1. The problem is to identify the equation of a cubic function given its graph characteristics. 2. The graph has x-intercepts at approximately $x = -2$, $x = 1$, and $x = 3$. This means the roots of the cubic polynomial are $-2$, $1$, and $3$. 3. The general form of a cubic polynomial with roots $r_1$, $r_2$, and $r_3$ is: $$y = (x - r_1)(x - r_2)(x - r_3)$$ 4. Using the roots from the graph, the equation should be: $$y = (x + 2)(x - 1)(x - 3)$$ 5. Let's check the options: - $y = (x + 2)(x - 1)(x - 3)$ matches the roots exactly. - $y = (x + 2)(x - 2)^2$ has roots at $-2$ and $2$ (with multiplicity 2), which does not match the graph. - $y = (x + 3)(x - 1)(x - 4)$ has roots at $-3$, $1$, and $4$, which does not match. - $y = (x - 2)(x + 1)(x + 3)$ has roots at $2$, $-1$, and $-3$, which does not match. 6. The graph also shows a local maximum near $x = -1$ and a local minimum near $x = 2$, consistent with the shape of $y = (x + 2)(x - 1)(x - 3)$. 7. Therefore, the correct equation for the graph is: $$y = (x + 2)(x - 1)(x - 3)$$