1. **State the problem:** We need to find which equation best matches a parabola that opens downward with vertex at approximately $(4,1)$. 2. **Recall the vertex form of a parabola:** The vertex form is $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex and $a$ determines the direction and width of the parabola. 3. **Analyze the vertex:** Given vertex is $(4,1)$, so $h=4$ and $k=1$. The equation must be of the form $$y = a(x - 4)^2 + 1$$ 4. **Determine the direction:** The parabola opens downward, so $a$ must be negative. 5. **Check the options:** - $y = -3(x - 4)^2 + 1$ matches vertex $(4,1)$ and opens downward. - $y = 3(x + 4)^2 + 1$ vertex at $(-4,1)$ and opens upward. - $y = -3(x + 4)^2 + 1$ vertex at $(-4,1)$ and opens downward. - $y = 3(x - 4)^2 + 1$ vertex at $(4,1)$ but opens upward. 6. **Conclusion:** The best match is $$\boxed{y = -3(x - 4)^2 + 1}$$ because it has the correct vertex and opens downward as described.