1. The problem asks to find the equation of the function $g(x)$ whose graph is a parabola opening downwards with vertex at $(0,3)$. 2. The parent function is $f(x) = x^2$, which is a parabola opening upwards with vertex at $(0,0)$. 3. To transform $f(x) = x^2$ to a parabola opening downwards, we multiply by $-1$, giving $-x^2$. 4. To shift the vertex from $(0,0)$ to $(0,3)$, we add 3 to the function: $-x^2 + 3$. 5. Therefore, the equation of the function graphed is: $$g(x) = -x^2 + 3$$ This represents a parabola opening downwards with vertex at $(0,3)$, matching the graph described.