To analyze the graph and write the solution in interval notation, I'll focus on identifying its important features:

1. Domain:

The domain represents all the possible $x$-values for which the function is defined. The graph seems to extend indefinitely in both directions along the $x$-axis. Thus, the domain is: $(-\infty, \infty)$

2. Range:

The range represents all the possible $y$-values that the function takes. From the graph, it appears that the graph goes downward to $-8$ and upward indefinitely as $y \to \infty$. Therefore, the range is: $(-8, \infty)$

3. Intervals of Increase:

The function increases where the graph moves upwards from left to right. By inspecting the graph, it increases in the intervals: $(-7, -1) \cup (4, \infty)$

4. Intervals of Decrease:

The function decreases where the graph moves downwards from left to right. It decreases in the intervals: $(-1, 1) \cup (2, 4)$

5. Local Maxima and Minima:

• Local maximum occurs around $x = 1$, where $y \approx 3$.
• Local minimum occurs around $x = 4$, where $y \approx -2$.

Would you like more details on any part of this analysis? Here are five related questions you might find useful:

1. How can we formally identify local maxima and minima using derivatives?
2. How do we interpret increasing and decreasing intervals using calculus?
3. Can we estimate more precise coordinates of the turning points?
4. How do inflection points influence the shape of a graph?
5. What can the behavior of the function at infinity tell us about its end behavior?

Tip: Understanding the intervals of increase and decrease helps in determining the overall behavior of the function and predicting its future behavior.