1. **Problem:** Find the equation of the parabola given the focus at (0, 2) and vertex at (0, 4). 2. **Formula and rules:** The vertex form of a parabola with vertical axis is $$ (x - h)^2 = 4p(y - k) $$ where $(h,k)$ is the vertex and $p$ is the distance from vertex to focus (positive if focus is above vertex, negative if below). 3. **Identify values:** Vertex $(h,k) = (0,4)$, focus $(0,2)$. 4. **Calculate $p$:** Distance from vertex to focus along y-axis is $p = 2 - 4 = -2$ (negative means parabola opens downward). 5. **Write equation:** Substitute into vertex form: $$ (x - 0)^2 = 4(-2)(y - 4) $$ $$ x^2 = -8(y - 4) $$ $$ x^2 = -8y + 32 $$ 6. **Rewrite:** $$ x^2 - 8y = 32 $$ 7. **Answer:** The equation of the parabola is $$ x^2 - 8y = 32 $$ which corresponds to choice (d).