1. **State the problem:** We are given a parabola with a focus at $ (3, 6) $ and a directrix $ y = -4 $. We need to find the equation of the parabola and describe its properties. 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 given focus and directrix:** If the focus is $ (h, k) $ and the directrix is $ y = d $, the parabola opens vertically and its equation is derived from the distance formula: $$ \sqrt{(x - h)^2 + (y - k)^2} = |y - d| $$ 4. **Apply the given values:** Here, $ h = 3 $, $ k = 6 $, and $ d = -4 $. 5. **Square both sides to remove the square root:** $$ (x - 3)^2 + (y - 6)^2 = (y + 4)^2 $$ 6. **Expand both sides:** $$ (x - 3)^2 + y^2 - 12y + 36 = y^2 + 8y + 16 $$ 7. **Simplify by canceling $ y^2 $ on both sides:** $$ (x - 3)^2 - 12y + 36 = 8y + 16 $$ 8. **Bring all terms involving $ y $ to one side:** $$ (x - 3)^2 + 36 - 16 = 8y + 12y $$ 9. **Simplify constants and combine $ y $ terms:** $$ (x - 3)^2 + 20 = 20y $$ 10. **Isolate $ y $:** $$ y = \frac{(x - 3)^2 + 20}{20} = \frac{(x - 3)^2}{20} + 1 $$ 11. **Interpretation:** The parabola opens upward (since the coefficient of $ (x - 3)^2 $ is positive), has vertex at $ (3, 1) $ (midpoint between focus and directrix), and the distance from vertex to focus (or vertex to directrix) is $ p = 5 $ (since $ 4p = 20 $). **Final answer:** $$ y = \frac{(x - 3)^2}{20} + 1 $$ This parabola opens upward with vertex at $ (3, 1) $, focus at $ (3, 6) $, and directrix $ y = -4 $.