Matching graphs with polynomial functions Consider the following polynomial functions. \[ \begin{array}{l}f(x)=(x+1)^{2}(x-3) \\ h(x)=x^{4}-3 x^{3}\end{array} \] Choose the graph of each function from the choices below.

To match the graphs with the polynomial functions \( f(x) \) and \( h(x) \), we need to analyze the characteristics of each function. ### Step 1: Analyze \( f(x) = (x+1)^{2}(x-3) \) 1. **Degree**: The degree of \( f(x) \) is \( 3 \) (since \( (x+1)^2 \) contributes \( 2 \) and \( (x-3) \) contributes \( 1 \)). 2. **Roots**: The roots of \( f(x) \) are: - \( x = -1 \) (with multiplicity \( 2 \)) - \( x = 3 \) (with multiplicity \( 1 \)) 3. **Behavior at Roots**: - At \( x = -1 \), the graph will touch the x-axis and turn around (since the root has even multiplicity). - At \( x = 3 \), the graph will cross the x-axis (since the root has odd multiplicity). 4. **End Behavior**: As \( x \to -\infty \), \( f(x) \to -\infty \) and as \( x \to +\infty \), \( f(x) \to +\infty \). ### Step 2: Analyze \( h(x) = x^{4} - 3x^{3} \) 1. **Degree**: The degree of \( h(x) \) is \( 4 \). 2. **Roots**: To find the roots, we can factor: \[ h(x) = x^{3}(x - 3) \] The roots are: - \( x = 0 \) (with multiplicity \( 3 \)) - \( x = 3 \) (with multiplicity \( 1 \)) 3. **Behavior at Roots**: - At \( x = 0 \), the graph will touch the x-axis and turn around (since the root has odd multiplicity). - At \( x = 3 \), the graph will cross the x-axis (since the root has odd multiplicity). 4. **End Behavior**: As \( x \to -\infty \), \( h(x) \to +\infty \) and as \( x \to +\infty \), \( h(x) \to +\infty \). ### Step 3: Summary of Characteristics - **\( f(x) \)**: - Degree: 3 - Roots: \( -1 \) (touch) and \( 3 \) (cross) - End behavior: Down to up - **\( h(x) \)**: - Degree: 4 - Roots: \( 0 \) (touch) and \( 3 \) (cross) - End behavior: Up to up ### Step 4: Match with Graphs Now, you would need to look at the provided graph options and match them based on the characteristics outlined above. - Look for a graph that has a cubic shape (for \( f(x) \)) with a turning point at \( x = -1 \) and crossing at \( x = 3 \). - Look for a graph that has a quartic shape (for \( h(x) \)) that touches the x-axis at \( x = 0 \) and crosses at \( x = 3 \), with both ends going upwards. If you provide the graph options, I can help you identify which graph corresponds to each function.