Transforming y=-x^2: Finding the Parabola Negative Everywhere Except at x=-4

To solve this problem, consider the following steps:

• Step 1: Recognize that translating a parabola horizontally involves modifying its vertex.
• Step 2: The original function $y = -x^2$ has its vertex at $(0, 0)$, and it is entirely non-positive.
• Step 3: To make the parabola non-negative only at $x = -4$, translate this parabola horizontally to have its vertex at $(-4, 0)$.

Let us translate $y = -x^2$ by moving it 4 units to the left, resulting in:

$y = -(x + 4)^2$

This new equation $y = -(x + 4)^2$ translates the parabola left by 4 units, positioning the vertex at $(-4, 0)$.

Since the coefficient of $(x + 4)^2$ is negative, the parabola is downward opening and is negative anywhere except at its vertex.

Therefore, the translated parabola is indeed negative throughout its domain except at $x = -4$, which satisfies the problem's conditions.

Thus, the parabola is correctly represented by the expression $\mathbf{y = -(x + 4)^2}$.