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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