Find the standard form of the equation of the parabola with the given characteristics. $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$
A) $( y - 1 ) ^ { 2 } = 3 ( x - 3 )$
B) $( x - 1 ) ^ { 2 } = - 20 ( y - 3 )$
C) $( y - 1 ) ^ { 2 } = - 20 ( x - 3 )$
D) $( x - 1 ) ^ { 2 } = 3 ( y - 3 )$
E) $( x - 1 ) ^ { 2 } = 20 ( y - 3 )$

Question

Quizplus · Accepted Answer

The standard form of a parabola that opens upwards or downwards is $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus or directrix. Given the vertex $(1, 3)$ and the directrix $y = -2$, the focus is above the directrix since parabolas open away from the directrix. The distance from the vertex to the directrix is $3 - (-2) = 5$, so $p = 5$. Substituting $h = 1$, $k = 3$, and $p = 5$ into the formula gives $(x - 1)^2 = 20(y - 3)$, which matches option E.

Find the Standard Form of the Equation of the Parabola Vertex: (1,3); directrix: y=−2\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2 Vertex: (1,3); directrix: y=−2

Find the Standard Form of the Equation of the Parabola $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$