To determine which of the graphs correctly represents the polynomial function $f(x)$, let's carefully analyze the given conditions:

1. $f(x)$ is positive on the intervals $(-3, -2)$ and $(0, \infty)$:

   • The graph should lie above the x-axis between $x = -3$ and $x = -2$, and for $x > 0$.

2. $f(x)$ is negative on the intervals $(-\infty, -3)$ and $(-2, 0)$:

   • The graph should lie below the x-axis for $x < -3$ and between $x = -2$ and $x = 0$.

3. $f(x)$ is increasing on the intervals $(-\infty, -2.5)$ and $(-1, \infty)$:

   • The graph should rise on the left for $x < -2.5$ and for $x > -1$.

4. $f(x)$ is decreasing on the interval $(-2.5, -1)$:

   • The graph should fall between $x = -2.5$ and $x = -1$.

Now, comparing this information to the graphs (a), (b), (c), and (d), let's identify the correct one:

• Graph (a): This graph fits the criteria:
  • It is positive on the intervals $(-3, -2)$ and $(0, \infty)$.
  • It is negative on the intervals $(-\infty, -3)$ and $(-2, 0)$.
  • It increases on $(-\infty, -2.5)$ and $(-1, \infty)$.
  • It decreases on $(-2.5, -1)$.

Therefore, Graph (a) is the correct graph.

Would you like further clarification or details on any of the intervals?

Here are five related questions:

1. How do you determine where a function is increasing or decreasing based on its derivative?
2. What are the key characteristics of polynomial functions based on their degree?
3. How can you find the exact points where a polynomial crosses the x-axis?
4. What is the significance of turning points in a polynomial's graph?
5. How does end behavior help in sketching polynomial graphs?

Tip: To analyze a function's behavior, it's helpful to find its critical points by setting its derivative equal to zero.