A parabola opening up or down has vertex \( (-4,0) \) and passes through \( \left(2,-\frac{9}{2}\right) \). Write its equation in vertex form. Simplify any fractions.

The vertex form of a parabola 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 \((-4, 0)\), so we have: \[ y = a(x + 4)^2 \] Next, we need to find the value of \(a\). We can do this by using the point \(\left(2, -\frac{9}{2}\right)\) that the parabola passes through. We substitute \(x = 2\) and \(y = -\frac{9}{2}\) into the equation: \[ -\frac{9}{2} = a(2 + 4)^2 \] Calculating \(2 + 4\): \[ 2 + 4 = 6 \] Now substituting this back into the equation: \[ -\frac{9}{2} = a(6)^2 \] Calculating \(6^2\): \[ 6^2 = 36 \] So we have: \[ -\frac{9}{2} = 36a \] To solve for \(a\), we divide both sides by 36: \[ a = -\frac{9}{2} \cdot \frac{1}{36} = -\frac{9}{72} = -\frac{1}{8} \] Now we can substitute \(a\) back into the vertex form equation: \[ y = -\frac{1}{8}(x + 4)^2 \] Thus, the equation of the parabola in vertex form is: \[ \boxed{y = -\frac{1}{8}(x + 4)^2} \]