1. **State the problem:** Determine the end behavior of the polynomial function graphed. 2. **Recall end behavior rules:** For a polynomial function $f(x)$, the end behavior depends on the degree and leading coefficient. - If the degree is even and the leading coefficient is positive, both ends go to $+\infty$. - If the degree is even and the leading coefficient is negative, both ends go to $-\infty$. - If the degree is odd and the leading coefficient is positive, left end goes to $-\infty$ and right end goes to $+\infty$. - If the degree is odd and the leading coefficient is negative, left end goes to $+\infty$ and right end goes to $-\infty$. 3. **Analyze the graph:** The graph crosses the x-axis at about $-3$, $0$, and $2$, indicating at least degree 3 (odd degree). 4. **Observe end behavior:** Both ends go downward to $-\infty$ as $x \to -\infty$ and as $x \to +\infty$. 5. **Interpret:** Since both ends go down, the leading coefficient is negative and the degree is even. But the graph has 3 real roots, so degree is at least 3 (odd). This suggests the polynomial has odd degree but the graph shows both ends down, which is unusual for odd degree polynomials. 6. **Conclusion:** The graph's end behavior is: - Right hand end behavior: As $x \to +\infty$, $f(x) \to -\infty$ - Left hand end behavior: As $x \to -\infty$, $f(x) \to -\infty$ This matches a polynomial with even degree and negative leading coefficient. **Final answer:** Right hand end behavior: As $x \to +\infty$, $f(x) \to -\infty$ Left hand end behavior: As $x \to -\infty$, $f(x) \to -\infty$