1. As shown in Figure 2, given the vertices of square $A B C D$ are $A(1,1)$, $B(3,1)$, $C(3,3)$, and $D(1,3)$, the line $y=2 x+b$ intersects $A B$ at point $E$ and $C D$ at point $F$. Then the range of the y-intercept $b$ of the line is
$$
\text { II. 1. }-3 \leqslant b \leqslant-1 \text {. }
$$

From the line $y=2 x+b$ moving parallel with different values of $b$, we know that when the line passes through point $A$, we get $b=-1$; when the line passes through point $C$, we get $b=-3$. Therefore, $-3 \leqslant b \leqslant-1$.