Find the Translation of y=-x² with Maximum Value at x=2

The problem requires us to find a translated version of the parabola $y = -x^2$ such that the resulting function is negative for all $x$ except for a specific point, $x = 2$. Here’s how we address this :

• The function $y = -x^2$ is an upside-down parabola centered at the origin $(0, 0)$.
• We want the vertex to be at $x = 2$. For a horizontal translation to the right by 2 units, we use $x - 2$ inside the function, leading to the new function $y = -(x - 2)^2$.
• This translation moves the vertex to the point $(2, 0)$ and ensures that the parabola will still open downwards, being negative for all $x \neq 2$.

Thus, the parabola described by the function $y = -(x - 2)^2$ is what fulfills all the required conditions, including negativity in all domains except at $x = 2$.

The correct choice, given the options, is $y = -(x - 2)^2$, identified as Choice 3.

Therefore, the solution to the problem is $y = -(x - 2)^2$.