Find the standard form of the parabola with the given characteristics.
directrix: $x = - 1$
vertex: $( 7 , - 1 )$
A) $( x + 7 ) ^ { 2 } = 32 ( y - 1 )$
B) $( x - 7 ) ^ { 2 } = 32 ( y + 1 )$
C) $\quad ( x - 1 ) ^ { 2 } = 32 ( y + 7 )$
D) $( y - 1 ) ^ { 2 } = 32 ( x + 7 )$
E) $( y + 1 ) ^ { 2 } = 32 ( x - 7 )$

Question

Quizplus · Accepted Answer

The standard form of a parabola that opens horizontally is $(y - k)^2 = 4p(x - h)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus or directrix. Given the directrix $x = -1$ and the vertex $(7, -1)$, the parabola opens to the right (since the directrix is to the left of the vertex), and $p = 7 - (-1) = 8$. Thus, $4p = 32$, and the equation becomes $(y + 1)^2 = 32(x - 7)$, matching choice E.

Find the Standard Form of the Parabola with the Given x=−1x = - 1x=−1

Find the Standard Form of the Parabola with the Given $x = - 1$