1. **Problem:** Find the equation of the parabola with directrix $y = -\frac{1}{2}$ and vertex at $(0,0)$. 2. **Formula and rules:** The vertex form of a parabola with vertical axis of symmetry is $$ (x - h)^2 = 4p(y - k) $$ where $(h,k)$ is the vertex and $p$ is the distance from the vertex to the focus (or directrix). The directrix is $y = k - p$. 3. **Identify values:** Here, vertex $(h,k) = (0,0)$ and directrix $y = -\frac{1}{2}$. So, $$ k - p = -\frac{1}{2} \implies 0 - p = -\frac{1}{2} \implies p = \frac{1}{2} $$ 4. **Write equation:** Substitute $h=0$, $k=0$, and $p=\frac{1}{2}$ into the formula: $$ x^2 = 4 \times \frac{1}{2} \times y = 2y $$ 5. **Interpretation:** This parabola opens upwards (since $p>0$) with vertex at the origin and directrix below the vertex at $y = -\frac{1}{2}$. **Final answer:** $$ x^2 = 2y $$