The moving circle below traces the parabola [math]x^2=8\left(y+2\right)[/math] through its center. The circle is required to pass through the origin and be tangent with the line [math]y=-4[/math] below. The distance of the circle's center to the origin and the its distance to the line are always equal to the radius of the circle.[br][br]The equality above satisfies the definition of a parabola in which any point in a parabola must be equidistant to a fixed point and a fixed line which are the focus and the directrix respectively. Changing the value of the slider [math]h[/math] allows one to relocate the center of the circle while still following the required conditions.[br][br]The parabolic path has a focus at [math]\left(0,0\right)[/math] and a directrix at [math]y=-4[/math] which indicates that the parabola should be opening upwards. This means that the standard form of this parabola is given by [math]\left(x-h\right)^2=4a\left(y-k\right)[/math] where [math]\left(h,k\right)[/math] is the location of the vertex and [math]a[/math] is the distance between the focus and the vertex.[br][br]By observation, the distance between the focus and vertex is equal to 2 which can be easily seen by letting [math]h[/math] be equal to 0. This means that [math]k=-2[/math] taken by subtracting [math]a[/math] to the [math]y[/math] coordinate of the focus. Substituting the know values gives the equation of the parabola above.