Deriving the Equation of the Parabola

Instructions: Move point A to adjust the position of the vertical line and move F along the x-axis as desired. Move C along the line and observe what happens.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Questions
1. What can you say about line a in relation to C and F?
2. Make a conjecture about segments PC and PF.
3.) Suppose P has coordinates (x,y), F has coordinates (2,0), and line d has equation x = 2, write the equation that would describe your conjecture in 2.
4.) Generalize the equation in 3 by changing the coordinates of F to (a,0) and line d has equation x = -d. Find the value of y.

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The Parabola in Vertex Form

1. Move sliders a, h, and k. What do you observe? 
2. How are a, h, and k relates to the graph's movement? 
3. Make a generalization about the relationship between the parameters a, h, and k and the movement of the appearance of the graph.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

The parabola has vertex form .

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Multiplication in Parabola

Move the points of on the parabola and observe the coordinates of the points on the axes.
What conjecture can you make from your observation?


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


Exercise: Prove your conjecture above.