Question

state whether the given graph could be the graph of a polynomial function. if it is, state whether it could be a polynomial function of degree 3, 4, or 5.
choose the correct answer.
a. not a polynomial function
b. polynomial function of degree 3, 4, or 5
c. polynomial function of degree 3 or 5
d. polynomial function of degree 4

Explanation:

Step1: Check polynomial graph traits

Polynomial graphs are smooth, continuous, no breaks/corners. The given graph meets this, so it is a polynomial.

Step2: Count turning points

Turning points are peaks/valleys. This graph has 3 turning points. For a polynomial, maximum turning points = degree - 1. So degree ≥ 3 + 1 = 4.

Step3: Analyze end behavior

As $x\to+\infty$, $y\to+\infty$; as $x\to-\infty$, $y\to-\infty$. This matches odd-degree polynomials (leading coefficient positive). Degree must be odd and ≥4, so possible degree is 5. Also, degree 3 has max 2 turning points (does not fit), degree 4 has even end behavior (opposite ends match, does not fit). So only degree 5 is valid, which falls into "3 or 5" (since 3 is invalid but 5 is, and option C includes the valid degree).

Answer:

C. polynomial function of degree 3 or 5