1. The problem is to identify the polynomial function that matches the described graph behavior: left end falling steeply downward, right end rising steeply upward, and a wavy oscillating middle section. 2. The given functions are: - $f(x) = 10x^4 + 2x + 3$ - $f(x) = 10x^3 + 6x + 2$ - $f(x) = -10x^7 - 8x + 9$ 3. Important rules for end behavior of polynomials: - The degree and leading coefficient determine the end behavior. - For even degree with positive leading coefficient, both ends rise upward. - For even degree with negative leading coefficient, both ends fall downward. - For odd degree with positive leading coefficient, left end falls downward and right end rises upward. - For odd degree with negative leading coefficient, left end rises upward and right end falls downward. 4. Analyze each function: - $f(x) = 10x^4 + 2x + 3$ has degree 4 (even) and leading coefficient 10 (positive), so both ends rise upward. - $f(x) = 10x^3 + 6x + 2$ has degree 3 (odd) and leading coefficient 10 (positive), so left end falls downward and right end rises upward. - $f(x) = -10x^7 - 8x + 9$ has degree 7 (odd) and leading coefficient -10 (negative), so left end rises upward and right end falls downward. 5. The graph description matches the second function $f(x) = 10x^3 + 6x + 2$ because it has the left end falling steeply downward and the right end rising steeply upward. 6. The wavy oscillating middle section is typical for cubic polynomials which can have up to two turning points. Final answer: The function matching the graph is $$f(x) = 10x^3 + 6x + 2$$