Find the vertex, focus, and directrix of the parabola with the given equation.
-$$( x + 1 ) ^ { 2 } = 12 ( y - 4 )$$
A) vertex: $( 1 , - 4 )$
focus: $( 1 , - 1 )$
$\text { directrix: } y=-7$

B) vertex: $( 4 , - 1 )$
focus: $( 4,2 )$
$\text { directrix: } y=-7$

C) vertex: $( - 1,4 )$
focus: $( - 1,1 )$
$\text { directrix: } y=-7$

D) vertex: $( - 1,4 )$
focus: $( - 1,7 )$
$\text { directrix: } y=-7$

Question

Find the vertex, focus, and directrix of the parabola with the given equation.
-$$( x + 1 ) ^ { 2 } = 12 ( y - 4 )$$
A) vertex: $( 1 , - 4 )$
focus: $( 1 , - 1 )$
$\text { directrix: } y=-7$

B) vertex: $( 4 , - 1 )$
focus: $( 4,2 )$
$\text { directrix: } y=-7$

C) vertex: $( - 1,4 )$
focus: $( - 1,1 )$
$\text { directrix: } y=-7$

D) vertex: $( - 1,4 )$
focus: $( - 1,7 )$
$\text { directrix: } y=-7$

Quizplus · Accepted Answer

The given equation is in the form $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex. Comparing, we find the vertex is $(-1, 4)$. The value of $4p = 12$, so $p = 3$. The focus is $p$ units above the vertex since the parabola opens upwards, making the focus $(-1, 4 + 3) = (-1, 7)$. The directrix is a line $p$ units below the vertex, which is $y = 4 - 3 = 1$. However, there's a mistake in my explanation regarding the directrix; it should be $y = 4 - 3 = 1$, but the correct answer choice indicates the directrix as $y = -7$, which doesn't align with the calculation. The correct directrix, based on the calculation, should be $y = 1$, indicating a mistake in my explanation. The correct elements based on the calculation are the vertex $(-1, 4)$ and the focus $(-1, 7)$, but the directrix provided in the options is incorrect based on the calculation.

Find the Vertex, Focus, and Directrix of the Parabola with the Given