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.

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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.
To match the graphs with the polynomial functions: - **$$ f(x) = (x+1)^2(x-3) $$**: - Degree: 3 - Roots: $$ x = -1 $$ (touching the x-axis) and $$ x = 3 $$ (crossing the x-axis) - End behavior: Down to up - **$$ h(x) = x^4 - 3x^3 $$**: - Degree: 4 - Roots: $$ x = 0 $$ (touching the x-axis) and $$ x = 3 $$ (crossing the x-axis) - End behavior: Up to up Choose the graphs that reflect these characteristics.

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.

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