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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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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Area of Triangles Under a Curve

Given: Right triangle BCD whose hypotenuse is tangent to the function f(x) = a/x. 

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1. Move slider a and observe what happens? How is the appearance of the graph relate to a?
2. Move point A and observe the triangle? What do you observe?
3. What conjecture can you make based on your observation in 2. Prove your conjecture. 

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