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To find an equation for this degree-four polynomial, we can analyze the graph's roots, behavior, and symmetry. Here's a step-by-step breakdown:

1. **Roots (x-intercepts):** The graph crosses the x-axis at \( x = -3 \), \( x = 0 \), and \( x = 2 \). Since this is a degree-four polynomial, we expect some of these roots to have even multiplicities.

   - The root at \( x = -3 \) crosses the x-axis with a smooth curve, suggesting a multiplicity of 2.
   - The root at \( x = 0 \) crosses the x-axis sharply, suggesting a multiplicity of 1.
   - The root at \( x = 2 \) also crosses sharply, suggesting a multiplicity of 1.

2. **Form of the Polynomial:** With these roots and multiplicities, we can propose a polynomial in the following form:
   \[
   y = a(x + 3)^2 x (x - 2)
   \]
   where \( a \) is a constant that will adjust the vertical stretch.

3. **Determining \( a \) Using a Point:** Looking at the graph, we see that when \( x = -1 \), the polynomial appears to reach its maximum value, around \( y = 4 \). Substituting \( x = -1 \) and \( y = 4 \) will allow us to solve for \( a \).

   Substituting \( x = -1 \) into the polynomial:
   \[
   4 = a(-1 + 3)^2 (-1)(-1 - 2)
   \]
   Simplifying,
   \[
   4 = a \cdot 4 \cdot (-1) \cdot (-3)
   \]
   \[
   4 = 12a
   \]
   \[
   a = \frac{1}{3}
   \]

4. **Final Equation:** Substituting \( a = \frac{1}{3} \) into the polynomial, we get:
   \[
   y = \frac{1}{3}(x + 3)^2 x (x - 2)
   \]

So, the equation for this polynomial is:
\[
y = \frac{1}{3}(x + 3)^2 x (x - 2)
\]

Would you like more details on the process or have any questions?

Here are five related questions you might find interesting:
1. How can we determine the degree of a polynomial based on its graph?
2. What does it mean when a root has an odd or even multiplicity?
3. How can symmetry in the graph help identify the polynomial's terms?
4. How do we calculate the vertical stretch factor for a polynomial?
5. Why does changing the multiplicity of a root affect the graph’s behavior at that root?

**Tip:** Always check the graph for any symmetry; it can simplify the form of your polynomial equation!

Write an equation for the degree-four polynomial graphed below.

Learn how to find the equation of a degree-four polynomial based on its graph. This example includes analyzing roots, identifying multiplicities, and using graph behavior to determine the polynomial’s form. Suitable for high school students in Grades 10-12.

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