1. The problem is to analyze the function $g(x) = -\frac{1}{3}x^2 - 2x + 1$ and determine which graph correctly represents it, including identifying whether it has a maximum or minimum. 2. The function is a quadratic of the form $ax^2 + bx + c$ where $a = -\frac{1}{3}$, $b = -2$, and $c = 1$. 3. Since $a < 0$, the parabola opens downward, so the function has a maximum point. 4. The vertex of a parabola given by $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. 5. Calculate the vertex $x$-coordinate: $$x = -\frac{-2}{2 \times -\frac{1}{3}} = -\frac{-2}{-\frac{2}{3}} = -\frac{-2}{-\frac{2}{3}} = -3$$ 6. Calculate the vertex $y$-coordinate by substituting $x = -3$ into $g(x)$: $$g(-3) = -\frac{1}{3}(-3)^2 - 2(-3) + 1 = -\frac{1}{3} \times 9 + 6 + 1 = -3 + 6 + 1 = 4$$ 7. So the vertex is at $(-3, 4)$ and since $a < 0$, this is a maximum point. 8. Among the given graphs, the one with a downward-opening parabola and vertex at $(-3, 4)$ is Graph 3. **Final answer:** Graph 3 shows the graph of $g(x)$ with a maximum at $(-3, 4)$.