Question

the focus, (0, - 2), and directrix, x = 6, of a parabola are graphed below. which equation represents the parabola? x = 1/12(y + 2)^2+3 x = 1/12(y - 2)^2 - 3 x=-1/12(y - 2)^2 - 3 x=-1/12(y + 2)^2+3

Explanation:

Step1: Recall the formula for a parabola

The standard - form equation of a parabola with a horizontal axis of symmetry 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 from the vertex to the directrix). The vertex of a parabola is the mid - point between the focus $(x_f,y_f)$ and the point on the directrix that is closest to the focus. The focus is $(0, - 2)$ and the directrix is $x = 6$. The $y$ - coordinate of the vertex is the same as the $y$ - coordinate of the focus, so $k=-2$. The $x$ - coordinate of the vertex $h$ is $\frac{0 + 6}{2}=3$. So the vertex is $(3,-2)$.

Step2: Calculate the value of $p$

The distance $p$ from the vertex $(3,-2)$ to the focus $(0,-2)$ is $p=0 - 3=-3$.

Step3: Substitute into the formula

Substitute $h = 3$, $k=-2$, and $p=-3$ into the equation $(y - k)^2=4p(x - h)$. We get $(y+2)^2=4\times(-3)(x - 3)$, which simplifies to $x=-\frac{1}{12}(y + 2)^2+3$.

Answer:

$x=-\frac{1}{12}(y + 2)^2+3$