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To determine the equation of the polynomial from the given graph, we need to analyze its roots (x-intercepts), turning points, and end behavior. Here’s what we can observe from the graph:

1. **Roots (x-intercepts)**:
   - The graph appears to intersect the x-axis at \( x = -4 \), \( x = 0 \), and \( x = 3 \). These are potential roots of the polynomial.

2. **Multiplicity of Roots**:
   - At \( x = -4 \), the graph crosses the x-axis, indicating that the root is likely of odd multiplicity (likely 1).
   - At \( x = 0 \), the graph crosses the x-axis as well, suggesting this root is of odd multiplicity (likely 1).
   - At \( x = 3 \), the graph touches the x-axis and then turns around, which suggests this root has even multiplicity (likely 2).

3. **End Behavior**:
   - The polynomial rises on both ends of the graph, indicating an even degree polynomial (e.g., degree 4 or 6).
   - Since there are three distinct roots (\( x = -4 \), \( x = 0 \), and \( x = 3 \)) and one of them has a multiplicity of 2, a degree-4 polynomial is the simplest form that matches this behavior.

4. **Forming the Polynomial**:
   Using the roots, we can construct the polynomial in the form:
   \[
   f(x) = a(x + 4)(x)(x - 3)^2
   \]
   Here, \( a \) is a leading coefficient that determines the vertical stretch of the polynomial.

5. **Finding \( a \)**:
   To find \( a \), we need a point that lies on the polynomial. For example, if the graph provides a clear point like \( (1, 2) \), we can plug this into the equation and solve for \( a \). If no such point is available, we can assume \( a = 1 \) for simplicity:
   \[
   f(x) = (x + 4)(x)(x - 3)^2
   \]

This is the general form of the polynomial based on the provided graph.

Would you like more details or have any questions?

### Related Questions:
1. How do we determine the degree of a polynomial from its graph?
2. What is the role of multiplicity in determining the shape of a polynomial graph?
3. How does the end behavior of a polynomial depend on its leading term?
4. How do you find the turning points of a polynomial function?
5. What are some common methods for graphing polynomials?

### Tip:
Always check the behavior near the x-intercepts to determine the multiplicity of roots; it helps in correctly constructing the polynomial's equation.

Write an equation for the polynomial graphed below.

Discover how to determine the equation of a polynomial from its graph by identifying its roots, multiplicities, and end behavior. This problem is suitable for Grades 11-12 students and covers key concepts in polynomial functions.

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