To find the polynomial function $f(x)$ of least degree based on the graph:

1. Identify the roots (zeros): The graph crosses the x-axis at $x = -3$, $x = 0$, and $x = 2$. These are the roots of the polynomial.

2. Multiplicity of the roots: The behavior of the graph at these roots will indicate their multiplicity:

   • At $x = -3$, the graph crosses the x-axis, indicating that the multiplicity of this root is odd (likely 1).
   • At $x = 0$, the graph bounces off the x-axis, which suggests a root of even multiplicity (likely 2).
   • At $x = 2$, the graph crosses the x-axis again, indicating this root has an odd multiplicity (likely 1).

3. Form of the polynomial: Using the identified roots and their multiplicities, we can write the polynomial as: $f(x) = a(x + 3)(x - 0)^2(x - 2)$ Simplifying: $f(x) = a(x + 3)x^2(x - 2)$ where $a$ is a constant that affects the vertical stretch of the graph.

4. Determine the leading coefficient $a$: We can use a known point from the graph to find $a$. From the graph, it appears that $f(1) = 4$. Substituting into the polynomial: $4 = a(1 + 3)(1^2)(1 - 2)$ Simplifying: $4 = a(4)(1)(-1)$ $4 = -4a$ Solving for $a$: $a = -1$

5. Final equation: The polynomial function is: $f(x) = -(x + 3)x^2(x - 2)$

Would you like more details or further clarification on any step?

Here are five related questions to further expand your understanding:

1. How do you determine the multiplicity of a root from a graph?
2. What is the difference between a graph crossing the x-axis versus bouncing off it?
3. How would the polynomial change if the graph passed through the x-axis differently?
4. Can you find the degree of the polynomial based on the number and types of roots?
5. How would the graph change if the leading coefficient was positive instead of negative?

Tip: The degree of a polynomial function is determined by the highest power of $x$, and it also indicates the maximum number of x-intercepts the graph may have.