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**Exploring
Parabolas (JavaSketchpad)
**

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:

- Directions and Recommendations
- Tips and Tricks working with JavaSketchpad
- JavaWindow and Simulation
- Questions and Explorations
- Definitions and Assistants
- Download GSP File
- 3 Dimensional Representation

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**Directions for Creating and Simulating a
Parabola in the JavaWindow:**

- Create a straight line across the bottom of the window named
the
**directrix****.** - 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.** - 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**. - 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. - 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. - 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.** - Try to formulate a
**proof**that this sketch is indeed a**parabola.**

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**Constructing A Parabola (Java
Window)**

**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.

If you have an other comments please contact me

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