Let’s go through each question based on the provided graph.

(a) Over which intervals is the function increasing?

To find the intervals where the function is increasing, we observe where the graph moves upwards as $x$ increases. Based on the graph:

1. The function is increasing between $x = -8$ and $x = -6$.
2. It also increases between $x = -2$ and $x = 3$.

Thus, the answer for (a) is:

• $(-8, -6)$ and $(-2, 3)$.

(b) At which $x$-values does the function have local maxima?

Local maxima occur at points where the graph changes from increasing to decreasing. From the graph:

• There is a local maximum near $x = -6$.
• Another local maximum near $x = 3$.

The answer for (b) is:

• $x = -6, 3$.

(c) What is the sign of the function's leading coefficient?

Since the graph falls to negative infinity as $x$ approaches positive infinity, this suggests a negative leading coefficient. So, the answer for (c) is:

• Negative.

(d) Which of the following is a possibility for the degree of the function?

The function has at least three turning points, suggesting a degree of at least 4. Higher degrees (e.g., 6 or 8) are also possible. Therefore, the plausible answers for the degree are:

• 4, 6, and 8.

Would you like a deeper explanation of any part?

Here are five related questions for further exploration:

1. How do you determine the intervals where a function is decreasing?
2. What differentiates a local maximum from a global maximum?
3. How does the end behavior of a polynomial function indicate the sign of the leading coefficient?
4. How can you determine the minimum possible degree of a polynomial given the number of turning points?
5. Why might some degrees (like 5 or 7) be unlikely for this function?

Tip: When analyzing polynomial graphs, the number of turning points can provide insight into the function’s minimum degree.