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Build & Fit - Parabola

A parabola can be defined symbolically by parameters a,b or by location of the vertex and the focus (and therefore the directrix). What are the relations between these representations? In the left panel you can define a blue parabola by varying the values of the a and b sliders. In the right panel you define the green parabola by adjusting the location of the vertex and the focus. Now try to vary the green parabola to match the blue one. Alternatively, start by defining a green parabola in right panel, and see if you can adjust the a and b parameters in the left panel so that the blue parabola matches the green one. What questions could/would you pose to your students based on this applet ?