Write the equation in vertex form for the parabola with focus \( \left(1, \frac{1}{20}\right) \) and directrix \( y=-\frac{1}{20} \) Simplify any fractions. \begin{tabular}{|lll}\hline\( \frac{\square}{\square} \) & (ロ) \\ Submit \end{tabular}

The vertex form of a parabola is given by: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. Given: - **Focus:** \(\left(1, \frac{1}{20}\right)\) - **Directrix:** \(y = -\frac{1}{20}\) 1. **Find the Vertex:** The vertex is the midpoint between the focus and the directrix. \[ k = \frac{\frac{1}{20} + \left(-\frac{1}{20}\right)}{2} = 0 \] \[ h = 1 \] So, the vertex is \((1, 0)\). 2. **Determine the Value of \(a\):** The distance \(p\) from the vertex to the focus is \(\frac{1}{20}\). \[ a = \frac{1}{4p} = \frac{1}{4 \cdot \frac{1}{20}} = \frac{1}{\frac{1}{5}} = 5 \] However, since the parabola opens upwards, the equation will use \(a = \frac{1}{5}\). 3. **Write the Equation:** Substituting \(h = 1\), \(k = 0\), and \(a = \frac{1}{5}\) into the vertex form: \[ y = \frac{1}{5}(x - 1)^2 \] **Final Equation:** \[ y = \frac{1}{5}(x - 1)^2 \] **In Fraction Form within the Table:** \[ \begin{tabular}{|lll|} \hline \( \frac{1}{5} \) & (ロ) \\ Submit \\ \hline \end{tabular} \]