Write the equation in vertex form for the parabola with focus \( (0,10) \) and directrix \( y=6 \). Simplify any fractions.

To find the equation of the parabola in vertex form given the focus \( (0,10) \) and the directrix \( y=6 \), follow these steps: 1. **Determine the Vertex:** - The vertex lies exactly midway between the focus and the directrix. - The y-coordinate of the vertex is \( \frac{10 + 6}{2} = 8 \). - Thus, the vertex is at \( (0, 8) \). 2. **Find the Value of \( p \):** - \( p \) is the distance from the vertex to the focus. - \( p = 10 - 8 = 2 \). 3. **Determine the Orientation:** - Since the focus is above the directrix, the parabola opens **upward**. 4. **Use the Vertex Form Equation:** - The vertex form of a parabola that opens upward is: \[ y = a(x - h)^2 + k \] where \( (h, k) \) is the vertex. - Here, \( h = 0 \) and \( k = 8 \), so: \[ y = a x^2 + 8 \] 5. **Calculate \( a \):** - The value of \( a \) is related to \( p \) by the formula \( a = \frac{1}{4p} \). - Substituting \( p = 2 \): \[ a = \frac{1}{4 \times 2} = \frac{1}{8} \] 6. **Write the Final Equation:** - Substitute \( a \) back into the vertex form: \[ y = \frac{1}{8}x^2 + 8 \] **Final Answer:** \[ y = \frac{1}{8}\,x^{2} + 8 \]