We need to construct a polynomial function that meets the given conditions.

Step-by-step solution:

1. Roots and Multiplicities:

   • Crosses the axis at $x = 2$ and $x = -3$. These have a multiplicity of 1.
   • Bounces at the axis at $x = 3$ and $x = -1$. These have a multiplicity of 2.

   Therefore, the polynomial can be written as: $f(x) = a(x - 2)(x + 3)(x - 3)^2(x + 1)^2$ where $a$ is a constant to be determined.

2. y-Intercept: The $y$-intercept is given as $(0, 3)$. Substituting $x = 0$ and $f(x) = 3$: $f(0) = a(0 - 2)(0 + 3)(0 - 3)^2(0 + 1)^2 = 3$ Simplify: $f(0) = a(-2)(3)(9)(1) = 3$ $-54a = 3$ Solve for $a$: $a = -\frac{1}{18}$

3. Final Polynomial Function: Substitute $a = -\frac{1}{18}$ into the polynomial: $f(x) = -\frac{1}{18}(x - 2)(x + 3)(x - 3)^2(x + 1)^2$

Final Answer:

$f(x) = -\frac{1}{18}(x - 2)(x + 3)(x - 3)^2(x + 1)^2$

Let me know if you need further explanation!

Related Questions:

1. What is the degree of the polynomial $f(x)$?
2. How would the graph of $f(x)$ behave near $x = -3$ and $x = 2$?
3. What does the constant $a$ signify in this context?
4. How do we determine whether a root has a "bounce" versus a "cross"?
5. How can you verify this function graphically or analytically?

Tip: Always verify your polynomial's behavior at the intercepts by substituting points back into the equation.