The activity that is presented here offers an opportunity to explore the properties of parabolas through a geometric perspective within your browser window.

There are five areas to explore within this activity:

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Directions for Creating and Simulating a Parabola in the JavaWindow:

1. Create a straight line across the bottom of the window named the directrix.
2. Construct a point on this directrix that can be moved about the line. Try moving it and notice that the line does not change direction or location but the point will move along this directrix.
3. Construct a point above the directrix anywhere in the window. This will be our focus. Now we need to construct a perpendicular bisector of a segment connecting our focus to the point on the directrix.
4. We can trace this line (click the appropriate button in the window). Now drag the point on the directrix back and forth and watch the formation that appears. This appears to be a parabola. Clear the screen using the red X in the bottom right of the JavaWindow. Now move the focus point further or closer from the directrix (click the red X again to clear the traced lines) and drag the point on the directrix again or press the show animate button-button and try the animation.
5. We recall the definition of a parabola and we try to show that this is indeed a construction of a parabola. Click the button that creates the locus point to represent all the points of the locus. Drag or animate the sketch again and notice where that point is. Click the Show measurements and Construct trace button. Notice that the distance from the focus to the point of the locus is equal to the distance from the locus point to the point on the directrix. (The measurements should be equal throughout the animation or movement). Check it for a few points. This is an important part of the definition.
6. Hide the traced line by clicking the necessary button and then hide all the buttons but the animate button. Animate the sketch and watch the sketch trace out a locus of points and create a parabola.
7. Try to formulate a proof that this sketch is indeed a parabola.

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Constructing A Parabola (Java Window)

 Sorry, this page requires a Java-compatible web browser.
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Tips and Tricks for Working with JavaSketchpad

• Remember that clicking the red X clears all the traces
• Try pressing return in the location bar at the top of your browser to reload the entire window (clear it all)
• Dragging the red points will result in changing the picture that the window shows you. Experiment with moving lines and points.
• Try to maximize the size of your browser window to see the entire picture
• The first time you load this page may take a few moments but wait and it will work fine.
• Clicking the Animate button once starts the animation. Clicking it again stops the animation.

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Questions and Explorations

• What does the perpendicular bisector represent when this activity is compared to the paper folding activity?
• Why is the distance from the directrix to the locus point measured using a perpendicular line?
• What happens to the parabola when the focus is further away from the directrix? What about closer to the directrix?
• What do the traced lines represent in relation to the parabola that is formed?
• What does the point on the directrix represent when it is moved back and forth?
• How can you construct a proof that this sketch does indeed represent a parabola? (Hint look at the triangles formed from the vertices of the focus, locus point, and point on the directrix)

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Definitions and Assistants

• locus of points is a collection of points or other objects that satisfies a particular requirement.
• directrix is a fixed line that serves as a guide in creating our parabola
• focus is the point used to determine the parabola's openness and distance from the directrix.
• parabola is a collection of points (a locus) such that the moving locus point is always equidistant from the focus and the directrix.