Question

the graph of a polynomial function f(x) is shown in the xy - coordinate plane.

select from the drop - down menus to correctly complete the sentence.
the graph of f(x) represent a ______.
(options: third degree polynomial (tercer grado), fourth degree polynomial (cuarto grado), fifth degree polynomial (quinto grado), sixth degree polynomial (sexto grado))

Explanation:

1. Observe the end behavior: Both ends of the graph point upward (as $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to+\infty$). This matches the end behavior of even-degree polynomials with a positive leading coefficient.
2. Count the number of turning points (local maxima/minima): The graph has 3 turning points. The maximum number of turning points of a polynomial is $\text{degree}-1$. So $\text{degree}-1\geq3$, meaning the degree is at least 4.
3. From the options, the only even-degree polynomial that meets this condition is the fourth degree polynomial.

Answer:

fourth degree polynomial (cuarto grado)