1. **State the problem:** We need to find the equation of a parabola given its focus at $(0, -5)$ and its directrix $y = 11$. 2. **Recall the definition of a parabola:** A parabola is the set of all points equidistant from the focus and the directrix. 3. **Formula for a parabola with vertical axis:** If the focus is at $(h, k)$ and the directrix is $y = d$, the parabola's equation is given by: $$ (x - h)^2 = 4p(y - m) $$ where $p$ is the distance from the vertex to the focus (or directrix), and the vertex $(h, m)$ lies midway between the focus and directrix. 4. **Find the vertex:** The vertex's $y$-coordinate is the midpoint between the focus $y$-coordinate $-5$ and directrix $y$-coordinate $11$: $$ m = \frac{-5 + 11}{2} = \frac{6}{2} = 3 $$ The vertex is at $(0, 3)$ since the focus $x$-coordinate is $0$. 5. **Calculate $p$:** $p$ is the distance from the vertex to the focus: $$ p = k - m = -5 - 3 = -8 $$ The negative sign indicates the parabola opens downward. 6. **Write the equation:** Using the formula: $$ (x - 0)^2 = 4 \times (-8) (y - 3) $$ which simplifies to: $$ x^2 = -32(y - 3) $$ 7. **Final answer:** The equation of the parabola is: $$ \boxed{x^2 = -32(y - 3)} $$