1. The problem asks to write a quadratic function in vertex form $f(x) = a(x - h)^2 + k$ for a parabola with vertex at $(4, 2)$ and opening downward. 2. The vertex form of a quadratic function is given by: $$f(x) = a(x - h)^2 + k$$ where $(h, k)$ is the vertex and $a$ determines the direction and width of the parabola. 3. Since the parabola opens downward, $a$ must be negative. 4. Given the vertex $(4, 2)$, substitute $h = 4$ and $k = 2$: $$f(x) = a(x - 4)^2 + 2$$ 5. The problem provides the function $f(x) = -1(x - 4)^2 + 2$, which matches the vertex form with $a = -1$. 6. Therefore, the quadratic function is: $$f(x) = -1(x - 4)^2 + 2$$ This function represents a downward opening parabola with vertex at $(4, 2)$ as required.