Determine the standard form of an equation of the parabola subject to the given conditions.
-Vertex: $( 0,0 )$; Directrix: $y = - 8$
A) $x ^ { 2 } = 32 y$
B) $y ^ { 2 } = 32 x$
C) $y ^ { 2 } = - 8 x$
D) $x ^ { 2 } = - 8 y$

Question

Quizplus · Accepted Answer

The vertex form of a parabola is $y = a(x - h)^2 + k$ or $x = a(y - k)^2 + h$, where $(h, k)$ is the vertex. Given the vertex is at the origin $(0,0)$ and the directrix is $y = -8$, the focus is at $y = 8$ (the same distance from the vertex as the directrix but in the opposite direction). This means the parabola opens upwards, and the standard form is $x^2 = 4py$, where $p = 8$. Substituting $p$ gives $x^2 = 32y$.

Determine the Standard Form of an Equation of the Parabola (0,0)( 0,0 )(0,0)

Determine the Standard Form of an Equation of the Parabola $( 0,0 )$