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# GeoGebraCommands32

last edited by 10 years, 5 months ago

Commands for GeoGebra 3.2

Angle Commands

Arc and Sector Commands

Boolean Commands

Conic Section Commands

Function Commands

General Construction

Geometric transformation commands

Line Commands

List and Sequence commands

Matrix Commands

Number Commands

Parametric Curve Commands

Random Variable Commands

Statistics Commands

Text Commands

Vector Commands

Angle Commands

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.

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 mode_angularbisector_32 Angle Bisector

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 mode_circlearc3_32 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.

Note: Also see tool mode_circlesector3_32 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 mode_circumcirclearc3_32 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 mode_circumcirclesector3_32 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 mode_semicircle_32 Semicircle

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.

IsInteger

IsInteger[Number]: Returns true or false depending whether the number is an integer or not.

Conic Section Commands

Area

Area[Conic c]: Calculates the area of a conic section c (circle or ellipse).

Axes

Axes[Conic]: Returns the major and minor axis of a conic section.

Center

UK English: Centre

Center[Conic]: Returns the center of a circle, ellipse, or hyperbola.

Note: Also see tool mode_midpoint_32 Midpoint or Center

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 mode_compasses_32 Compass, mode_circle2_32 Circle with Center through Point, mode_circlepointradius_32 Circle with Center and Radius, and mode_circle3_32 Circle through Three Points

Circumference

Circumference[Conic]: Returns the circumference of a circle or ellipse.

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 mode_conic5_32 Conic through Five Points

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.

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 mode_ellipse3_32 Ellipse

Focus

Focus[Conic]: Yields (all) foci of the conic section.

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 mode_hyperbola3_32 Hyperbola

Intersect

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.

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.

MajorAxis

MajorAxis[Conic]: Returns the major axis of the conic section.

MinorAxis

MinorAxis[Conic]: Returns the minor axis of the conic section.

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 mode_parabola_32 Parabola

Parameter

Parameter[Parabola]: Returns the parameter of the parabola, which is the distance of directrix and focus.

Polar

Polar[Point, Conic]: Creates the polar line of the given point relative to the conic section.

Note: Also see tool mode_polardiameter_32 Polar or Diameter Line

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.

Vertex

Vertex[Conic]: Returns (all) vertices of the conic section.

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.

Curvature

Curvature[Point, Function]: Calculates the curvature of the function in the given point.

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

Extremum

UK English: TurningPoint

Extremum[Polynomial]: Yields all local extrema of the polynomial function as points on the function graph.

Factor

UK English: Factorise

Factor[Polynomial]: Factors the polynomial.

Example: Factor[x^2 + x - 6] gives you f(x) = (x-2)(x+3)

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].

InflectionPoint

InflectionPoint[Polynomial]: Yields all inflection points of the polynomial as points on the function graph.

Integral

Integral[Function]: Yields the indefinite integral for the given function.

Note: Also see command for Definite integral

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

Intersect

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.

Length

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.

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.

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.

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 s

Root[Function, Number a, Number b]: Yields one root of the function in the interval [a, b] (regula falsi).

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.

Tangent

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 mode_tangent_32 Tangents

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.

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.

General Construction 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 mode_delete_16 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.

Geometric Transformation Commands

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).

Dilate

UK English: Enlarge

Dilate[Point A, Number, Point S]: Dilates point A from point S using the given factor.

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 mode_dilatefrompoint_16 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 mode_mirroratpoint_16 Reflect Object about Point; mode_mirroratline_16 Reflect Object about Line;

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.

Note: Also see tool mode_rotatebyangle_16 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 mode_translatebyvector_16 Translate Object by Vector

Line Commands

Asymptote

Asymptote[Hyperbola]: Yields both asymptotes of the hyperbola.

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.

Note: Also see tool mode_join_32 Line through Two Points

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 mode_orthogonal_32 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 mode_linebisector_32 Perpendicular Bisector

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 mode_polygon_32 Polygon and mode_regularpolygon_32 Regular Polygon

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 mode_ray_32 Ray through Two Points

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 mode_segment_32 Segment between Two Points and mode_segmentfixed_32 Segment with Given Length from Point

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 mode_slope_32 Slope

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 mode_tangent_32 Tangents

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.

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}.

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 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.

Matrix commands

Row

Row[Spreadsheet Cell]: Returns the row number of a spreadsheet cell (starting at 1).

Example: Row[B3] gives you number a = 3.

Determinant

Determinant[Matrix]: Returns the determinant of the matrix.

Example: Determinant[{{1, 2}, {3, 4}}] gives you the number a = -2.

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}}.

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,…

· In order to calculate the area between two function graphs, you need to use the command Integral.

· Also see tool mode_area_32 Area

BinomialCoefficient

BinomialCoefficient[Number n, Number r]: Calculates the binomial coefficient

n choose r.

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].

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 mode_distance_32 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.

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.

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).

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.

Perimeter

Perimeter[Polygon]: Returns the perimeter of the polygon.

Commands for Parametric Curves

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.

Note: Using the mouse you can also place a point on a curve using tool mode_point_16 New Point or command Point. Since the parameters a and b are dynamic you could use slider variables as well (see tool mode_slider_32.gif Slider).

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.

Point Commands

Centroid

Centroid[Polygon]: Returns the centroid of the polygon.

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.

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[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.

Note: Also see tool mode_intersect_16 Intersect two Objects

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 mode_locus_32 Locus

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 mode_midpoint_32 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 mode_point_16 New Point

Random Variable 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.

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.

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}]

· 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 mode_fitline_32 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 x^(-kx)).

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.

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.

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.

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

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

· "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 gives you the text "a".

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".

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.

ote: A line with equation ax + by = c has the direction vector (b, - a).

Length

Length[Vector]: Yields the length of the vector.

Length[Point A]: Yields the length of the position vector of the given point .

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 mode_vector_32 Vector between Two Points