Let's analyze the polynomial graph from the image you provided.

1. Degree of the Polynomial:

The degree of a polynomial is the highest power of $x$ that appears in the function. Based on the graph, the polynomial has three turning points, which suggests it is of degree 4. This is because the degree of the polynomial is always one more than the number of turning points.

Thus:

• The degree of the polynomial is 4.

2. Leading Coefficient:

The leading coefficient determines the end behavior of the graph. Since the graph falls on both the left and right ends (as $x \to \pm \infty$, $y \to -\infty$), this implies that the leading coefficient is negative.

Therefore:

• The leading coefficient of the polynomial is negative.

3. End Behavior:

To describe the end behavior in words:

• As $x \to -\infty$, $y \to -\infty$.
• As $x \to +\infty$, $y \to -\infty$.

This matches the observation that the graph falls on both the far left and far right sides.

Would you like more details or have any other questions?

Here are 5 follow-up questions to deepen your understanding:

1. How can you determine the leading coefficient just from the graph?
2. What role do turning points play in identifying the degree of a polynomial?
3. How would the end behavior change if the leading coefficient were positive?
4. How does the number of x-intercepts relate to the degree of a polynomial?
5. What can you say about the symmetry of the graph?

Tip: The end behavior of a polynomial is always determined by its leading term (highest degree term). For even-degree polynomials, both ends will go in the same direction, while for odd-degree polynomials, they will go in opposite directions.