Write the equation in vertex form for the parabola with focus \( (0,-7) \) and directrix \( y=9 \) Simplify any fractions. \( \begin{array}{lll}\text { Submit }\end{array} \)

To find the vertex form of the parabola with focus \( (0, -7) \) and directrix \( y = 9 \), follow these steps: 1. **Determine the Vertex:** - The vertex lies halfway between the focus and the directrix. - The distance between the focus \( y = -7 \) and the directrix \( y = 9 \) is \( 9 - (-7) = 16 \). - Therefore, the vertex is \( 8 \) units above the focus: \( y = -7 + 8 = 1 \). - **Vertex:** \( (0, 1) \). 2. **Determine the Orientation and \( p \):** - Since the directrix is above the focus, the parabola opens downward. - The value of \( p \) (distance from the vertex to the focus) is \( -8 \). 3. **Write the Vertex Form:** - The standard vertex form for a vertically oriented parabola is: \[ (x - h)^2 = 4p(y - k) \] where \( (h, k) \) is the vertex. - Plugging in the vertex \( (0, 1) \) and \( p = -8 \): \[ (x - 0)^2 = 4(-8)(y - 1) \] \[ x^2 = -32(y - 1) \] 4. **Final Equation:** \[ x^2 = -32(y - 1) \] **Answer:** \[ x^{2} = -32\,\left( y\;-\;1 \right) \]