which function is negative for the interval $-1, 1$?

Explanation:

Step1: Define interval [-1,1]

We analyze each graph for $x \in [-1, 1]$ (all x-values between -1 and 1, inclusive), checking if $y<0$ (function is negative) across the entire range.

Step2: Evaluate first graph

For $x \in [-1,1]$, the graph has $y<0$ only between $x=0$ and $x=1$; at $x=-1$, $y=0$, so it is not negative for the full interval.

Step3: Evaluate second graph

For $x \in [-1,1]$, the graph has $y<0$ only between $x=-1$ and $x=1$ except endpoints, but at $x=-1$ and $x=1$, $y=0$, so it is not negative for the full interval.

Step4: Evaluate third graph

For $x \in [-1,1]$, the graph has $y>0$ between $x=-1$ and $x=1$ (peaks above the x-axis), so it is positive here, not negative.

Step5: Evaluate fourth graph

For $x \in [-1,1]$, every point on the graph lies below the x-axis ($y<0$) across the entire interval $[-1,1]$.

Answer:

The fourth graph (rightmost option, with the curve below the x-axis for all $x$ between -1 and 1)