## Friday, December 13, 2013

### Pythagorean Theorem

Irina Boyadzhiev, 13 December 2013, Created with GeoGebra

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Labels:
Pythagorean theorem

## Thursday, October 3, 2013

### 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.

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Irina Boyadzhiev's GeoGebra Applets

- In the input box enter a polynomial equation of
*x*and*y*in the form "**polynomial = number**".

For example, to graph*y = x*, enter^{2}*y - x*^{2}= 0 - Drag the slider to use the vertical line test.

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Labels:
function,
is a function,
is not a function,
Vertical Line Test

## Thursday, September 26, 2013

### Inverse Function

This applet can be used to study inverse functions.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

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- 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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Labels:
function,
inverse functions,
one-to-one

## Friday, September 20, 2013

### 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”.

- 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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Labels:
horizontal line test,
inverse functions,
one-to-one

## Thursday, May 2, 2013

### 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

Think of each cube as a collection of

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*natural numbers.***n**- Drag the slider “NumberCubes” to create up to seven cubes.

Think of each cube as a collection of

*x***n n***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.***n**- Drag the slider “Rearrange” or press the “animate” button.

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## Monday, April 15, 2013

### Octagon to Square Dissection

The applet below shows the octagon to square dissection.

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

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

## Monday, April 8, 2013

### 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

Irina Boyadzhiev, 6 April 2013, Created with GeoGebra

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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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Irina Boyadzhiev's GeoGebra Applets

Labels:
algebra errors,
Square root,
square root of sum

## Thursday, March 7, 2013

### 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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Irina Boyadzhiev, 6 March 2013, Created with GeoGebra

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**See the general case - Arithmetic Series.**

Irina's GeoGebra Applets

Labels:
natural numbers,
sum,
sum of natural numbers

## Sunday, March 3, 2013

### 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.

Irina Boyadzhiev, 13 February 2013, Created with GeoGebra

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Irina's GeoGebra Applets

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

- Click on Show the formulas to see the result.

Irina Boyadzhiev, 13 February 2013, Created with GeoGebra

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Irina's GeoGebra Applets

Labels:
natural numbers,
series,
squares,
sum,
sum of squares

## Wednesday, January 23, 2013

### 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.

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Irina Boyadzhiev - GeoGebra Applets

- 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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## Sunday, January 13, 2013

### Domain and Range of a Function

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

Irina Boyadzhiev, 12 January 2013, Created with GeoGebra

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Irina Boyadzhiev - GeoGebra Applets

- 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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## Wednesday, January 2, 2013

### 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 V*ector between Two Points*tool is used to translate objects and create a tessellation.
The Step by Step instructions on how to create the applet above can be found

**here**.
Labels:
semiregular tessellation,
tessellation

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