Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior to match the function with its graph. \( f(x)=4 x^{3}-3 x^{2}-2 x+2 \)

To determine the end behavior of the polynomial function \( f(x) = 4x^{3} - 3x^{2} - 2x + 2 \) using the **Leading Coefficient Test**, follow these steps: ### 1. Identify the Leading Term The **leading term** of a polynomial is the term with the highest power of \( x \). For \( f(x) \): - **Leading Term:** \( 4x^{3} \) ### 2. Determine the Degree and Leading Coefficient - **Degree:** The highest power of \( x \) is **3**. - **Leading Coefficient:** The coefficient of the leading term is **4** (which is positive). ### 3. Apply the Leading Coefficient Test The Leading Coefficient Test helps determine the end behavior based on the degree and the leading coefficient: - **If the degree is odd and the leading coefficient is positive:** - As \( x \to +\infty \), \( f(x) \to +\infty \). - As \( x \to -\infty \), \( f(x) \to -\infty \). - **If the degree is odd and the leading coefficient is negative:** - As \( x \to +\infty \), \( f(x) \to -\infty \). - As \( x \to -\infty \), \( f(x) \to +\infty \). - **If the degree is even and the leading coefficient is positive:** - As \( x \to \pm\infty \), \( f(x) \to +\infty \). - **If the degree is even and the leading coefficient is negative:** - As \( x \to \pm\infty \), \( f(x) \to -\infty \). ### 4. Determine the End Behavior for \( f(x) = 4x^{3} - 3x^{2} - 2x + 2 \) Given that: - **Degree:** 3 (which is **odd**) - **Leading Coefficient:** 4 (which is **positive**) **End Behavior:** - **As \( x \to +\infty \):** \( f(x) \to +\infty \) - **As \( x \to -\infty \):** \( f(x) \to -\infty \) ### 5. Matching the Function with Its Graph Based on the determined end behavior: - **Left Side of the Graph (\( x \to -\infty \)):** The graph will descend towards negative infinity. - **Right Side of the Graph (\( x \to +\infty \)):** The graph will rise towards positive infinity. When matching with possible graphs: - **Look for a graph where:** - The **left end** falls downwards. - The **right end** rises upwards. - The graph may have **turning points** or **intercepts** corresponding to the roots of the polynomial, but the primary feature to match is the end behavior described above. ### Visualization While I cannot display a graph here, you can visualize the general shape of \( f(x) = 4x^{3} - 3x^{2} - 2x + 2 \) as follows: 1. **Starts** from the bottom left (since \( f(x) \to -\infty \) as \( x \to -\infty \)). 2. **Rises** to the top right (since \( f(x) \to +\infty \) as \( x \to +\infty \)). 3. **Crosses** the y-axis at \( f(0) = 2 \). 4. **Has Turning Points:** As it's a cubic polynomial, it can have up to two turning points. When presented with multiple graphs, select the one that aligns with this end behavior pattern.