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An interactive model designed for students studying parabolas in algebra.


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Translate up or down, and open and close the parabola by dragging the buttons.


Two equations for a parabola

The code uses two forms of the equation of a curve. The first is the standard equation:

  • y = a x2 + bx + c

The second is the vertex form:

  • y = a (x - h)2 + k

To translate between these two equations we need write the equations to translate the coefficients:

  • To go from vertex form to standard form:
    • a = a
    • b = -2ah
    • c = ah2 + k
  • To go from standard form to vertex form:
    • a = a
    • h = -b/(2a)
    • k = c - b/(4a)


You can use the controls to slide the parabola all around the graph and squeeze or open the tightness of the parabola by changing the a coefficient. I chose to allow that control to move in a circular arc centered around the vertex of the parabola. This required determining the intersection point between the parabola and the half circle.

First off, to simplify the problem, let's act as though the parabola's vertex is at the origin (0,0), and therefore the center of the circle is also at (0,0). So we create the local coordinates:


  • For some reason if the a coefficient is too high the graph disappears.
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