# Applications of GeoGebra

GeoGebra is a dynamic and interactive mathematics software for geometry, algebra, calculus, trigonometry and statistics. Tools in GeoGebra are helpful in various constructions and calculations. Entry of equations and mapping of various variables can be done using the tools, input bar, CAS and spreadsheet views. Interactive explorations can be done using the tools in 2D and 3D Graphics modes. GeoGebra is a very useful tool to learn and teach different branches of mathematics. GeoGebra desktop application are available for Windows, Mac OS and Linux and tablet appls are available for Android, iPad and Windows. Its web app is based on HTML5 technology. GeoGebra was created by Markus Hohenwarter and started as part of his master’s thesis at the University of Salzburg, continuing at Florida Atlantic University, Florida State University, and then at the University of Linz with the help of open-source developers and translators all over the world. Bernard Parisses’ Giac was integrated into GeoGebra’s CAS view in 2013. Both commercial and not-for-profit entities work together to expand the software and cloud services for users.

Contributors, Content Editors and Reviewers The Spoken Tutorial Effort for Applications for GeoGebra is being contributed by Dr. Vidhya Iyer from IIT Bombay. Ms. Madhuri Ganapathi from IIT Bombay created two tutorials and prepared the outline along with Ms. Shruti Arya and Mr. Kiran Eranki. Other contributors include Ms. Sandhya Punekar and Ms. Madhuri Ganapathi, who contributed as domain reviewers, Ms. Minal Sathaye, who was the novice and Ms. Nancy Varkey, the administrative reviewer.

**Learners**: UG/PG students (12th standard and above) as well as Research Scholars.

## Contents

## Basic Level

- Vectors and Matrices
- Define a vector
- Change the Font Size using Options menu
- Magnitude and direction of a vector
- Relation between vectors and a parallelogram
- Arithmetic operations on vectors
- To create a matrix
- Arithmetic operations on matrices
- Transpose of a matrix
- Determinant of a matrix
- Inverse of a matrix

- Introduction to Trigonometry using GeoGebra
- Construct a circle of variable radius
- Construct a right triangle inside a unit circle
- Create a slider to change angle in right triangle
- Change properties (labels, colors and styles) of the right triangle
- Change x axis values to radians

- Trigonometric Ratios and Graphs
- The concept of a unit circle to find trigonometric ratios
- Conversion of degrees into radians to look at periodicity of functions
- Creation of a slider to change angle alpha to look at trigonometric ratios
- Changing the appearance of graphs
- Sine function
- Cosine function
- Tangent function
- Effects of rotation of the point around the circle on periodicity of functions
- Looking at co-ordinates of points tracing graphs of above trigonometric functions
- Corresponding graphs for sine, cosine and tangent functions

- Inverse Trigonometric Functions
- Creation of a slider to change angle alpha to look at trigonometric ratios
- Sine function
- Cosine function
- Tangent function
- Inverse sine function
- Inverse cosine function
- Inverse tangent function
- Fixing the domain to look at trigonometric and inverse trigonometric functions
- Co-ordinates of points tracing graphs of trigonometric and inverse trigonometric functions
- Creation of check boxes to show or hide function

- Roots of Polynomials
- Binomial theorem and polynomials
- Quadratic polynomials: real roots and complex roots
- Finding roots and determinants
- Quadratic functions: parabolic graphs, extremum
- Complex numbers in XY plane
- Complex numbers in complex plane
- Cubic polynomials: roots and extrema
- Point of inflection
- Plotting polynomials using input bar in GeoGebra
- Plotting polynomials in Computer Algebra system (CAS) in GeoGebra

- Complex Roots of Quadratic Equations
- Complex numbers in XY plane
- Complex numbers in complex plane
- Quadratic polynomials and parabolic functions
- Using sliders to look at different quadratic equations in GeoGebra
- Effects of quadratic equation coefficients on parabolic functions
- Finding roots using coefficients in quadratic equations
- Finding vertex (extremum) from formula for roots of quadratic equations
- Real roots of a quadratic polynomial using Spreadsheet view in GeoGebra
- Complex roots of a quadratic polynomial using Spreadsheet view in GeoGebra
- Changes in determinants for real and complex roots of quadratic equations

- Conic Sections-Parabola
- Definition, parts and properties of a parabola
- Construction of a parabola in GeoGebra
- Changing the appearance of a parabola in GeoGebra
- The standard equation of a parabola, (x-a)
^{2}= 4p (y-b) in GeoGebra - The standard equation of a parabola, (y-l)
^{2}= 4p (x-m) in GeoGebra - Differentiating the two forms of standard equations in terms of direction of axes of symmetry
- Creation of sliders to change coefficients in standard equations in GeoGebra
- The effects of the coefficients in standard equations on the parabola in GeoGebra
- Finding foci, vertex and equations of the axis of symmetry and directrix of a parabola
- Calculating length of latus rectum of a parabola

- Conic Sections-Ellipse
- Definition, parts and properties of an ellipse
- Construction of an ellipse in GeoGebra by adapting the Arcs of Circle method
- Changing the appearance of an ellipse in GeoGebra
- The standard equation of an ellipse, (x-h)
^{2}/a^{2}+(y-k)^{2}/b^{2}=1, in GeoGebra - The standard equation of an ellipse, (x-p)
^{2}/b^{2}+(y-q)^{2}/a^{2}=1, in GeoGebra - Differentiating the two forms of standard equations in terms of direction of major and minor axes
- Creation of sliders to change coefficients in standard equations in GeoGebra
- The effects of the coefficients in standard equations on the ellipse in GeoGebra
- Finding foci, vertices, co-vertices, center and eccentricity of an ellipse
- Calculating length of major and minor axes, and latus recti of an ellipse

- Conic Sections-Hyperbola
- Definition, parts and properties of a hyperbola
- Construction of a hyperbola in GeoGebra
- Changing the appearance of a hyperbola in GeoGebra
- The standard equation of a hyperbola, (x-h)
^{2}/a^{2}-(y-k)^{2}/b^{2}=1, in GeoGebra - The standard equation of a hyperbola, (x-p)
^{2}/b^{2}-(y-q)^{2}/a^{2}=1, in GeoGebra - Differentiating the two forms of standard equations in terms of direction of transverse and conjugate axes
- Creation of sliders to change coefficients in standard equations in GeoGebra
- The effects of the coefficients in standard equations on the hyperbola in GeoGebra
- Finding foci, vertices, center and eccentricity of a hyperbola
- Calculating length of transverse and conjugate axes, and latus recti of a hyperbola

## Intermediate Level

- 3D Geometry
- The rectangular co-ordinate system
- Drawing a line in 3D
- Drawing a plane in 3D
- Drawing a sphere in 3D
- Drawing a pyramid in 3D
- Drawing a double-napped cone in 3D
- Visualizing intersection of line with a plane and sphere in 3D
- Visualizing intersection of a double-napped cone with a plane in 3D as a parabola
- Visualizing the solid obtained by rotation of a polynomial about the x axis
- Trigonometric functions in 3D

- Limits and Continuity of Functions
- The concept of limits
- Evaluation of functions
- Left hand limit
- Right hand limit
- Limits that DNE
- Using a spreadsheet to look at limits
- Limits at infinity
- Limits of rational polynomial functions
- Continuous functions, their limits
- Discontinuous functions, their limits

- Differentiation using GeoGebra
- Differentiation using first principles
- Differentiation to find maxima of a function
- Differentiation to find minima of a function
- Derivative of a trigonometric function
- Addition rule of differentiation
- Subtraction rule of differentiation
- Product rule of differentiation
- Chain rule of differentiation
- Quotient rule of differentiation
- Practical application of differentiation

- Integration using GeoGebra
- Indefinite integrals
- Definite integrals
- Concept of area under curve (AUC)
- Upper, lower Riemann and trapezoidal sums to estimate AUC
- Integration as an estimation of AUC
- Relationship between number of rectangles/trapezoids and AUC
- Double integrals as area between two functions
- Using input bar to find area between two functions in GeoGebra
- Using CAS to find area between two functions in GeoGebra
- Integration as anti-differentiation

- Statistics using GeoGebra
- Pasting data into Spreadsheet in GeoGebra
- Measures of central tendency: arithmetic mean, median and mode
- Coefficient of variation to compare data series
- Measures of dispersion: range, quartiles, mean and standard deviation
- Box plot = 5 member summary: minimum and maximum values, first and third quartiles, median
- One Variable Analysis—bar chart, box plot, histogram representation
- Least squares linear regression (LSLR)—coefficient of determination R
^{2}, regression coefficient - Two Variable Regression Analysis: Scatterplot and residual plot
- Multiple Variable Analysis tool
- Stacked box plots

- Probability and Distributions using GeoGebra
- Pasting data into Spreadsheet in GeoGebra
- Hypothesis testing: null and alternative hypotheses
- z-, T- and F- tests
- Multiple Variable Analysis tool
- Stacked box plots
- Analysis of Variance (ANOVA)
- T-tests: Unpaired and Paired
- Probability Calculator tool in GeoGebra
- Probability
- Distributions

Contributors and Content Editors Vidhya Iyer, Madhuri Ganapathi, Sandhya Punekar