A parabola opening up or down has vertex \( (-5,-7) \) and passes through \( \left(4, \frac{25}{8}\right) \). Write its equation in vertex form. Simplify any fractions.

The vertex form of a parabola that opens up or down is given by the equation: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((-5, -7)\), so we can substitute \(h = -5\) and \(k = -7\) into the equation: \[ y = a(x + 5)^2 - 7 \] Next, we need to find the value of \(a\). We know that the parabola passes through the point \(\left(4, \frac{25}{8}\right)\). We can substitute \(x = 4\) and \(y = \frac{25}{8}\) into the equation to solve for \(a\): \[ \frac{25}{8} = a(4 + 5)^2 - 7 \] Calculating \(4 + 5\): \[ 4 + 5 = 9 \] Now substituting this back into the equation: \[ \frac{25}{8} = a(9^2) - 7 \] Calculating \(9^2\): \[ 9^2 = 81 \] So we have: \[ \frac{25}{8} = 81a - 7 \] Next, we need to isolate \(a\). First, add \(7\) to both sides: \[ \frac{25}{8} + 7 = 81a \] To add \(\frac{25}{8}\) and \(7\), we convert \(7\) to a fraction with a denominator of \(8\): \[ 7 = \frac{56}{8} \] Now we can add the fractions: \[ \frac{25}{8} + \frac{56}{8} = \frac{81}{8} \] So we have: \[ \frac{81}{8} = 81a \] Now, divide both sides by \(81\): \[ a = \frac{81}{8} \div 81 = \frac{81}{8} \cdot \frac{1}{81} = \frac{1}{8} \] Now that we have \(a\), we can substitute it back into the vertex form equation: \[ y = \frac{1}{8}(x + 5)^2 - 7 \] Thus, the equation of the parabola in vertex form is: \[ \boxed{y = \frac{1}{8}(x + 5)^2 - 7} \]