Pythagorean Theorem

Irina Boyadzhiev, 13 December 2013, Created with GeoGebra

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Vertical Line Test

This applet can be used to determine whether one relation is a function or not by using the Vertical Line Test. The Vertical Line Test says that if some vertilcal line meets the graph in more than one point then the relation is not a function.

•  In the input box enter a polynomial equation of x and y in the form "polynomial = number".
  For example, to graph y = x², enter y - x² = 0 
• Drag the slider to use the vertical line test.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

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Inverse Function

This applet can be used to study inverse functions.

• Enter a function of x in the input box f(x).
• Set the domain of the function by dragging the endpoints of the blue line at the bottom of the window.
• Click on the “Horizontal Line Test” and drag the vertical slider. Is f(x) a one-to-one function?
• Click on the “Reflect f(x)” button.
• Drag the horizontal slider to run the Vertical Line Test. Is the reflection a graph of a function?
• Drag the endpoints of the blue segment under the graphs to restrict the domain of f(x) in such a way that the reflected graph is a graph of a function.
• Hide all points created by the horizontal and vertical line tests by clicking the respective checkboxes.
• Click the “Show Corresponding Points”. Drag the orange point along the graph of f(x) and notice its reflection on the graph of the inverse function.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

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Horizontal Line Test

This applet can be used to determine whether a function is one-to-one or not, and also to restrict the domain to some interval where the function is one-to-one.

• Enter a function of x in the input box. 
• Drag the vertical slider up or down (or press the Play button) to find the intersecting points of the graph and the horizontal line. 
• To see the x-coordinates of the intersection points, check the “show coordinates”. 

If we want to define an inverse function then we have to find an interval where f(x) is one-to-one.

•  Drag the left and/or the right side of the blue segment under the graph to restrict the domain.

Use the second button on the toolbar to drag the viewing window, zoom in or zoom out if necessary.

Irina Boyadzhiev, 19 September 2013, Created with GeoGebra

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The Sum of the Cubes of the First N Natural Numbers Dynamic Geometric Proof

This applet shows a dynamic geometric proof of the formula for the sum of the cubes of the first n natural numbers.

• Drag the slider “NumberCubes” to create up to seven cubes.

Think of each cube as a collection of n nxn square layers of unit cubes. To find a formula for the sum of the cubes of the first natural numbers we will rearrange the square layers.

• Drag the slider “Rearrange” or press the “animate” button.

Irina Boyadzhiev, 29 April 2013, Created with GeoGebra

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Octagon to Square Dissection

The applet below shows the octagon to square dissection.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Reference:
"Bennett's Octagon-to-Square Dissection" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/BennettsOctagonToSquareDissection/ Contributed by: Izidor Hafner Based on work by: Greg N. Frederickson

Square Root of a Sum? - Geometric Interpretation

We know that for non-negative numbers  Can we apply the same for the sum? 
In the example below a and b are the legs of a right triangle.  According to the Pythagorean Theorem the hypotenuse is

• Drag to the left the blue point A to rotate the leg AC around point C until the two legs form one segment.
  
• Drag to the left the orange point A to rotate the hypotenuse around point B until it comes to the base of the triangle.
• Compare the lengths  

Irina Boyadzhiev, 6 April 2013, Created with GeoGebra

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The Sum of the First n Natural Numbers - Dynamic Geometric Proof

This applet gives a dynamic proof of the formula for the sum of the first n natural numbers.


Irina Boyadzhiev, 6 March 2013, Created with GeoGebra

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• See the general case - Arithmetic Series.

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The Sum of Squares - dynamic proof

This applet gives a  dynamic proof of the formula for the sum of the squares of the first n natural numbers.  We start with three times the sum of the squares and rearrange the parts of one of the sums.

• Click on "Color the third column".
• Drag the slider to rearrange the parts of the third column.

The area of the formed rectangle equals three times the sum of the squares.

• Click on  Show the formulas  to see the result.

Irina Boyadzhiev, 13 February 2013, Created with GeoGebra

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Domain and Range of a Function by Flattening to the Axis

This applet can be used to introduce the concept of Domain and Range of a function by flattening the graph of the function over the coordinate axes.

• Click on a button to select one of the three functions.
• Drag the "domain" slider. The graph of the function is flattened to the x-axis. The trace that is left on the x-axis is the domain of the function.
• Drag the "range" slider. The graph of the function is flattened to the y-axis.  The trace left on the y-axis is the range of the function. 

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Domain and Range of a Function

This applet can be used to demonstrate the concept of Domain and Range of a function.

• Click on a button to select one of the three functions.
• Drag the "domain" slider. A vertical ray will scan the viewing window and will project the intersection point of the ray and  the function to the x-axis. The domain appears as red interval(s) on the x-axis.
• Drag the "range" slider. A horizontal ray will scan the viewing window and will project its intersection  points with the function to the y-Axis. The range appears as green interval(s) on the y-axis.

Irina Boyadzhiev, 12 January 2013, Created with GeoGebra
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Square Octagon Tessellation

One of the objectives of GeoGebra Applet Central this year is to update the output files of my GeoGebra Tutorial Series. The previous outputs were still in version 3.2. The applet below is the output of GeoGebra Tutorial 10 - Vectors and Tessellation. I just changed the color of the vectors to make it more visible. In this tutorial, the Vector between Two Points tool is used to translate objects and create a tessellation.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com The Step by Step instructions on how to create the applet above can be found here.