Question

which graph shows the polynomial function $g(x) = -x^4 + x + 2$?

Explanation:

Step1: Check end behavior

For $g(x)=-x^4+x+2$, the leading term is $-x^4$. Since the degree (4) is even and the leading coefficient (-1) is negative, as $x\to\pm\infty$, $g(x)\to-\infty$. This eliminates the two graphs that open upward.

Step2: Find y-intercept

Substitute $x=0$ into $g(x)$:
$g(0)=-(0)^4+0+2=2$.
The y-intercept is $(0,2)$, which eliminates the graph with y-intercept at $(0,-2)$.

Step3: Verify key point

Substitute $x=1$ into $g(x)$:
$g(1)=-(1)^4+1+2=-1+1+2=2$.
The point $(1,2)$ lies on the remaining graph, confirming it matches.

Answer:

The top-left graph (the one with downward opening, y-intercept at (0,2), and passing through (1,2))