GeoGebra Quick Start SM025

SM025

GeoGebra Classic 6.0

Quick Start
Basic tips for a simpler way in
teaching Mathematics

G-maths
Kedah Matriculation College

Kolej Matrikulasi Kedah
Kementerian Pendidikan Malaysia
Kedah Darul Aman
Malaysia

© G-Maths 2019
First published 2019

All rights reserved.
No part of this publication may be reproduced,
Stored in a retrieval system, or transmitted in any form or
By any means, electronic, mechanical, photocopying, recording
Or otherwise, without the prior permission of G-Maths

Printed in Malaysia by 1
G-Maths

G-MATHS

What is GeoGebra Classic?

The “GeoGebra Classic” is a free dynamic mathematics apps for all levels of education that
brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one
easy-to-use package. Interactive learning, teaching and evaluation resources created with
GeoGebra can be shared and used by everyone at tube.geogebra.org.

GeoGebra Classic features

• Geometry, Algebra and Spreadsheet are connected and fully dynamic
• Computer Algebra System for symbolic calculations
• 3D Graphics View available
• Easy-to-use interface, yet many powerful features
• Authoring tool to create interactive learning materials as web pages
• Available in many languages for millions of users around the world
• Open source software freely available for non-commercial users

Ready to get started?
We hope you enjoy working in GeoGebra Classic!

Sincerely,
G-Maths
The Quick Start Team

G-MATHS 2

Table of Contents 4
4
INTEGRATION 5
Integral() 6
IntegralBetween() 7
Restricting the Domain of a Function 7
Surface() 9
Copying the Graphics View to Clipboard
Hands-on Chapter 1 10
12
VECTORS 12
Abs() 13
Length() 13
UnitVector() 14
Dot() 15
Cross() 17
Line() 18
Curve() 21
Plane() 23
PerpendicularVector() 24
Intersect()
Angle() 26

Appendix 3

G-MATHS

1)Integration

a) Integral(<Function>)

Gives the indefinite integral with respect to the main variable.

Example 1

Mathematics: ∫

: ( )

:


b) Integral(<Function>,<variable>)

Gives the definite integral over the interval [Start x-Value, End x-Value] with respect
to the main variable. This command also shades the area between the function
graph of f and the x-axis.

Example 2

Mathematics: ∫( + )

: ( + , )

: +


c) Integral(<Function>,<Start x-Value>, <End x-Value>)

Gives the partial integral with respect to the given variable.

Example 3

Mathematics: ∫
: ( , , )
: .

G-MATHS 4

d) IntegralBetween(<Function 1>,<Function 2>,<Number 1>, <Number 2>)

Gives the definite integral of the difference ( ) − ( ) of two function
over the interval [ , ], where is the first number and b the second, with respect
to the main variable.

Example 4

Mathematics: = ∫ ( ) − ( )
: ( ( ), ( ), , )
Or ( ( ), ( ), , °)
:

Note:

1. This command also shades the area between the function graphs of .
2. Press <Alt><O> to get the symbol of degree(°).
3. Press <Alt><P> to get the symbol of pi( ).

G-MATHS 5

e) Restricting the Domain of a Function
: ( ) = ( )

: ( ) = ( ), ( ≤ ≤ )

G-MATHS 6

f) Surface(<Function>,<Angle>,<Axis>)
Creates a surface of revolution, rotating the given Functions from 0 to given
Angle around the x-axis.
* Turn on <3D Graphics View> before using <Surface> Command

Menu

3D Graphics

Example 5
Mathematics: = ∫ ( )
: ( ( ), °, )
:

g) Copying the Graphics View to Clipboard

In the Desktop version, we can copy the graphics to the Clipboard in the following
ways:

G-MATHS 7

1. In the menu <Edit>, select the option <Graphics View to Clipboard>

Menu

Edit

Graphics View
to Clipboard

2. In the menu <File>, first select <Export> and then select <Graphics View to
Clipboard>.

3. In the menu <File>, <Export Image>, Select <Copy to Clipboard>

Try This:
:

Turn on <3D Graphics>
= ( °, °)
: = ( )
= ( , , )
Move around the slider a and see the effect

G-MATHS 8

Hands-on Chapter 1

1. Sketch the following function and shade the bounded area:

No Function Command and Output

1 Sketch the area of the region bounded by

the curve ( ) = 1 , = 2, = 5 −

−1

. Command:


= −


( − , , )

Output:
1.39

2 Sketch the curve ( ) = ( − 2)( − 5).
Hence, shade the region bounded by the
curve, = 0, = 5 and the −

Command:
( ( − )( − ), , )

3 Sketch the area of the region bounded by

the curves = sin and = cos for = Command:
0 = .

4

( ) = ( ), ( ≤ ≤ )

( ) = ( ), ( ≤ ≤ )


( , , , )

G-MATHS 9

2) Vectors

* Turn on <3D Graphics View> for vector in 3D
Points and vectors may enter via<Input Bar> in Cartesian. Points can also be
created using Point tools and vectors can be created using Vector from Point
Tool or the Vector Tools.

Point
Tools

Vector

Vector
from
Point

Upper case label denoted points, whereas lower case labels refer to vectors.

Example:

Task Commands
(Examples)
a) To enter a point P in 2D in Cartesian form = ( , )
b) To enter a point Q in 3D in Cartesian form = ( , , )
c) To enter a vector u in 2D = ( , )
d) To enter a vector v in 3D = ( , , )
e) To enter a point R in 2D polar form = ( ; °)
f) To enter a point S in 3D spherical coordinates = ( ; °; °)

G-MATHS 10

Notes:

1. Use semicolon (;) to separate polar coordinates. If the command without the

symbol of degree, GeoGebra will treat the angle as radians.

2. Coordinates of points and vectors can be accessed using predefine functions
(), () ().

3. Magnitude of a point Q can be obtained using ( ).
4. Midpoint M of any two points and can be obtained using = ( + )/2.
5. Length of a vector using ℎ( ).

Examples:

If = (3,4), = (7,6) and = (2,4,7).

a) ( ) returns the value of x for . ➔ ( ) = 3

b) ( ) returns the value of y for . ➔ ( ) = 4

c) ( ) returns the value of z for P. ➔ ( ) = 7

d) ( ) returns the magnitude of . ➔ ( ) = 5

e) = ( + )/2 returns the midpoint of P and Q. ➔(5,5)

f) ℎ( ) returns the length of vector ➔ 8.31

G-MATHS 11

Basic Operations 12
Magnitude and Unit Vector

a) Abs(<x>)
Gives the magnitude of a vector.
Example
Mathematics: | + + |
: = ( , , )
( )
: .

b) Length(<Object>)
Gives the magnitude of a vector.
Example

Mathematics: | + + |
: = ( , , )

( )
: .

G-MATHS

c) UnitVector(<Object>)

Yields a vector with length1, which has the same direction and orientation as the

given vector. The vector must be defined first.

Example

Mathematics: = + +
: = ( , , )

( )
.
: ( . )
.

Addition/Subtraction of vectors (+/-)
Example

Mathematics: = + + = + + .
( + ) ( − ).

: = ( , , )
= ( , , )
+
−


: + ⟹ ( )


−
− ⟹ (− )

−

Scalar Product (Dot)

a) Dot(<Vector 1>, <Vector 2>)

Returns the value of dot product (scalar product) of two vectors.

Example

Mathematics: ( + + ). ( + + )
: = ( , , )

G-MATHS 13

= ( , , )
( , )

:

Vector Product(Cross, ⊗)
a) Cross(<Vector 1>, <Vector 2>)
Calculates the cross product of two vectors.
Example

Mathematics: ( + + ) ( + + )

: = ( , , )

= ( , , )

( , )

: ( )
−

Note:

<Alt><Shift><8> to get the symbol of vector product, ⊗.

Example:

If = + 2 + 3 and = 4 + 5 + 6 , find .

: = ( , , )
= ( , , )
⊗

−
: ( )

−

or
: { , , } ⊗ { , , }
: {− , , − }

G-MATHS 14

Line
a) = + ∗
Mathematics: = ( + + ) + ( + + ))
: ( , , ) + ∗ ( , , )

b) Line(<Point 1>, <Point 2>)
Creates a line through two points A and B
Example
Mathematics: Find the equation of a line passing through
( , , ) ( , , ).
: (( , , ), ( , , ))
: = ( , , ) + ( , , )

c) Line(<Point>, <Parallel Line>) 15
Creates a line through the given point parallel to the given line.
Example
Mathematics: Find the equation of a line passing through
( , , ) ( , , ).
: (( , , ), ( , , ))

G-MATHS

: Same as above

d) Line(<Point>, <Direction Vector>)

Creates a line through the given point with direction vector v.
Example

Mathematics: Find the equation of a line passing through ( , , ) with the
direction vector = + + .

:
= ( , , )
= ( , , )
( , )

:

G-MATHS 16

e) Curve(<Expression>,<Expression>,<Parameter Variable>,<Start Value>, <End Value>)

Yields the 3D Cartesian parametric curve for the given x-expression, y-expression and z-
expression within the given interval.

Example

Mathematics: Plot the equation of line:
= +
= −
= +

:
( + , − , + , , − , )

:

G-MATHS 17

Plane
a) + + =
Plot a plane.
Example
Mathematics: Plot the plane with the equation of + + =
: + + =
:

b) Plane(<Point>,<Plane>)
Creates a plane through the given point, parallel to the given plane.
Example
Mathematics: Find the equation of the plane passes through ( , , ) and
parallel to the plane + + = .
: (( , , ), ( + + = ))
: + + =

G-MATHS 18

c) Plane(<Point>,<Line>)

Creates the plane through the given point and line.

Example

Mathematics: Find the equation of the plane passes through ( , , ) and
line = ( , , ) + ( , , )

: = ( , , )
: = ( , , ) + ∗ ( , , )
( , )

: − + − =

d) Plane(<Line>,<Line>) 19
Creates the plane through the lines (if the lines are in the same plane).

Example

Mathematics: Find the equation of the plane which contains the
line = (− + + ) + ( + − ) and
line = ( + ) + (− − + )

: : = (− , , ) + ∗ ( , , − )
: = ( , , ) + ∗ (− , − , )
( , )

: − − − = −

G-MATHS

e) Plane(<Point>,<Point>,<Point>)
Creates a plane through three points.
Example

Mathematics: Find the equation of the plane which contains the points
( , , ), ( , , ) ( , , ).

: = ( , , )
= ( , , )
= ( , )
( , , )

: − − + =

G-MATHS 20

f) Plane(<Point>,<Vector>,<Vector>)

Creates the plane through the given point and vectors.

Example

Mathematics: Find the equation of the plane which contains the points
( , , ), = + = + + .

: = ( , , )
= ( , )
= ( , , )
( , , )

: − + =

g) PerpendicularVector(<Plane>)

Creates a perpendicular vector to a plane.

Example

Mathematics: Determine the normal vector of the plane 3x+2y-4z=6.
: : + − =

( )

: ( )
−

G-MATHS 21

G-MATHS 22

Intersection between Line and Plane

a) Intersect(<Object>,<Object>)

Yields the intersection points of two objects.

Example

Mathematics: Find the intersection point between a line = + ( + +

) and a plane + − = .

: : + − =
: = ( , , ) + ∗ ( , , )
( , )

: ( , , )

G-MATHS 23

Angles
a) Angle(<Vector 1>, <Vector 2>)

Returns the angle between two vectors depending on the default angle unit.

Mathematics: Find the angle between = 2 + 2 + 4 and = 5 + 6 + 7 .
:

(( , , ), ( , , ))
: { . °}

b) Angle (<Line 1>, <Line 2>)

Returns the angle between direction vectors of two lines depending on the default angle
unit.

Mathematics:
Line A passes through the points ( , , ) and ( , , ) and line B
connecting points ( , , ) and ( , , ). Find the angle between the
line A and line B.

:
( (( , , ), ( , , )), (( , , ), ( , , )))

: { . °}

c) Angle(<Line>, <Plane>)

Returns the angle between the line and the plane.

Mathematics:
Find the angle between the line = ( ) + ( + + ) and the plane +
− = .
:
= ( , , ) + ∗ ( , , )
: + − =
( , )
: { . °}

G-MATHS 24

d) Angle(<Plane>, <Plane>)
Returns the angle between planes.
Mathematics:
Find the angle between the plane + − = and the plane + − =
.
:
( + − = , + − = )
: { . °}

References

1. https://www.geogebra.org/m/XUv5mXTm#material/seuyvj5u
2. Piecewise functions https://www.youtube.com/watch?v=cg5DVoYEDxE

G-MATHS 25

Appendix

No. Mathematics GeoGebra Keyboard
Notation Command
+
1+ -
*
2− /
+−
3 <Alt>-<>>
<Alt>-<<>
4÷
<Alt>-<A>
5± <Alt>-<B>
<Alt>-<G>
6≥ >= <Alt>-<T>
<Alt>-<L>
7≤ <= <Alt>-<M>
<Alt>-<P>
8≠ /= <Alt>-<U>
<Alt>-<E>
9 <Alt>-<O>
^
10 <Alt>-<R><x>

11 ^ − 1
∗
12 < >< >< ℎ >< 8 >< >

13

14

15

16 ∞

17

18 °

19

20 √ ( )
21 | | ( )

22 log (a, b)

23 + +

24 −1 ( )

25 . ( , )

26 ( , )

G-MATHS 26