Since the parabola opens to the left, the standard form of the equation is:
(y - k)^2 = -4p(x - h)
where (h, k) is the vertex and p is the distance from the vertex to the focus.
We are given that the vertex is (2, 9), so h = 2 and k = 9.
We are also given that the focal diameter is 32, which means that the distance between the focus and the directrix is 16.
Since the parabola opens to the left, the focus is located at (h - p, k), and the directrix is a vertical line located p units to the right of the vertex.
Therefore, we have:
h - p = 2 - p = -14 (since the distance between the focus and the directrix is 16)
p = 16/2 = 8
Substituting the values of h, k, and p into the standard form of the equation, we get:
(y - 9)^2 = -4(8)(x - 2)
Simplifying the right-hand side, we get:
(y - 9)^2 = -32(x - 2)
Therefore, the equation of the parabola is (y - 9)^2 = -32(x - 2).