https://file.aiquickdraw.com/mathbot/file/172590041886798covkwh.jpg

The graph shown in the image appears to represent a cubic or quartic polynomial. It has the following key features based on the visual dataThe graph shown in the image appears to represent a cubic or quartic polynomial. It has the following key features based on the visual 

1. **X-Intercepts (Roots)**:
   - There is a root at \(x = -3\), where the curve touches the x-axis but does not cross it. This suggests a **double root**.
   - There is another root at \(x = 2\), where the curve crosses the x-axis. This suggests a **single root**.

2. **Behavior at Roots**:
   - At \(x = -3\), the curve touches and turns around at the x-axis, indicating this is a root with even multiplicity (most likely 2).
   - At \(x = 2\), the curve crosses the x-axis, indicating a root with odd multiplicity (most likely 1).

3. **Leading Coefficient**:
   - The curve falls to negative infinity on the left and rises to positive infinity on the right. This indicates that the leading coefficient of the polynomial is **positive**.

### Factored Form of the Polynomial:
The factored form would be:
\[
y = a(x + 3)^2(x - 2)
\]
Where \(a\) is the leading coefficient. Based on the general shape of the graph, it seems \(a = 1\), but this needs to be confirmed by checking additional points (e.g., y-intercept).

Would you like more details on how to find the exact value of \(a\), or do you have any other questions?

### Related Questions:
1. How do we confirm the leading coefficient \(a\)?
2. Why does the curve touch the x-axis at \(x = -3\) but cross at \(x = 2\)?
3. What does the multiplicity of a root tell us about the behavior of the graph?
4. Can this function be written in standard form from its factored form?
5. How do different values of \(a\) affect the steepness of the graph?

#### Tip:
Always check if a root has odd or even multiplicity by observing if the graph crosses or merely touches the x-axis.

Write the equation of the graph shown below in factored form.

This problem involves finding the factored form of a polynomial function based on its graph. Key concepts include identifying roots, multiplicities, and interpreting the graph's behavior. Suitable for students in Grades 9-12.

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