f(x)=-x^2-4x+1

To find the vertex of the quadratic function f(x) = -x^2 - 4x + 1, we first need to rewrite the function in vertex form.

The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Given f(x) = -x^2 - 4x + 1, we can rewrite it as f(x) = -(x^2 + 4x) + 1.

To complete the square, we need to find the value that completes the square for x^2 + 4x.

To find this value, we take half of the coefficient of x (which is 4) and square it. Half of 4 is 2, and 2 squared is 4.

Therefore, we add and subtract 4 inside the parentheses: f(x) = -(x^2 + 4x + 4 - 4) + 1.

Now we can rewrite the function as f(x) = -(x + 2)^2 + 1 - 4.

Simplifying further, we get f(x) = -(x + 2)^2 - 3.

Therefore, the vertex of the function f(x) = -x^2 - 4x + 1 is at (-2, -3).