https://file.aiquickdraw.com/mathbot/file/1727226065279gsh0om60.jpg

The table in part **b)** provides possible values for the leading coefficient **a** of the polynomial function \( m(x) \) in its factored form. Based on the graph shown in the figure, you're supposed to determine which value of **a** would result in the polynomial matching the behavior of the graph.

### Steps to Analyze:

1. The given graph is a cubic polynomial because it crosses the x-axis three times.
2. The factored form of the polynomial given is \( a(x + 2)^2(x - 3) \), which means:
   - There's a double root at \( x = -2 \) (since \( (x + 2)^2 \)).
   - A single root at \( x = 3 \) (since \( (x - 3) \)).
   
3. Now, we need to pick the right value of **a** so that the graph of \( m(x) \) matches the provided graph.

   - If **a** is positive, the end behavior of the cubic function would go from \( -\infty \) to \( +\infty \), meaning the graph rises to the right.
   - If **a** is negative, the end behavior would be reversed (from \( +\infty \) to \( -\infty \)), meaning the graph falls to the right.

### Analysis of the Graph:
- The graph in the image **falls to the right**, meaning the leading coefficient **a** must be **negative**.

### Conclusion:
From the possible values in the table (0.3, 0.2, -0.2, -0.3), the correct value for **a** would likely be **-0.3**, since the graph has steep behavior and requires a stronger negative coefficient to match the steepness and shape of the graph.

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Do you have any questions or need more details?

Here are 5 related questions for further exploration:
1. What is the general form of a cubic polynomial?
2. How do double roots affect the shape of the graph of a polynomial?
3. Why does the leading coefficient affect the end behavior of a polynomial graph?
4. How can you determine the multiplicity of roots from a graph?
5. What role does the degree of a polynomial play in its graph?

**Tip:** Always observe the direction of the graph as it moves towards infinity to identify the sign of the leading coefficient.

Learn how to determine the leading coefficient of a polynomial function from its graph. Explore the factored form, end behavior, and the role of roots and their multiplicities in shaping the graph. Suitable for Grades 10-12 students studying polynomial functions.

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