Question

question 6 of 10
what is the equation of the parabola, in vertex form, with vertex at (-2,-4) and directrix x = -6?
a. (x + 4)^2 = 8(y + 2)
b. (y + 4)^2 = - 8(x + 6)
c. (x + 2)^2 = 8(y + 4)
d. (y + 4)^2 = 16(x + 2)

Explanation:

Step1: Determine the orientation

Since the directrix is a vertical line $x = - 6$, the parabola opens horizontally. The vertex - form of a horizontal parabola is $(y - k)^2=4p(x - h)$, where $(h,k)$ is the vertex. Here, $(h,k)=(-2,-4)$.

Step2: Calculate the value of $p$

The distance between the vertex $(h,k)=(-2,-4)$ and the directrix $x=-6$ is given by $|h - x_{directrix}|$. So, $p=| - 2-(-6)|=4$.

Step3: Write the equation

Substitute $h=-2$, $k = - 4$, and $p = 4$ into the vertex - form $(y - k)^2=4p(x - h)$. We get $(y+4)^2=4\times4(x + 2)$, which simplifies to $(y + 4)^2=16(x + 2)$.

Answer:

D. $(y + 4)^2=16(x + 2)$