Question

which graph has the same end behavior as the graph of $f(x)=-3x^4 - x^2 + 1$?

Explanation:

Step1: Identify degree and leading coefficient

The function is $f(x) = -3x^4 - x^2 + 1$. It is a 4th-degree (even) polynomial, with leading coefficient $-3$ (negative).

Step2: Define end behavior for even degree, negative leading coefficient

For even-degree polynomials with negative leading coefficients:
As $x \to +\infty$, $f(x) \to -\infty$;
As $x \to -\infty$, $f(x) \to -\infty$.

Step3: Match to given graphs

The top graph has ends going $+\infty$ (both sides), which is for positive leading coefficient even degree. The bottom graph has both ends going $-\infty$, matching the required end behavior.

Answer:

The bottom graph (the one where both ends of the curve point downward towards negative infinity) has the same end behavior as $f(x) = -3x^4 - x^2 + 1$.