https://r2.erweima.ai/mathbot/image/1719121381962cpkx3pu1.png

To solve this problem, we need to determine the polynomial function that corresponds to the given graph.

1. **Determine the roots and their multiplicities:**
   - The graph crosses the x-axis at \( x = -2 \) and \( x = 2 \), indicating these are roots.
   - The graph touches and bounces off the x-axis at \( x = 0 \), indicating this root has even multiplicity (likely multiplicity 2).

2. **Write the polynomial function:**
   - Given roots at \( x = -2 \), \( x = 0 \) (multiplicity 2), and \( x = 2 \):
     \[
     f(x) = a(x + 2)(x - 0)^2(x - 2)
     \]
   - Simplifying, we get:
     \[
     f(x) = a(x + 2)x^2(x - 2)
     \]

3. **Determine the correct sign and leading coefficient (a):**
   - Given that the polynomial function passes through the point \((-1, -9)\):
     \[
     -9 = a(-1 + 2)(-1)^2(-1 - 2)
     \]
     Simplifying this:
     \[
     -9 = a(1)(1)(-3)
     \]
     \[
     -9 = -3a
     \]
     Solving for \(a\):
     \[
     a = 3
     \]

   Thus, the polynomial function is:
   \[
   f(x) = 3(x + 2)x^2(x - 2)
   \]

4. **Matching the function to the given options:**
   - The correct answer must match this function.
   - Since \(f(x) = 3(x + 2)x^2(x - 2)\), it simplifies and checks our options:
     - Correct option should have similar factor structure and matching sign.
     - Answer E: \( f(x) = x(x - 2)^2(x + 2) \)

Therefore, the correct answer is:

\[
\boxed{\text{E}}
\]

Do you have any questions or need further details?

Here are 5 questions you can ask next:
1. Can you explain how to determine the roots of a polynomial from its graph?
2. What are the multiplicities of roots in a polynomial function, and how do they affect the graph?
3. How can we confirm the leading coefficient of a polynomial function using a specific point?
4. Can you provide more examples of polynomial functions and their graphs?
5. What are some common mistakes to avoid when interpreting polynomial graphs?

**Tip:** When analyzing polynomial graphs, always look for where the graph touches or crosses the x-axis to determine the roots and their multiplicities.

Learn how to solve polynomial functions based on their graphs with this detailed example. Explore concepts such as roots, multiplicities, and function construction. Suitable for high school students.

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