To sketch a possible graph based on the given characteristics, let's break down the conditions:

1. Increases in the interval $x \in (-\infty, -2) \cup (3, \infty)$:

   • The graph should be rising (moving upwards) for $x < -2$ and $x > 3$.

2. Decreases in the interval $x \in (-2, 3)$:

   • The graph should be falling (moving downwards) between $x = -2$ and $x = 3$.

3. Has a y-intercept of $y = -1$:

   • The graph should pass through the point $(0, -1)$.

How to sketch the graph:

• For $x < -2$, the graph increases, so it moves upwards.
• At $x = -2$, the graph changes behavior and starts decreasing until $x = 3$.
• At $x = 3$, the graph starts increasing again.
• The graph passes through $(0, -1)$, meaning the curve should hit that point.

Now, I'll describe the structure of the graph:

• From left to right: Start with an upward slope for $x < -2$, then at $x = -2$, the graph slopes down until it reaches $x = 3$. After $x = 3$, the graph increases again.
• Ensure the curve crosses the y-axis at $y = -1$.

Would you like a digital sketch of this graph? Let me know if you'd like more details or have any questions!

Related Questions:

1. How would the graph look if the y-intercept was changed to $y = 2$?
2. What happens to the graph if the decreasing interval is extended to $(-3, 3)$?
3. How do you determine concavity from the graph of a function?
4. Can you sketch a graph with the same intervals but different y-intercepts?
5. What is the significance of critical points at $x = -2$ and $x = 3$?

Tip:

When sketching graphs, always look for key points like intercepts and intervals of increasing or decreasing behavior to structure the sketch easily.