Find the vertex,focus,and directrix of the parabola. $y = - 3 x ^ { 2 }$
A)Vertex: (0,0);Focus: $\left( 0 , - \frac { 1 } { 12 } \right)$ ;Directrix: $y = - \frac { 1 } { 12 }$
B)Vertex:(0,0);Focus: $\left( 0 , - \frac { 1 } { 12 } \right)$ ;Directrix: $y = \frac { 1 } { 12 }$
C)Vertex: (0,0);Focus: $\left( 0 , \frac { 1 } { 3 } \right)$ ;Directrix: $y = \frac { 1 } { 3 }$
D)Vertex: (0,0);Focus: $\left( 0 , \frac { 1 } { 12 } \right)$ ;Directrix: $y = \frac { 1 } { 12 }$
E)Vertex: (0,0);Focus: $\left( 0 , - \frac { 1 } { 3 } \right)$ ;Directrix: $y = \frac { 1 } { 3 }$

Question

Find the vertex,focus,and directrix of the parabola.​ $y = - 3 x ^ { 2 }$ ​
A)Vertex: (0,0);Focus: $\left( 0 , - \frac { 1 } { 12 } \right)$ ;Directrix: $y = - \frac { 1 } { 12 }$
B)Vertex:(0,0);Focus: $\left( 0 , - \frac { 1 } { 12 } \right)$ ;Directrix: $y = \frac { 1 } { 12 }$
C)Vertex: (0,0);Focus: $\left( 0 , \frac { 1 } { 3 } \right)$ ;Directrix: $y = \frac { 1 } { 3 }$
D)Vertex: (0,0);Focus: $\left( 0 , \frac { 1 } { 12 } \right)$ ;Directrix: $y = \frac { 1 } { 12 }$
E)Vertex: (0,0);Focus: $\left( 0 , - \frac { 1 } { 3 } \right)$ ;Directrix: $y = \frac { 1 } { 3 }$

Quizplus · Accepted Answer

The vertex of the parabola $y = -3x^2$ is at the origin (0,0) because there is no horizontal or vertical shift in the equation. The focus and directrix of a parabola are determined by the formula $4p(y-k) = (x-h)^2$ for a parabola that opens up or down, where (h,k) is the vertex and p is the distance from the vertex to the focus (if p is positive, the parabola opens upwards; if p is negative, it opens downwards). Here, comparing $y = -3x^2$ to the standard form, we find $4p = -3$, so $p = -\frac{3}{4}$. Therefore, the focus is $\left(0, -\frac{1}{4}\right)$, and the directrix is $y = \frac{1}{4}$. However, there was a mistake in my calculation of p; correctly, $p = -\frac{1}{4}$, leading to the focus being $\left(0, -\frac{1}{12}\right)$ and the directrix being $y = \frac{1}{12}$, which matches choice B.

Find the Vertex,focus,and Directrix of the Parabola y=−3x2y = - 3 x ^ { 2 }y=−3x2

Find the Vertex,focus,and Directrix of the Parabola $y = - 3 x ^ { 2 }$