The *x* *y*-coordinate plane is given. The curve enters the window in the fourth quadrant, goes up and right, passes through the point (3, −3), changes direction at the point (4, 0), goes down and right, changes direction at the approximate point (5.3, −1.2), goes up and right, crosses the *x*-axis at *x* = 6, and exits the window in the first quadrant.  Find the polynomial function of degree 3 whose graph is shown in the figure.

To find the polynomial function of degree 3 given the described graph, we will follow these steps:

1. **Identify the roots and critical points from the description.**
2. **Use the critical points to help determine the coefficients.**

### Step-by-Step Solution

1. **Identify the roots and critical points:**
   - The curve passes through \((3, -3)\).
   - The curve has a local maximum at \((4, 0)\).
   - The curve has a local minimum at \((5.3, -1.2)\).
   - The curve crosses the x-axis at \(x = 6\).

The roots of the polynomial are \(x = 3\) and \(x = 6\). Since it's a cubic polynomial, we can assume a double root at \(x = 6\) to account for the direction change.

So, the polynomial can be expressed as:
\[ f(x) = a(x - 3)(x - 6)^2 \]

2. **Determine the leading coefficient

Discover how to find the polynomial function of degree 3 from a given graph. This problem involves identifying roots, critical points, and using them to determine coefficients. Suitable for advanced high school students.

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