Question

select the correct answer from each drop - down menu.
the degree of the polynomial function represented by the graph is ▼, and its leading coefficient is ▼.
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Explanation:

Step1: Analyze End Behavior

The graph of the polynomial goes down on both ends (as \( x \to +\infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to -\infty \)). For a polynomial, the end behavior is determined by the degree (even or odd) and the leading coefficient (positive or negative). If the degree is even, both ends go in the same direction; if odd, opposite. Here, same direction (both down), so degree is even. Also, since both ends go down, leading coefficient is negative (for even degree, negative leading coefficient means both ends down).

Step2: Count Turning Points

The number of turning points (local maxima/minima) is related to the degree. A polynomial of degree \( n \) has at most \( n - 1 \) turning points. Looking at the graph, there are 3 turning points (two minima, one maximum). So \( n - 1 \geq 3 \implies n \geq 4 \). Since degree is even, possible degrees: 4, 6, etc. But with 3 turning points, the smallest even degree with at least 3 turning points is 4? Wait, no: degree 4 has at most 3 turning points. Wait, degree \( n \): number of turning points \( \leq n - 1 \). So if there are 3 turning points, \( n - 1 \geq 3 \implies n \geq 4 \). And since end behavior is both down (even degree, negative leading coefficient), let's check: degree 4, leading coefficient negative. Let's confirm: for degree 4, even, negative leading coefficient: as \( x \to \pm\infty \), \( y \to -\infty \), which matches the graph. So degree is 4 (even, at least 3 turning points, so degree 4), leading coefficient negative.

Answer:

The degree of the polynomial function represented by the graph is \( \boldsymbol{4} \) (or another even number like 6, but 4 is the smallest even with 3 turning points), and its leading coefficient is \( \boldsymbol{\text{negative}} \).

(Note: Assuming the drop-down options for degree include 4 (or similar even numbers) and for leading coefficient include "negative" (or similar). If options were, e.g., degree: 3,4,5,6 and leading coefficient: positive, negative, then degree is 4, leading coefficient negative.)