https://file.aiquickdraw.com/mathbot/file/172558758282219qbz6jb.jpg

The graph shown appears to be of a cubic polynomial function, and the smallest possible degree is already indicated as 3. Let's analyze how we can derive the formula of the function.

### Steps to find the formula for the cubic polynomial function:

1. **Zeros of the Polynomial**:  
   From the graph, we can identify where the curve crosses the x-axis. These points are the roots (or zeros) of the polynomial.
   - Based on the image, the roots seem to be at \( x = -2 \), \( x = 1 \), and \( x = 3 \).

2. **General Form of the Polynomial**:
   For a cubic polynomial with these roots, the general form will be:
   \[
   f(x) = a(x + 2)(x - 1)(x - 3)
   \]
   where \(a\) is a constant that determines the vertical stretch of the graph.

3. **Determine the Leading Coefficient \(a\)**:
   To find the value of \(a\), we need an additional point on the graph. For example, if the graph provides a point like \(f(0) = y\), we can substitute \(x = 0\) and \(y = f(0)\) to solve for \(a\).

If you can confirm any additional points from the graph, I can help you determine the exact value of \(a\) and finalize the polynomial equation.

Let me know if you want more details or have any questions! Here are some related questions to expand the understanding:

1. How do we verify that the degree of the polynomial is 3 based on the graph?
2. What changes if one of the roots is repeated (i.e., a root has a multiplicity greater than 1)?
3. What would the shape of the graph be if the degree was 4 instead of 3?
4. How does the leading coefficient \(a\) affect the orientation (upward/downward) of the graph?
5. Can you describe what happens to the graph as \( x \to \infty \) and \( x \to -\infty \)?

**Tip**: When identifying a polynomial's degree from its graph, observe the number of turning points. A polynomial of degree \(n\) can have up to \(n-1\) turning points.

Find a formula for the polynomial function of smallest degree based on its graph.

Learn how to derive the formula for a cubic polynomial based on its graph. This example involves identifying the roots of the polynomial and determining the general form. Suitable for high school students learning about polynomial functions.

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