Answer:
Starting with the given function:
f(x) = -2x^2 - 4x + 1
To write this in the form f(x) = a(x-h)^2 + k, we need to complete the square. We can do this by adding and subtracting a constant term that will make the expression a perfect square trinomial.
f(x) = -2(x^2 + 2x) + 1
f(x) = -2(x^2 + 2x + 1) + 2 + 1 (adding and subtracting 1)
f(x) = -2(x + 1)^2 + 3
Therefore, the equation in the form f(x) = a(x-h)^2 + k is f(x) = -2(x + 1)^2 + 3, where a = -2, h = -1, and k = 3.
The vertex of the parabola is at the point (-1, 3).