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The problem asks about how the multiplicity of a zero in a polynomial function affects the behavior of the graph at that zero. The provided polynomial has zeros at \( -6 \) (multiplicity of 3), \( 0 \) (multiplicity of 3), \( 1 \) (multiplicity of 2), and \( 4 \) (multiplicity of 1).

### Key concepts:
1. **Multiplicity and Behavior:**
   - **Multiplicity 1**: The graph crosses the x-axis.
   - **Multiplicity 2 (even multiplicity)**: The graph touches the x-axis but does not cross it (it "bounces").
   - **Multiplicity 3 (odd multiplicity)**: The graph crosses the x-axis but flattens or has an inflection at the zero.

### Let's analyze each zero:
- **At \( x = -6 \)** (multiplicity 3): The graph **crosses** the x-axis but flattens.
- **At \( x = 0 \)** (multiplicity 3): The graph **crosses** the x-axis with flattening (similar behavior as at \( -6 \)).
- **At \( x = 1 \)** (multiplicity 2): The graph **touches** the x-axis but does not cross it (bounces).
- **At \( x = 4 \)** (multiplicity 1): The graph **crosses** the x-axis.

### Completing the statements:
1. **The graph of the function crosses the x-axis at \( -6 \).**
2. **The graph of the function crosses the x-axis at 0.**
3. **The graph of the function touches the x-axis at 1.**
4. **The graph of the function crosses the x-axis at 4.**

Do you have any further questions, or would you like more details?

Here are some related questions for practice:
1. What is the general behavior of a graph at a zero with even multiplicity?
2. How does the graph behave for a zero with odd multiplicity greater than 1?
3. What is the significance of the degree of the polynomial in terms of graph shape?
4. How do you find the multiplicities of the zeros given the polynomial equation?
5. How does the end behavior of a polynomial change with its degree?

**Tip**: For quick identification, remember that zeros with odd multiplicities allow the graph to cross the x-axis, while zeros with even multiplicities cause the graph to touch but not cross it.

How does the multiplicity of a zero determine the behavior of the graph at that zero? A seventh degree polynomial function has zeros of -6 (multiplicity of 3), 0 (multiplicity of 3), 1 (multiplicity of 2), and 4 (multiplicity of 1).

Understand how the multiplicity of zeros affects the graph of a seventh-degree polynomial. This problem involves zeros at -6, 0, 1, and 4 with varying multiplicities, demonstrating the graph's behavior at each zero.

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