Question

derive the equation of a parabola with a focus at (-2, -3) and a directrix at y = 1. type your answers in the boxes. be sure to include the correct signs and operation symbols. y = □(x □)^2□

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus and the directrix. The x - coordinate of the vertex is the same as the x - coordinate of the focus, so $x=-2$. The y - coordinate of the vertex is $\frac{-3 + 1}{2}=\frac{-2}{2}=-1$. So the vertex is $(-2,-1)$.

Step2: Calculate the value of $p$

The distance $p$ from the vertex to the focus (or from the vertex to the directrix) is the absolute value of the difference in their y - coordinates. Since the vertex is $(-2,-1)$ and the focus is $(-2,-3)$, $p=-1-(-3)=2$ (or the distance from the vertex $(-2,-1)$ to the directrix $y = 1$ is $1-(-1)=2$). For a parabola opening downwards (since the focus is below the directrix), $p=-2$.

Step3: Use the vertex - form of the parabola equation

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex and $a=\frac{1}{4p}$. Substituting $p = - 2$ into $a=\frac{1}{4p}$, we get $a=-\frac{1}{8}$. And $h=-2,k=-1$. So the equation of the parabola is $y=-\frac{1}{8}(x + 2)^2-1$.

Answer:

$y=-\frac{1}{8}(x + 2)^2-1$