43-320.jpg)
![44
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 p]: Generates a random
number from a binomial distribution with n trials and probability p.
RandomNormal[Mean, Standard 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 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-44-320.jpg)
![45
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 v1, 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 angle enclosed by BA and BC
(between 0 and 360°), where point B is the apex.
Angle[Point A, Point B, Angle α]: Returns the angle of size α drawn from point A
with apex B.
Note: The point Rotate[A, α, 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 0 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 Angle and 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-45-320.jpg)
![46
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 the 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 h]: 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 n]: Yields the nth
intersection point of the line and
the conic section.
Intersect[Conic c1, Conic c2]: Yields all intersection points of conic sections c1
and c2 (max. 4).
Intersect[Conic c1, Conic c2, Number n]: Yields the nth
intersection point of
conic sections c1 and c2.
Intersect[Polynomial f1, Polynomial f2]: Yields all intersection points of
polynomials f1 and f2.
Intersect[Polynomial f1, Polynomial f2, Number n]: Yields the nth
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 nth
intersection point of the
polynomial and the line.
Intersect[Function f, Function g, Point A]: Calculates the intersection point
of functions 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-46-320.jpg)
![47
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
Point[Line]: Returns a point on the line.
Point[Conic]: Returns a point on the conic section.
Point[Function]: Returns a point on the function.
Point[Polygon]: Returns a point on the polygon.
Point[Vector ]: Returns a point on the vector.
Point[Point, Vector]: Creates a new point by adding the vector to the given point.
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 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).](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-47-320.jpg)
![48
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
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](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-48-320.jpg)
![49
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 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 two 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-49-320.jpg)
![50
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 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 the x‐coordinate of point A.
Tangent[Point, Curve]: Creates the tangent to the curve in the given point.
Note: Also see tool Tangents](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-50-320.jpg)
![51
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 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 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 > 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.
Note: Also see tool Ellipse
Hyperbola
Hyperbola[Point F, Point G, Number a]: Creates a hyperbola with focal points F
and G and semimajor axis length a.
Note: Condition: 0 < 2a < 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](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-51-320.jpg)
![52
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 Parabola
3.3.12. Function Commands
Conditional Functions
You can use the Boolean command If 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:
• f(x) = If[x < 3, sin(x), x^2] gives you a function that equals sin(x) for
x < 3 and x2
for x ≥ 3.
• a 3 ˄ b ≥ 0 tests whether “a equals 3 and b is greater than or equal to 0”.
Note: Symbols for conditional statements (e. g., , ˄, ≥) can be found in the list next to the
right of the Input Bar.
Derivative
Derivative[Function]: Returns the derivative of the function.
Derivative[Function, Number n]: Returns the nth
derivative of the function.
Note: You can use f'(x) instead of Derivative[f]as well as f''(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) = x2
‐ 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)](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-52-320.jpg)
![53
Function
Function[Function, Number a, Number b]: Yields a function graph, that is equal to
f on 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 x2
in the
interval [‐1, 1]. If you then type in g(x) = 2 f(x) you will get the function g(x) = 2 x2
, 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 x2
‐ 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:
• Simplify[x + x + x] gives you a function f(x) = 3x.
• Simplify[sin(x) / cos(x)] gives you a function f(x) = tan(x).
• 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 e1, 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-53-320.jpg)
![54
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 nth
derivative of the parametric curve.
Length[Curve, Number t1, Number t2]: Yields the length of the curve between the
parameter 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:
• Circle: (r cos(t), r sin(t)) where r is the circle's radius.
• 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-54-320.jpg)
![55
Note: Also see tool Circular Sector with Center between Two Points
CircumcircularArc
CircumcircularArc[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 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 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 t1, 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:
• Circle: (r cos(t), r sin(t)) where r is the circle's radius.
• 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 x2
, then FormulaText[f] returns 2 x2
(as a LaTeX
text).
FormulaText[Object, Boolean]: Returns the formula for the object as LaTeX text. The
Boolean 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 x2
, then](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-55-320.jpg)
![56
FormulaText[f, true] returns 2 x2
(as a LaTeX text).
FormulaText[f, false] returns a x2
(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: If a: 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:
• TableText[{x^2, 4}, {x^3, 8}, {x^4, 16}]
creates a table as a text object with three rows and two columns. All items of
the table are left aligned.
• TableText[Sequence[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".
• "v" = vertical, i. e. lists are columns
• "h" = horizontal, i. e. lists are rows
• "l" = left aligned](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-56-320.jpg)
![57
• "r" = right aligned
• "c" = centered
Examples:
• TableText[{1,2,3,4},{1,4,9,16},"v"]
creates a text with two columns and four rows whose elements are left
aligned.
• TableText[{1,2,3,4},{1,4,9,16},"h"]
creates a text with two rows and four columns whose elements are left
aligned.
• TableText[{11.2,123.1,32423.9,"234.0"},"vr"]
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 = a2
, 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 text (false).
Example: If a = 2 and c = a2
, then
Text[c, true] returns the text "4".
Text[c, false] returns the text "a2
".
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:
• TextToUnicode["Some text"] gives you the list of Unicode numbers
{83, 111, 109, 101, 32, 116, 101, 120, 116}.
• If text1 is "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".](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-57-320.jpg)
![58
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 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}.
CountIf
CountIf[Condition, List]: Counts the number of elements in the list satisfying the
condition.
Examples:
• CountIf[x < 3, {1, 2, 3, 4, 5}] gives you the number 2.
• CountIf[x<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 nth
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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-58-320.jpg)
![59
Insert
Insert[Object, List, Position]: Inserts the object in the list at the given position.
Example: Insert[x^2, {1, 2, 3, 4, 5}, 3] places x2
at the third position
and gives you the list {1, 2, x2
, 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 x2
at the end of the list
and gives you the list {1, 2, 3, 4, 5, x2
}.
Insert[List 1, List 2, Position]: Inserts all elements of list1 in list2 at the given
position.
Example: Insert[{11, 12}, {1, 2, 3, 4, 5}, 3] places the elements of
list1 at the third (and following) position(s) of list2 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
IterationList[Function, Number x0, Number n]:
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:
• Join[{{1, 2}}] creates the list {1, 2}.
• Join[{{1, 2, 3}, {3, 4}, {8, 7}}] creates the list
{1, 2, 3, 3, 4, 8, 7}.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-59-320.jpg)
![60
KeepIf
KeepIf[Condition, List]: Creates a new list that only contains those elements of the
initial list that fulfill the condition.
Example: KeepIf[x<3, {1, 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 i 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 the index i 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](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-60-320.jpg)
![61
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:
• Sort[{3, 2, 1}] gives you the list {1, 2, 3}.
• Sort[{"pears", "apples", "figs"}] gives you the list elements in
alphabetical order.
• 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:
• Sum[{1, 2, 3}] gives you a number a = 6.
• Sum[{x^2, x^3}] gives you f(x) = x2
+ x3
.
• Sum[Sequence[i,i,1,100]] gives you a number a = 5050.
• Sum[{(1, 2), (2, 3)}] gives you a point A = (3, 5).
• Sum[{(1, 2), 3}] gives you point B = (4, 2).
• 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.
Take
Take[List, Start Position m, End Position n]: Returns a list containing the
elements from positions m to 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-61-320.jpg)
![62
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 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[Image, Line]: Reflects the image about the line.
Reflect[Point, Circle]: Inverts the point in the circle.
Note: Also see tools Reflect Object about Point; 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-62-320.jpg)
![63
Note: Also see tool Rotate Object around Point by Angle
Translate
Translate[Point, Vector ]: Translates the point by the vector.
Translate[Line, 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
c to number d.
Example: If p = 0.1, q = 0.9, and n = 10 are numbers, then
BarChart[ -0.5, n + 0.5,
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]: 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: BarChart[ {1,1,1,2,2,2,2,2,3,3,3,5,5,5,5}, 1]
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:
• BarChart[{10,11,12,13,14}, {5,8,12,0,1}]
• BarChart[{5, 6, 7, 8, 9}, {1, 0, 12, 43, 3}]](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-63-320.jpg)
![64
• BarChart[{0.3, 0.4, 0.5, 0.6}, {12, 33, 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:
• BarChart[{10,11,12,13,14}, {5,8,12,0,1}, 0.5] leaves gaps
between bars.
• 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, Q3, End Value
b]: Creates 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 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 ^( ‐kx)).](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-64-320.jpg)
![65
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 xb
.
Note: All points used need to be in the first quadrant of the coordinate system.
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}, {2, 6, 8, 3, 1}] creates a
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: Histogram[{1, 2, 3, 4},{1.0, 1.1, 1.1, 1.2, 1.7, 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 μ, Standard Deviation σ, Probability P]: Calculates
the function Φ‐1
(P) * σ + μ where Φ ‐1
is the inverse of the probability density
function Φ for N(0,1).
Note: Returns the x‐coordinate with the given probability to the left under the
normal distribution curve.
Mean commands
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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-65-320.jpg)
![66
Mode
Mode[List of Numbers]: Determines the mode(s) of the list elements.
Examples:
• Mode[{1,2,3,4}] returns an empty list {}.
• Mode[{1,1,1,2,3,4}] returns the list {1} .
• Mode[{1,1,2,2,3,3,4}] returns the list {1, 2, 3}.
Normal
Normal[Mean μ, Standard Deviation σ, Variable Value x]: Calculates the
function Φ((x – μ) / σ) where Φ 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
Q1[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-Coordinates, 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 Σ(x2
) ‐ Σ(x) * Σ(x)/n.
Sxx[List of Points]: Calculates the statistic Σ(x2
) ‐ Σ(x) * Σ(x)/n using the x‐coordinates
of the given points.
Sxy[List of Numbers, List of Numbers]: Calculates the statistic
Σ(xy) ‐ Σ(x) * Σ(y)/n.
Sxy[List of Points]: Calculates the statistic Σ(xy) ‐ Σ(x) * Σ(y)/n.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-66-320.jpg)
![67
Syy[List of Points]: Calculates the statistic Σ(y2
) ‐ Σ(y) * Σ(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
Sxy[list] / sqrt(Sxx[list] Syy[list]).
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 corresponding spreadsheet cells:
A1 = 1, A2 = 4, A3 = 9. The command CellRange[A1, 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[{{1, 2}, {3, 4}}] gives you the number a = ‐2.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-67-320.jpg)
![68
Invert
Invert[Matrix]: Inverts the given matrix.
Example: Invert[{{1, 2}, {3, 4}}] gives you the inverse matrix
{{‐2, 1}, {1.5, ‐0.5}}.
Transpose
Transpose[Matrix]: Transposes the matrix.
Example: Transpose[{{1, 2}, {3, 4}}] gives you the matrix {{1, 3}, {2, 4}}.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-68-320.jpg)
![82
There are different ways of making sure that your user defined tools are displayed in the
Toolbar of a new GeoGebra window:
• After creating a new user defined tool you can save your settings using item Save
Settings from the Options menu. From now 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 >. Don’t forget to save your settings after removing the custom tool.
• 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 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:
• Create three sliders a, b, and c with an interval from 0 to 1.
• Create a polygon whose color should be influenced by the slider values.
• Open the Properties Dialog for the polygon poly1 and enter the names of the three
sliders into the text boxes for the color components.
• 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.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-82-320.jpg)
![83
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
Key [plain]
Ctrl
(Mac OS: Cmd)
Ctrl‐Shift
(Mac OS:
Cmd‐Shift)
Alt
(Mac OS: Ctrl)
A Select All
Show / hide
Algebra View
alpha α
B beta β
C
Copy
(Spreadsheet
View)
Export
Graphics View
to Clipboard
D delta δ
E Properties Dialog Euler ℯ
F Refresh views phi φ
G gamma γ
H
I
J
K
L
Select current
layer
lambda λ
M mu μ
N New window
O Open degree symbol °
P Print preview
Export
Graphics
View as
Picture
(png, eps)…
pi π
Q
Select
descendants
Select
ancestors
R
S Save
Show / hide
Spreadsheet
View
sigma σ
T
Export as
PSTricks
theta θ
U
V
Paste
(Spreadsheet
View)](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-83-320.jpg)
![84
Key [plain]
Ctrl
(Mac OS: Cmd)
Ctrl‐Shift
(Mac OS:
Cmd‐Shift)
Alt
(Mac OS: Ctrl)
W
Close
(Mac OS only)
Export
Dynamic
Worksheet
as Webpage
(html)
omega ω
X
Y Redo
Z Undo
0 Exponent 0
1 Exponent 1
2 Exponent 2
3 Exponent 3
4 Exponent 4
5 Exponent 5
6 Exponent 6
7 Exponent 7
8 Exponent 8
9 Exponent 9
‐
Decrease selected
number / angle
Zoom Out minus‐or‐plus
+
Increase selected
number / angle
Zoom In plus‐or‐minus ±
=
Increase selected
number / angle
Zoom In not‐equal‐to ≠
< less‐than‐or‐equal‐to ≤
, (comma) less‐than‐or‐equal‐to ≤
>
greater‐than‐or‐equal‐
to ≥
. (period)
greater‐than‐or‐equal‐
to ≥
*
F1 Help
F2
Start editing
selected object
(Algebra View)
F3
Insert definition of
selected object
(Input Bar)
F4
Insert value of
selected object
(Input Bar)](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-84-320.jpg)
![85
Key [plain]
Ctrl
(Mac OS: Cmd)
Ctrl‐Shift
(Mac OS:
Cmd‐Shift)
Alt
(Mac OS: Ctrl)
F5
Insert name of
selected object
(Input Bar)
F9
Update random
numbers
Enter
Toggle focus
between Graphics
View and Input Bar
Left‐click
Right‐click
(Mac OS:
Ctrl‐click) in
Graphcis
View
Click:
Open Context Menu
(on object)
Properties Dialog
of Graphics View
(on background)
Click and drag:
Fast Drag Mode
(on object)
Zoom rectangle
(on background)
Scroll Wheel Zoom in / out
Zoom in / out
(Applet)
Accelerated Zoom in /
Zoom out
Delete
Delete current
selection
Backspace
Delete current
selection
Up arrow ↑
Increase selected
number / angle
Move selected
point up
Go to prior entries
in Input Bar
history
Go up in
Construction Protocol
x10
speed
multiplier
x0.1 speed
multiplier
(press Shift
only)
x100 speed multiplier](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-85-320.jpg)
![86
Key [plain]
Ctrl
(Mac OS: Cmd)
Ctrl‐Shift
(Mac OS:
Cmd‐Shift)
Alt
(Mac OS: Ctrl)
Right arrow
→
Increase selected
number/angle
Move selected
point to the right
Go up in
Construction Protocol
x10
speed
multiplier
x0.1 speed
multiplier
(press Shift
only)
x100 speed multiplier
Left arrow ←
Decrease selected
number/angle
Move selected
point to the left
Go down in
Construction Protocol
x10
speed
multiplier
x0.1 speed
multiplier
(press Shift
only)
x100 speed multiplier
Down arrow
↓
Decrease selected
number/angle
Move selected
point down
Go to newer entry
in Input Bar
history
Go down in
Construction Protocol
x10
speed
multiplier
x0.1 speed
multiplier
(press Shift
only)
x100 speed multiplier
Home/PgUp
Go to first item in
Construction Protocol
End/PgDn
Go to last item in
Construction Protocol
Additional keyboard commands:
• Alt‐Shift (Mac OS: Ctrl‐Shift): Upper‐case Greek letters
• Spreadsheet View: Ctrl‐Alt‐C copies values (not the formulae)
Note: The degree symbol ° (Alt‐O, Mac OS: Ctrl‐O) and the symbol π for pi (Alt‐P, Mac OS:
Ctrl‐P) can also be used in the slider dialog window for interval (min, max) and the
increment.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-86-320.jpg)
![88
Further use of layers:
• For SVG export objects are grouped by layer.
• 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:
• 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.
• Select the Move tool and double click on any object in the Graphics View. This
opens the Redefine dialog and allows you to redefine the object.
• Change any object by entering its name and the new definition into the Input Bar.
• 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:
• 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).
• 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 Refresh Views in the View menu clears all traces.](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-88-320.jpg)
![89
You can also let GeoGebra automatically create the locus of a point by either using tool
Locus with the mouse, or enter 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:
• Create a segment a between the points A = (‐1, ‐1) and B = (1, ‐1).
• Place a point C on the segment, so it is restricted to move along segment a.
• Create a point P that depends on point C (e. g., P = (x(C), x(C)^2)).
• Use either tool or command Locus in order to create the locus of point P in
dependence on point C:
o Tool 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.
• Note: The locus created in this example is the graph of a parabola on the interval
[‐1, 1].](https://image.slidesharecdn.com/manualdegeogebra-101024142231-phpapp01/85/Manual-de-geogebra-89-320.jpg)
Manual de geogebra
- 3. 3 Contents 1. WHAT IS GEOGEBRA? .......................................................................................................... 6 1.1. Multiple Views for Mathematical Objects ....................................................................................... 6 1.1.1. Graphics View ................................................................................................................................... 6 1.1.2. Algebra View .................................................................................................................................... 7 1.1.3. Spreadsheet View ............................................................................................................................. 8 1.2. GeoGebra as a Tool for Teaching and Learning Mathematics ........................................................... 8 1.2.1. Customizing the User Interface ........................................................................................................ 8 1.2.2. The Properties Dialog ..................................................................................................................... 10 1.2.3. The Context Menu .......................................................................................................................... 10 1.3. GeoGebra as a Presentation Tool .................................................................................................. 11 1.3.1. The Navigation Bar ......................................................................................................................... 11 1.3.2. The Construction Protocol .............................................................................................................. 11 1.3.3. Customize the Settings ................................................................................................................... 12 1.4. GeoGebra as an Authoring Tool .................................................................................................... 13 1.4.1. Printing Options ............................................................................................................................. 13 1.4.2. Creating Pictures of the Graphics View ........................................................................................... 13 1.4.3. Creating Interactive Webpages ...................................................................................................... 14 2. GEOMETRIC INPUT ............................................................................................................ 16 2.1. General Notes .............................................................................................................................. 16 2.2. Construction Tools ........................................................................................................................ 16 2.2.1. General Tools ................................................................................................................................. 17 2.2.2. Point Tools ..................................................................................................................................... 18 2.2.3. Vector Tools ................................................................................................................................... 19 2.2.4. Segment Tools ................................................................................................................................ 19 2.2.5. Ray Tool .......................................................................................................................................... 20 2.2.6. Polygon Tools ................................................................................................................................. 20 2.2.7. Line Tools ....................................................................................................................................... 20 2.2.8. Conic Section Tools ......................................................................................................................... 21 2.2.9. Arc and Sector Tools ....................................................................................................................... 22 2.2.10. Number and Angle Tools ............................................................................................................ 23 2.2.11. Boolean Variable Tool ................................................................................................................ 25 2.2.12. Locus Tool .................................................................................................................................. 25 2.2.13. Geometric Transformation Tools ............................................................................................... 25 2.2.14. Text Tool .................................................................................................................................... 26 2.2.15. Image Tool ................................................................................................................................. 28 3. ALGEBRAIC INPUT ............................................................................................................. 30 3.1. General Notes .............................................................................................................................. 30 3.2. Direct Input .................................................................................................................................. 32
- 4. 4 3.2.1. Numbers and Angles ...................................................................................................................... 32 3.2.2. Points and Vectors .......................................................................................................................... 33 3.2.3. Lines and Axes ................................................................................................................................ 33 3.2.4. Conic Sections ................................................................................................................................ 34 3.2.5. Functions of x ................................................................................................................................. 34 3.2.6. Pre‐defined Functions and Operations ........................................................................................... 35 3.2.7. Boolean Variables and Operations ................................................................................................. 36 3.2.8. List Objects and Operations ............................................................................................................ 37 3.2.9. Matrix Objects and Operations ....................................................................................................... 38 3.2.10. Complex Numbers and Operations ............................................................................................ 38 3.3. Commands ................................................................................................................................... 39 3.3.1. General Commands ........................................................................................................................ 40 3.3.2. Boolean Commands ....................................................................................................................... 40 3.3.3. Number Commands ....................................................................................................................... 41 3.3.4. Angle Command ............................................................................................................................. 45 3.3.5. Point Commands ............................................................................................................................ 45 3.3.6. Vector Commands .......................................................................................................................... 47 3.3.7. Segment Command ........................................................................................................................ 48 3.3.8. Ray Command ................................................................................................................................ 48 3.3.9. Polygon Command ......................................................................................................................... 49 3.3.10. Line Commands ......................................................................................................................... 49 3.3.11. Conic Section Commands ........................................................................................................... 51 3.3.12. Function Commands .................................................................................................................. 52 3.3.13. Parametric Curve Command ...................................................................................................... 53 3.3.14. Arc and Sector Commands ......................................................................................................... 54 3.3.15. Text Commands ......................................................................................................................... 55 3.3.16. Locus Command ........................................................................................................................ 58 3.3.17. List and Sequence Commands ................................................................................................... 58 3.3.18. Geometric Transformation Commands ...................................................................................... 61 3.3.19. Statistics Commands .................................................................................................................. 63 3.3.20. Spreadsheet Commands ............................................................................................................ 67 3.3.21. Matrix Commands ..................................................................................................................... 67 4. MENU ITEMS ........................................................................................................................ 69 4.1. File Menu ..................................................................................................................................... 69 4.2. Edit Menu ..................................................................................................................................... 71 4.3. View Menu ................................................................................................................................... 73 4.4. Options Menu .............................................................................................................................. 74 4.5. Tools Menu .................................................................................................................................. 76 4.6. Window Menu .............................................................................................................................. 77 4.7. Help Menu ................................................................................................................................... 77 5. SPECIAL GEOGEBRA FEATURES ..................................................................................... 79 5.1. Animation .................................................................................................................................... 79 5.1.1. Automatic Animation ..................................................................................................................... 79 5.1.2. Manual Animation .......................................................................................................................... 79
- 5. 5 5.2. Conditional Visibility ..................................................................................................................... 80 5.3. User Defined Tools ....................................................................................................................... 81 5.4. Dynamic Colors ............................................................................................................................. 82 5.5. JavaScript Interface ...................................................................................................................... 82 5.6. Keyboard Shortcuts ...................................................................................................................... 83 5.7. Labels and Captions ...................................................................................................................... 87 5.8. Layers ........................................................................................................................................... 87 5.9. Redefine ....................................................................................................................................... 88 5.10. Trace and Locus ............................................................................................................................ 88 6. INDEX ..................................................................................................................................... 90
- 6. 6 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 of the representations, no matter how they were initially created. 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. Algebra View Graphics View Spreadsheet View Input Bar Toolbar Toolbar Help
- 7. 7 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 Algebra 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 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 ). 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 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.
- 8. 8 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 Graphics 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, spreadsheet 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. • You may use tool Show/Hide Object to show or hide objects. • Open the Context Menu and select item Show Object to change the visibility status of the selected object. • In the 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. • 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 Drawing Pad and use the following ways of zooming: • You may use the tools Zoom In and Zoom Out in order to zoom in the Graphics View. Note: The position of your click determines the center of zoom. • You may use the scroll wheel of your mouse in order to zoom in the Graphics View.
- 9. 9 • You may use keyboard shortcuts to zoom in (Ctrl +) and to zoom out (Ctrl ‐). • After right clicking (Mac OS: Ctrl‐click) on an empty spot on the drawing pad a Context Menu appears which allows you to Zoom. • 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: Ctrl‐click) on the drawing pad and selecting the corresponding items 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: Ctrl‐click) on the drawing pad, you can open this dialog window by selecting Properties from the appearing Context Menu of the Graphics View. • On tab 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. • 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 Isometric. 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 tool 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 > 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.
- 10. 10 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: • Right click (Mac OS: Ctrl‐click) on an object and select Properties… from the appearing Context Menu. • Select item Properties from the Edit menu. • 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: Ctrl‐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 Rename, Delete, Trace On, Animation On, or 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 Properties… in the Context Menu opens the Properties Dialog, where you can change the properties of all objects used.
- 11. 11 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): • button: go back to step 1 • button: go back step by step • button: go forward step by step • button: go to the last step • 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 Play button. • Pause: pause the automatic play feature Note: This button only appears after you click on the Play button. • button: This button opens the Construction 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: • Use the ↑ up arrow of your keyboard to go to the previous construction step. • Use the ↓ down arrow of you keyboard to go to the next construction step. • Use the Home key to go to the beginning of the Construction Protocol. • Use the End key to go to the end of the Construction Protocol. • 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: • Double click a row in order to select a construction step. • Double click the header of any column in order to go to the start of the Construction Protocol.
- 12. 12 • 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. • Right click a row in order to open the Context Menu for the object of this construction 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 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.
- 13. 13 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. 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… • …use tools Move Drawing Pad, Zoom In, and/or 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 mouse. • … use the selection rectangle in order to specify which part of the Graphics View should be exported and saved as a picture.
- 14. 14 • 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 Export1 and Export2 must be within the visible area of the Graphics View. In the File menu, select item Export before clicking on item 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 ‐ 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: • In the Edit menu, you may select item Graphics View to Clipboard. • In the File menu, you first need to select item Export, before you can click on item Graphics View to Clipboard. • In the 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 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 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 Dynamic Worksheet as Webpage (html). This opens the export dialog window for Dynamic Worksheets: • At the top of the export window you can enter the Title, Author, and a Date for your Dynamic Worksheet. • 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. • 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).
- 15. 15 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: • HTML file (e. g. circle.html) – this file includes the worksheet itself • GGB file (e. g. circle.ggb) – this file includes your GeoGebra construction • 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.
- 16. 16 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 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 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 Rename dialog for this object. Then, type in the new name of the selected object and click on the OK button.
- 17. 17 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. Move Drag and drop free objects with the mouse. If you select an object by clicking on it in Move mode, you may… • … delete the object by pressing the Delete‐key • … 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 Drawing Pad Drag and drop the drawing pad in the Graphics View to change its visible area. Note: • 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. • In this mode you can also scale each of the axes by dragging it with the mouse. 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: GeoGebra will use the first two empty columns of the Spreadsheet View to record the values of the selected objects. Relation Select two objects to get information about their relation in a pop‐up window (also see command Relation).
- 18. 18 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). Show / Hide Label Click on an object to show or hide its label. 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. 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. 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 Intersect Two Objects Intersection points of two objects can be created in two ways (also see command Intersect). • Selecting two objects creates all intersection points (if possible). • 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.
- 19. 19 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). 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: • 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). • Clicking on the intersection of two objects creates this intersection point (also see tool Intersect Two Objects and command Intersect). 2.2.3. Vector Tools Vector between Two Points Select the starting point and then the end point of the vector (also see command Vector). 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 Segment between Two Points Select two points A and B in order to create a segment between A and B (also see command Segment). Note: In the Algebra View, the segment's length is displayed. 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.
- 20. 20 2.2.5. Ray Tool 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 Algebra View the equation of the corresponding line is displayed. 2.2.6. Polygon Tools 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 Angle Bisector Angle bisectors can be defined in two ways (also see command AngleBisector): • Selecting three points A, B, and C produces the angle bisector of the enclosed angle, where point B is the apex. • Selecting two lines produces their two angle bisectors. Note: The direction vectors of all angle bisectors have length 1. Best Fit Line Create the best fit line for a set of points in the following ways (also see command FitLine): • Create a selection rectangle that contains all points. • Select a 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).
- 21. 21 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. 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). 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). • Select a point and a conic section to get the polar line. • Select a line or a vector and a conic section to get the diameter line. Tangents Tangents to a conic section can be produced in several ways (also see command Tangent): • Selecting a point A and a conic c produces all tangents through A to c. • Selecting a line g and a conic c produces all tangents to c that are parallel to line g. • 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).
- 24. 24 • 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 the Properties Dialog. 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 α, where α is the angle ABC. 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). 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). 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 (π) for the interval and increment by using the following keyboard shortcuts: • Alt‐O (Mac OS: Ctrl‐O) for the degree symbol ° • Alt‐P (Mac OS: Ctrl‐P) for the pi symbol π 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 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).
- 25. 25 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 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 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: • Type f(x) = x^2 – 2 x – 1 into the Input Bar and press the Enter‐key. • Place a new point A on the x‐axis (see tool New Point; see command Point). • Create point B = (x(A), f'(x(A))) that depends on point A. • Select tool Locus and successively click on point B and point A. • 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. 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)).
- 27. 27 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 Description This is static text Static text A Dynamic text (if point A exists) "Point A = " + A Two‐part mixed text using the value of point A "a = " + a + "cm" Three‐part mixed text using the value of number a Note: If an object with the name xx already exists and you want to create a static text using the object’s name, you need to enter it with quotation marks ("xx"). Otherwise, GeoGebra will automatically create a dynamic text that gives you the value of object xx instead of its name. However, you can type any text that doesn’t match any existing object’s name without the quotation marks. Note: Within a mixed text, the static part needs to be in between a pair of quotation marks. Different parts of a text (e. g., static and dynamic parts) 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 $sqrt{ 2 }$. 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 within the formula, you need to click on an object causing GeoGebra to insert its name as well as the syntax for mixed text.
- 28. 28 Some important LaTeX commands are explained in following table. Please have a look at any LaTeX documentation for further information. LaTeX input Result a cdot b ba ⋅ frac{a}{b} b a sqrt{x} x sqrt[n]{x} n x vec{v} v r overline{AB} AB x^{2} 2 x a_{1} 1a sinalpha + cosbeta βα cossin + int_{a}^{b} x dx ∫ b a xdx sum_{i=1}^{n} i^2 ∑= n i i1 2 2.2.15. Image Tool 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: • Click in the Graphics View to specify the position of the image’s lower left corner. • 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 Insert Image, you can use the keyboard shortcut Alt‐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). • Corner 1: position of the lower left corner of the image
- 29. 29 • Corner 2: position of the lower 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. • 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. • Set point A as the first and point B as the second corner of your image. By dragging points A and B in Move mode you can explore their influence. • 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. • 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: • Set Corner 1 to A • Set Corner 2 to A + (3, 0) • Set Corner 4 to A + (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 Properties… 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 0 % and 100 % on tab Style of the Properties Dialog.
- 30. 30 3. Algebraic Input 3.1. General Notes The algebraic representations of mathematical objects (e. g., values, coordinates, equations) are shown in the Algebra 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 Input Bar: • 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 • 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 • 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: x^2 – y^2 = 2 • Functions: You can name functions by typing, for example, f(x) = or g(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: • If you don’t manually assign a name to an object, GeoGebra assigns the names of new objects in alphabetical order. • You can create indices within the names of objects by using an underscore. For example A1 is entered as A_1 and sAB is entered as s_{AB}. Change Values There are two ways of manipulating a free object’s value: • Change the value of the object by entering its name and the new value in the Input Bar (see section Direct Input).
- 31. 31 Example: If you want to change the value of an existing number a = 3, type a = 5 into the Input Bar and press the Enter‐key. • 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 the ↑ up and ↓ 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. • Right click (Mac OS: Ctrl‐click) on the object and select item Copy to Input Bar from the appearing Context Menu. • 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 Input 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 Input Bar. • Alt click on the object to insert the object’s definition and delete whatever input might have been in the Input Bar before. • Activate tool Move and select the object whose definition you want to insert into the Input Bar. Then, 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.
- 32. 32 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 typing in r = 5.32). Note: In GeoGebra, numbers and angles use a period (.) as a decimal point. You can also use the constant π and 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 π is useful for radian values and can also be entered as pi. Note: You can enter a degree symbol (°) or the pi symbol (π) by using the following keyboard shortcuts: • Alt‐O (Mac OS: Ctrl‐O) for the degree symbol ° • Alt‐P (Mac OS: Ctrl‐P) for the pi symbol π Example: You can enter an angle α in degree (e. g., α = 60°) or in radians (e. g., α = pi/3). Note: GeoGebra does all internal calculations in radians. The degree symbol (°) is nothing but the constant π/180 used to convert degree into radians. Examples: • If a = 30 is a number, then α = a° converts number a to an angle α = 30°, without changing its value. • If you type in b = α / °, the angle α is converted back to the number b = 30, without changing its value.
- 33. 33 Sliders and Arrow Keys Free numbers and angles can be displayed as sliders in the Graphics View (see tool 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: • To enter a point P or a vector v in Cartesian coordinates you may use P = (1, 0) or v = (0, 5). • In order to use polar coordinates type in P = (1; 0°) or v = (5; 90°). Note: You 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: • You can create the midpoint M of two points A and B by entering M = (A + B) / 2 into the Input Bar. • You may calculate the length of a vector v using length = sqrt(v * v) 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 Input Bar. In both cases previously defined variables (e. g. numbers, points, vectors) can be used within the equation. Note: You can enter a line’s name at the beginning of the input followed by a colon. Examples: • Type in g: 3x + 4y = 2 to enter line g as a linear equation. • Define a parameter t (e. g., t = 3) before entering line g in parametric form using g: X = (-5, 5) + t (4, -3).
- 34. 34 • Define the parameters m = 2 and b = -1. Then, you can enter the equation h: y = m*x + b to get 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 section’s name can be entered at the beginning of the input followed by a colon. Examples: • Ellipse ell: ell: 9 x^2 + 16 y^2 = 144 • Hyperbola hyp: hyp: 9 x^2 – 16 y^2 = 144 • Parabola par: par: y^2 = 4 x • Circle c1: c1: x^2 + y^2 = 25 • Circle c2: c2: (x – 5)^2 + (y + 2)^2 = 25 Note: If you define two parameters a = 4 and b = 3 in advance, you may enter for example an ellipse as ell: b^2 x^2 + a^2 y^2 = a^2 b^2. 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: • Function f: f(x) = 3 x^3 – x^2 • Function g: g(x) = tan(f(x)) • 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 f''(x),… in order to get the derivatives of a previously defined function f(x). Example: Define function f as f(x) = 3 x^3 – x^2. Then, you can type in g(x) = cos(f' (x + 2)) in order to get function g.
- 35. 35 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 Operations 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 Addition + Subtraction - Multiplication * or Space key Scalar product * or Space key Division / Exponentiation ^ or 2 Factorial ! Gamma function gamma( ) Parentheses ( ) x‐coordinate x( ) y‐coordinate y( ) Absolute value abs( ) Sign sgn( ) Square root sqrt( ) Cubic root cbrt( ) Random number between 0 and 1 random( ) Exponential function exp( ) or ℯx Logarithm (natural, to base e) ln( ) or log( ) Logarithm to base 2 ld( ) Logarithm to base 10 lg( ) Cosine cos( ) Sine sin( ) Tangent tan( ) Arc cosine acos( ) Arc sine asin( ) Arc tangent atan( ) Hyperbolic cosine cosh( ) Hyperbolic sine sinh( ) Hyperbolic tangent tanh( ) Antihyperbolic cosine acosh( ) Antihyperbolic sine asinh( )
- 36. 36 Operation / Function Input Antihyperbolic tangent atanh( ) Greatest integer less than or equal floor( ) Least integer greater than or equal ceil( ) Round round( ) 3.2.7. Boolean Variables and Operations You can use the Boolean variables true and false in GeoGebra. Just type, for example, a = true or b = false into the Input Bar and press the Enter‐key. Check Box and Arrow Keys Free Boolean variables can be displayed as check boxes in the Graphics View (see tool Check Box to Show/Hide objects). By using the arrow keys of your keyboard you may also change Boolean variables in the Algebra View (see section Manual Animation). Note: You may also use Boolean variables like numbers (value 0 or 1). This allows you to use a checkbox as the dynamic speed of an animated slider allowing you to start and stop the animation. In this case, the animation button is only shown in the Graphics View if there is also an animated slider with static (i. e. non‐dynamic) speed. Operations You can use the following operations for Boolean variables and conditions in GeoGebra by either selecting them from the list next to the Input Bar or by entering them using the keyboard: List Keyboard Example Object types Equal == a b or a == b numbers, points, lines, conics a, b Unequal ≠ != a ≠ b or a != b numbers, points, lines, conics a, b Less than < < a < b numbers a, b Greater than > > a > b numbers a, b Less or equal than ≤ <= a ≤ b or a <= b numbers a, b Greater or equal than ≥ >= a ≥ b or a >= b numbers a, b And && a b or a && b Booleans a, b Or || a b or a || b Booleans a, b Not ¬ ! ¬a or !a Boolean a Parallel a b lines a, b Perpendicular a b lines a, b
- 37. 37 3.2.8. List Objects and Operations Using curly braces you can create a list of several objects (e. g. points, segments, circles). Examples: • L = {A, B, C} gives you a list consisting of three prior defined points A, B, and C. • L = {(0, 0), (1, 1), (2, 2)} produces a list that consists of the entered points, as well as these nameless points. Note: By default, the elements of this list are not shown in the Graphics View. Compare Lists of Objects You can compare two lists of objects by using the following syntax: • List1 == List2: Checks if the two lists are equal and gives you true or false as a result. • List1 != 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: • List1 + List2: Adds corresponding elements of two lists. Note: The two lists need to be of the same length. • List + Number: Adds the number to every element of the list. • List1 – 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. • List – Number: Subtracts the number from every element of the list. Multiplication and Division examples: • List1 * 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. • List * Number: Multiplies every list element with the number. • List1 / 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. • List / Number: Divides every list element by the number. • Number / List: Divides the number by every element of the list. Examples using functions: • List^2: Squares every element of the list. • sin(List): Applies the sine function to every element of the list.
- 38. 38 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. Example: In GeoGebra, {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} represents the matrix ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ 987 654 321 . Matrix Operations Addition and subtraction examples: • Matrix1 + Matrix2: Adds the corresponding elements of two compatible matrices. • Matrix1 – Matrix2: Subtracts the corresponding elements of two compatible matrices. Multiplication examples: • Matrix * Number: Multiplies every element of the matrix by the given number. • Matrix1 * 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}, {4, 5, 6}} gives you the matrix {{9, 12, 15}, {19, 26, 33}, {29, 40, 51}}. • 2x2 Matrix * Point (or Vector): Multiplies the matrix with the given point/vector and gives you a point as a result. Example: {{1, 2}, {3, 4}} * (3, 4) gives you the point A = (11, 25). • 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}, {4, 5, 6}, {0, 0, 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: {{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * {1, 2, 1}. Other examples (see also section Matrix Commands): • Determinant[Matrix]: Calculates the determinant for the given matrix. • Invert[Matrix]: Inverts the given matrix • 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.
- 39. 39 Example: If you enter the 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 Algebra 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 0 + 1i. This also means, that you can use this variable i in order to type complex numbers into the Input Bar (e. g., q = 3 + 4i). Addition and subtraction examples: • (2 + 1i) + (1 – 2i) gives you the complex number 3 – 1i. • (2 + 1i) - (1 – 2i) gives you the complex number 1 + 3i. Multiplication and division examples: • (2 + 1i) * (1 – 2i) gives you the complex number 4 – 3i. • (2 + 1i) / (1 – 2i) gives you the complex number 0 + 1i. Note: The usual multiplication (2, 1)*(1, -2) gives you the scalar product of the two vectors. Other examples: GeoGebra also recognizes expressions involving real and complex numbers. • 3 + (4 + 5i) gives you the complex number 7 + 5i. • 3 - (4 + 5i) gives you the complex number ‐1 ‐ 5i. • 3 / (0 + 1i) gives you the complex number 0 ‐ 3i. • 3 * (1 + 2i) gives you the complex number 3 + 6i. 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 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: A1 is entered as A_1 while SAB 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
- 40. 40 the command into the Input Bar, GeoGebra displays the first command of an alphabetically sorted list that starts with these letters. • In order to accept this suggestion and place the cursor in between the brackets, hit the Enter‐key. • If the 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 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 point 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.
- 41. 41 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 λ of three collinear points A, B, and C, where C = A + λ * 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: • In order to calculate the area between two function graphs, you need to use the command Integral. • Also see tool 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). BinomialCoefficient 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 λ of four collinear points A, B, C, and D, where λ = AffineRatio[B, C, D] / AffineRatio[A, C, D].
- 42. 42 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 0. 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 [a , b]. 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 Indefinite Integral Iteration Iteration[Function, Number x0, Number n]: Iterates the function n times using the given start value x0. Example: After defining f(x) = x^2 the command Iteration[f, 3, 2] gives you the result (32 )2 = 81.
- 43. 43 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 x1, 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 t1, Number t2]: Yields the length of the curve between the parameter 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. 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.
- 44. 44 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 p]: Generates a random number from a binomial distribution with n trials and probability p. RandomNormal[Mean, Standard 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 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.
- 45. 45 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 v1, 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 angle enclosed by BA and BC (between 0 and 360°), where point B is the apex. Angle[Point A, Point B, Angle α]: Returns the angle of size α drawn from point A with apex B. Note: The point Rotate[A, α, 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 0 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 Angle and 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.
- 46. 46 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 the 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 h]: 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 n]: Yields the nth intersection point of the line and the conic section. Intersect[Conic c1, Conic c2]: Yields all intersection points of conic sections c1 and c2 (max. 4). Intersect[Conic c1, Conic c2, Number n]: Yields the nth intersection point of conic sections c1 and c2. Intersect[Polynomial f1, Polynomial f2]: Yields all intersection points of polynomials f1 and f2. Intersect[Polynomial f1, Polynomial f2, Number n]: Yields the nth 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 nth intersection point of the polynomial and the line. Intersect[Function f, Function g, Point A]: Calculates the intersection point of functions 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.
- 47. 47 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 Point[Line]: Returns a point on the line. Point[Conic]: Returns a point on the conic section. Point[Function]: Returns a point on the function. Point[Polygon]: Returns a point on the polygon. Point[Vector ]: Returns a point on the vector. Point[Point, Vector]: Creates a new point by adding the vector to the given point. 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 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).
- 48. 48 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 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
- 49. 49 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 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 two 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.
- 50. 50 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 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 the x‐coordinate of point A. Tangent[Point, Curve]: Creates the tangent to the curve in the given point. Note: Also see tool Tangents
- 51. 51 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 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 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 > 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. Note: Also see tool Ellipse Hyperbola Hyperbola[Point F, Point G, Number a]: Creates a hyperbola with focal points F and G and semimajor axis length a. Note: Condition: 0 < 2a < 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
- 52. 52 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 Parabola 3.3.12. Function Commands Conditional Functions You can use the Boolean command If 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: • f(x) = If[x < 3, sin(x), x^2] gives you a function that equals sin(x) for x < 3 and x2 for x ≥ 3. • a 3 ˄ b ≥ 0 tests whether “a equals 3 and b is greater than or equal to 0”. Note: Symbols for conditional statements (e. g., , ˄, ≥) can be found in the list next to the right of the Input Bar. Derivative Derivative[Function]: Returns the derivative of the function. Derivative[Function, Number n]: Returns the nth derivative of the function. Note: You can use f'(x) instead of Derivative[f]as well as f''(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) = x2 ‐ 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)
- 53. 53 Function Function[Function, Number a, Number b]: Yields a function graph, that is equal to f on 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 x2 in the interval [‐1, 1]. If you then type in g(x) = 2 f(x) you will get the function g(x) = 2 x2 , 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 x2 ‐ 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: • Simplify[x + x + x] gives you a function f(x) = 3x. • Simplify[sin(x) / cos(x)] gives you a function f(x) = tan(x). • 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 e1, 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.
- 54. 54 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 nth derivative of the parametric curve. Length[Curve, Number t1, Number t2]: Yields the length of the curve between the parameter 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: • Circle: (r cos(t), r sin(t)) where r is the circle's radius. • 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.
- 55. 55 Note: Also see tool Circular Sector with Center between Two Points CircumcircularArc CircumcircularArc[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 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 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 t1, 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: • Circle: (r cos(t), r sin(t)) where r is the circle's radius. • 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 x2 , then FormulaText[f] returns 2 x2 (as a LaTeX text). FormulaText[Object, Boolean]: Returns the formula for the object as LaTeX text. The Boolean 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 x2 , then
- 56. 56 FormulaText[f, true] returns 2 x2 (as a LaTeX text). FormulaText[f, false] returns a x2 (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: If a: 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: • TableText[{x^2, 4}, {x^3, 8}, {x^4, 16}] creates a table as a text object with three rows and two columns. All items of the table are left aligned. • TableText[Sequence[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". • "v" = vertical, i. e. lists are columns • "h" = horizontal, i. e. lists are rows • "l" = left aligned
- 57. 57 • "r" = right aligned • "c" = centered Examples: • TableText[{1,2,3,4},{1,4,9,16},"v"] creates a text with two columns and four rows whose elements are left aligned. • TableText[{1,2,3,4},{1,4,9,16},"h"] creates a text with two rows and four columns whose elements are left aligned. • TableText[{11.2,123.1,32423.9,"234.0"},"vr"] 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 = a2 , 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 text (false). Example: If a = 2 and c = a2 , then Text[c, true] returns the text "4". Text[c, false] returns the text "a2 ". 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: • TextToUnicode["Some text"] gives you the list of Unicode numbers {83, 111, 109, 101, 32, 116, 101, 120, 116}. • If text1 is "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".
- 58. 58 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 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}. CountIf CountIf[Condition, List]: Counts the number of elements in the list satisfying the condition. Examples: • CountIf[x < 3, {1, 2, 3, 4, 5}] gives you the number 2. • CountIf[x<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 nth 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.
- 59. 59 Insert Insert[Object, List, Position]: Inserts the object in the list at the given position. Example: Insert[x^2, {1, 2, 3, 4, 5}, 3] places x2 at the third position and gives you the list {1, 2, x2 , 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 x2 at the end of the list and gives you the list {1, 2, 3, 4, 5, x2 }. Insert[List 1, List 2, Position]: Inserts all elements of list1 in list2 at the given position. Example: Insert[{11, 12}, {1, 2, 3, 4, 5}, 3] places the elements of list1 at the third (and following) position(s) of list2 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 IterationList[Function, Number x0, Number n]: 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: • Join[{{1, 2}}] creates the list {1, 2}. • Join[{{1, 2, 3}, {3, 4}, {8, 7}}] creates the list {1, 2, 3, 3, 4, 8, 7}.
- 60. 60 KeepIf KeepIf[Condition, List]: Creates a new list that only contains those elements of the initial list that fulfill the condition. Example: KeepIf[x<3, {1, 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 i 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 the index i 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
- 61. 61 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: • Sort[{3, 2, 1}] gives you the list {1, 2, 3}. • Sort[{"pears", "apples", "figs"}] gives you the list elements in alphabetical order. • 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: • Sum[{1, 2, 3}] gives you a number a = 6. • Sum[{x^2, x^3}] gives you f(x) = x2 + x3 . • Sum[Sequence[i,i,1,100]] gives you a number a = 5050. • Sum[{(1, 2), (2, 3)}] gives you a point A = (3, 5). • Sum[{(1, 2), 3}] gives you point B = (4, 2). • 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. Take Take[List, Start Position m, End Position n]: Returns a list containing the elements from positions m to 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.
- 62. 62 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 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[Image, Line]: Reflects the image about the line. Reflect[Point, Circle]: Inverts the point in the circle. Note: Also see tools Reflect Object about Point; 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.
- 63. 63 Note: Also see tool Rotate Object around Point by Angle Translate Translate[Point, Vector ]: Translates the point by the vector. Translate[Line, 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 c to number d. Example: If p = 0.1, q = 0.9, and n = 10 are numbers, then BarChart[ -0.5, n + 0.5, 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]: 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: BarChart[ {1,1,1,2,2,2,2,2,3,3,3,5,5,5,5}, 1] 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: • BarChart[{10,11,12,13,14}, {5,8,12,0,1}] • BarChart[{5, 6, 7, 8, 9}, {1, 0, 12, 43, 3}]
- 64. 64 • BarChart[{0.3, 0.4, 0.5, 0.6}, {12, 33, 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: • BarChart[{10,11,12,13,14}, {5,8,12,0,1}, 0.5] leaves gaps between bars. • 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, Q3, End Value b]: Creates 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 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 ^( ‐kx)).
- 65. 65 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 xb . Note: All points used need to be in the first quadrant of the coordinate system. 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}, {2, 6, 8, 3, 1}] creates a 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: Histogram[{1, 2, 3, 4},{1.0, 1.1, 1.1, 1.2, 1.7, 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 μ, Standard Deviation σ, Probability P]: Calculates the function Φ‐1 (P) * σ + μ where Φ ‐1 is the inverse of the probability density function Φ for N(0,1). Note: Returns the x‐coordinate with the given probability to the left under the normal distribution curve. Mean commands 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.
- 66. 66 Mode Mode[List of Numbers]: Determines the mode(s) of the list elements. Examples: • Mode[{1,2,3,4}] returns an empty list {}. • Mode[{1,1,1,2,3,4}] returns the list {1} . • Mode[{1,1,2,2,3,3,4}] returns the list {1, 2, 3}. Normal Normal[Mean μ, Standard Deviation σ, Variable Value x]: Calculates the function Φ((x – μ) / σ) where Φ 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 Q1[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-Coordinates, 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 Σ(x2 ) ‐ Σ(x) * Σ(x)/n. Sxx[List of Points]: Calculates the statistic Σ(x2 ) ‐ Σ(x) * Σ(x)/n using the x‐coordinates of the given points. Sxy[List of Numbers, List of Numbers]: Calculates the statistic Σ(xy) ‐ Σ(x) * Σ(y)/n. Sxy[List of Points]: Calculates the statistic Σ(xy) ‐ Σ(x) * Σ(y)/n.
- 67. 67 Syy[List of Points]: Calculates the statistic Σ(y2 ) ‐ Σ(y) * Σ(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 Sxy[list] / sqrt(Sxx[list] Syy[list]). 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 corresponding spreadsheet cells: A1 = 1, A2 = 4, A3 = 9. The command CellRange[A1, 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[{{1, 2}, {3, 4}}] gives you the number a = ‐2.
- 68. 68 Invert Invert[Matrix]: Inverts the given matrix. Example: Invert[{{1, 2}, {3, 4}}] gives you the inverse matrix {{‐2, 1}, {1.5, ‐0.5}}. Transpose Transpose[Matrix]: Transposes the matrix. Example: Transpose[{{1, 2}, {3, 4}}] gives you the matrix {{1, 3}, {2, 4}}.
- 70. 70 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. Print Preview Keyboard shortcut: Ctrl‐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 ‐ Dynamic Worksheet 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. Export ‐ Graphics View as Picture (png, eps)… Keyboard shortcut: Ctrl‐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: • 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. • 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). 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.
- 71. 71 • 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 Bézier curves (this guarantees that the text looks the same even if the correct font is not installed). • SVG – Scaleable Vector Graphic (see EPS format above) • 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. Close Keyboard shortcut: Alt‐F4 (Mac OS: Cmd‐W) This menu item closes the GeoGebra window. If you didn’t save your construction prior to selecting Close, you are asked if you would like to do so. 4.2. Edit Menu 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. Redo Keyboard shortcut: Ctrl‐Y (Mac OS: Cmd‐Shift‐Z)
- 77. 77 Note: The current Toolbar setting is saved with your construction in a GGB file. 4.6. Window 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 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 are 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: • 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. • If 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. www.geogebra.org If you have Internet access, this menu item opens the GeoGebra webpage in your default web browser (http://www.geogebra.org).
- 79. 79 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: Ctrl‐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: • <=> Oscillating: The animation cycle alternates between Decreasing and Increasing. • => 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. • <= 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.
- 80. 80 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: • Shift + arrow key gives you a step width of 0.1 units • 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 also 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 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., ≠, ≥, ˄, ǁ) from the drop down menu in order to create your conditional statements. Examples: • If a is a slider, then the conditional statement a < 2 means that the corresponding object is only shown in the Graphics View if the slider’s value is less than 2. • If b is 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. • If g and h are two lines and you would like a text to be shown whenever these lines are parallel, then you could use g h as a conditional statement for the text.
- 81. 81 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 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. • 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. • Select Create New Tool in the Tools menu. • 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. • 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 construction. • 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. • You may also enter text to be shown in the Toolbar Help. • 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 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.
- 82. 82 There are different ways of making sure that your user defined tools are displayed in the Toolbar of a new GeoGebra window: • After creating a new user defined tool you can save your settings using item Save Settings from the Options menu. From now 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 >. Don’t forget to save your settings after removing the custom tool. • 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 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: • Create three sliders a, b, and c with an interval from 0 to 1. • Create a polygon whose color should be influenced by the slider values. • Open the Properties Dialog for the polygon poly1 and enter the names of the three sliders into the text boxes for the color components. • 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.
- 83. 83 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 Key [plain] Ctrl (Mac OS: Cmd) Ctrl‐Shift (Mac OS: Cmd‐Shift) Alt (Mac OS: Ctrl) A Select All Show / hide Algebra View alpha α B beta β C Copy (Spreadsheet View) Export Graphics View to Clipboard D delta δ E Properties Dialog Euler ℯ F Refresh views phi φ G gamma γ H I J K L Select current layer lambda λ M mu μ N New window O Open degree symbol ° P Print preview Export Graphics View as Picture (png, eps)… pi π Q Select descendants Select ancestors R S Save Show / hide Spreadsheet View sigma σ T Export as PSTricks theta θ U V Paste (Spreadsheet View)
- 84. 84 Key [plain] Ctrl (Mac OS: Cmd) Ctrl‐Shift (Mac OS: Cmd‐Shift) Alt (Mac OS: Ctrl) W Close (Mac OS only) Export Dynamic Worksheet as Webpage (html) omega ω X Y Redo Z Undo 0 Exponent 0 1 Exponent 1 2 Exponent 2 3 Exponent 3 4 Exponent 4 5 Exponent 5 6 Exponent 6 7 Exponent 7 8 Exponent 8 9 Exponent 9 ‐ Decrease selected number / angle Zoom Out minus‐or‐plus + Increase selected number / angle Zoom In plus‐or‐minus ± = Increase selected number / angle Zoom In not‐equal‐to ≠ < less‐than‐or‐equal‐to ≤ , (comma) less‐than‐or‐equal‐to ≤ > greater‐than‐or‐equal‐ to ≥ . (period) greater‐than‐or‐equal‐ to ≥ * F1 Help F2 Start editing selected object (Algebra View) F3 Insert definition of selected object (Input Bar) F4 Insert value of selected object (Input Bar)
- 85. 85 Key [plain] Ctrl (Mac OS: Cmd) Ctrl‐Shift (Mac OS: Cmd‐Shift) Alt (Mac OS: Ctrl) F5 Insert name of selected object (Input Bar) F9 Update random numbers Enter Toggle focus between Graphics View and Input Bar Left‐click Right‐click (Mac OS: Ctrl‐click) in Graphcis View Click: Open Context Menu (on object) Properties Dialog of Graphics View (on background) Click and drag: Fast Drag Mode (on object) Zoom rectangle (on background) Scroll Wheel Zoom in / out Zoom in / out (Applet) Accelerated Zoom in / Zoom out Delete Delete current selection Backspace Delete current selection Up arrow ↑ Increase selected number / angle Move selected point up Go to prior entries in Input Bar history Go up in Construction Protocol x10 speed multiplier x0.1 speed multiplier (press Shift only) x100 speed multiplier
- 86. 86 Key [plain] Ctrl (Mac OS: Cmd) Ctrl‐Shift (Mac OS: Cmd‐Shift) Alt (Mac OS: Ctrl) Right arrow → Increase selected number/angle Move selected point to the right Go up in Construction Protocol x10 speed multiplier x0.1 speed multiplier (press Shift only) x100 speed multiplier Left arrow ← Decrease selected number/angle Move selected point to the left Go down in Construction Protocol x10 speed multiplier x0.1 speed multiplier (press Shift only) x100 speed multiplier Down arrow ↓ Decrease selected number/angle Move selected point down Go to newer entry in Input Bar history Go down in Construction Protocol x10 speed multiplier x0.1 speed multiplier (press Shift only) x100 speed multiplier Home/PgUp Go to first item in Construction Protocol End/PgDn Go to last item in Construction Protocol Additional keyboard commands: • Alt‐Shift (Mac OS: Ctrl‐Shift): Upper‐case Greek letters • Spreadsheet View: Ctrl‐Alt‐C copies values (not the formulae) Note: The degree symbol ° (Alt‐O, Mac OS: Ctrl‐O) and the symbol π for pi (Alt‐P, Mac OS: Ctrl‐P) can also be used in the slider dialog window for interval (min, max) and the increment.
- 87. 87 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: • Select the tool Show / Hide Label and click on the object whose label you would like to show or hide. • Open the Context Menu for the desired object and select Show Label. • 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 & 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 0 to 9) and higher numbered 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 0 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.
- 88. 88 Further use of layers: • For SVG export objects are grouped by layer. • 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: • 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. • Select the Move tool and double click on any object in the Graphics View. This opens the Redefine dialog and allows you to redefine the object. • Change any object by entering its name and the new definition into the Input Bar. • 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: • 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). • 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 Refresh Views in the View menu clears all traces.
- 89. 89 You can also let GeoGebra automatically create the locus of a point by either using tool Locus with the mouse, or enter 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: • Create a segment a between the points A = (‐1, ‐1) and B = (1, ‐1). • Place a point C on the segment, so it is restricted to move along segment a. • Create a point P that depends on point C (e. g., P = (x(C), x(C)^2)). • Use either tool or command Locus in order to create the locus of point P in dependence on point C: o Tool 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. • Note: The locus created in this example is the graph of a parabola on the interval [‐1, 1].