Write the equation in vertex form for the parabola with focus \( \left(7,-\frac{21}{4}\right) \) and directrix \( y=-\frac{11}{4} \) Simplify any fractions.

To express the equation of a parabola in vertex form given the focus and directrix, we start by finding the vertex, which is the midpoint between the focus and the directrix. The focus is at \( \left(7, -\frac{21}{4}\right) \) and the directrix is the line \( y = -\frac{11}{4} \). The vertex's y-coordinate is the average of the y-coordinates of the focus and directrix: \[ \text{Vertex y-coordinate} = \frac{-\frac{21}{4} + -\frac{11}{4}}{2} = \frac{-\frac{32}{4}}{2} = -4. \] Hence, the vertex is at \( \left(7, -4\right) \). The distance \( p \) from the vertex to the focus can be calculated as follows: \[ p = \text{focus y} - \text{vertex y} = -\frac{21}{4} - (-4) = -\frac{21}{4} + \frac{16}{4} = -\frac{5}{4}. \] Since the focus is below the vertex, the parabola opens downward. The vertex form of a parabola that opens downwards is given by the equation: \[ (y - k) = -\frac{1}{4p}(x - h)^2, \] where \( (h, k) \) is the vertex. Substituting in our values \( h = 7 \), \( k = -4 \), and \( p = \frac{5}{4} \) (note the negative sign for opening downward): \[ y + 4 = -\frac{1}{5}(x - 7)^2. \] Rearranging gives us the equation in vertex form: \[ y = -\frac{1}{5}(x - 7)^2 - 4. \] Thus, the equation of the parabola in vertex form is \[ y = -\frac{1}{5}(x - 7)^2 - 4. \]