https://file.aiquickdraw.com/mathbot/file/17304190624809k0zsnrf.jpg

Let's analyze the questions based on the graph:

### (a) Intervals where the function is decreasing
To determine where the function is decreasing, we look for intervals where the graph slopes downward as \( x \) increases. Observing the graph, the function decreases in the following intervals:
- From \( x = -6 \) to \( x = -2 \)
- From \( x = 3 \) to \( x = 7 \)

So, the correct intervals to select are:
- \((-6, -2)\)
- \((3, 7)\)

### (b) \( x \)-values of local maxima
Local maxima occur at points where the graph changes from increasing to decreasing. Based on the graph:
- There is a local maximum around \( x = -6 \)
- Another local maximum around \( x = 3 \)

Thus, the answer is:
- \( x = -6, 3 \)

### (c) Sign of the function's leading coefficient
The end behavior of the graph indicates the sign of the leading coefficient:
- As \( x \to -\infty \), \( f(x) \to -\infty \)
- As \( x \to +\infty \), \( f(x) \to +\infty \)

Since the ends go in opposite directions, this suggests an **odd degree** polynomial with a **positive leading coefficient**.

### (d) Possible degree of the function
The graph shows three turning points, which implies at least a fourth-degree polynomial (since an \( n \)-degree polynomial has at most \( n - 1 \) turning points). Thus, possible degrees are:
- 4
- 6
- 8

Answers:
1. **Intervals of decrease**: \((-6, -2)\) and \((3, 7)\)
2. **Local maxima at \( x \)-values**: \( -6, 3 \)
3. **Leading coefficient sign**: Positive
4. **Possible degrees**: 4, 6, 8

Would you like further details or have any questions?

### Additional Questions:
1. What does it mean for a function to be increasing or decreasing on an interval?
2. How can the degree of a polynomial be inferred from the number of turning points?
3. Why does the sign of the leading coefficient affect the end behavior of a polynomial?
4. What are local maxima and minima, and how are they identified on a graph?
5. Can a polynomial function have a degree higher than the number of its turning points?

### Tip:
For polynomial graphs, the end behavior often reveals the degree and sign of the leading coefficient.

Below is the graph of a polynomial function with real coefficients. Use the graph to answer the following questions:

(a) Over which intervals is the function decreasing? Choose all that apply.
(b) At which x-values does the function have local maxima? If there is more than one value, separate them with commas.
(c) What is the sign of the function's leading coefficient?
(d) Which of the following is a possibility for the degree of the function? Choose all that apply.

Learn to analyze a polynomial function's graph by identifying intervals where the function decreases, locating local maxima, determining the sign of the leading coefficient, and inferring possible degrees based on turning points. This detailed solution is suitable for high school math students studying polynomial functions.

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