What equation represents the parabola?

The first option, x² = -8(y-2).

Explanation:

The first step is to find the vertex.  We define the vertex as the point on a parabola that is equidistant from the focus and directrix.  The focus is at (0,0) and the directix has the equation of y = 4, so we can easily find that the vertex is at (0,2).  Vertex form is y = a(x-h)²+k, where (h,k) is the vertex of the parabola, so we can plug in (0,2) to get y = a(x-0)²+2 ⇒ y = ax²+2.  Secondly, we know that a = , where p is defined as the distance (and direction) from the vertex to the focus.  We know that the vertex is at (0,2) and the focus is at (0,0), so the distance is 2.  Because the focus is below the vertex, though, the parabola must open downwards, and therefore, p = -2.  a = ⇒ .  Now, the equation is y = x²+2.  None of the answers are in this particular form, though.  The entire equation is multiplied by -8 to produce -8y = x²-16.  16 is added on both sides to get x² = -8y+16.  This is simplified to x² = -8(y-2), and we can see that the answer is the first option.

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