Square Root Function: Find Algebraic Expression from Graph with Point (-2)

We begin by analyzing the shape and nature of the parabola described. From the visual description, the parabola opens downwards, indicating that the leading coefficient of the quadratic must be negative.

Notice that the vertex of the parabola sits on the negative direction of the y-axis, which is consistent with the vertex form $y = ax^2 + k$. For a parabola opening downwards, we have $a < 0$.

Given that the vertex appears at the value $y = -2$ on the y-axis, we can leverage the standard form:

The function base form becomes $y = -x^2 + c$.

The detail given suggests a vertex directly on $y = -2$ (as one of the intersecting point/vertex specifics), hence: $y = -x^2 - 2$.

The mathematical representation of this function, aligned with the vertex downwards and y-intercept at -2, is therefore $y = -x^2 - 2$.