II. (20 points) As shown in Figure 4, the side length of the equilateral $\triangle ABC$ is $1, D$ and $E$ are points on sides $AB$ and $AC$ respectively. When the triangle is folded along $DE$, vertex $A$ exactly lands on point $F$ on side $BC$, with $BD=x$ and $BF=y$. Find the relationship between $x$ and $y$, and sketch the approximate graph in the Cartesian coordinate system. Determine the range of $x$.
II. Connect $D F$.
In $\triangle B D F$, by the cosine rule we have
$$
D F^{2}=B D^{2}+B F^{2}-2 B D \cdot B F \cos 60^{\circ} \text {, }
$$

which is $(1-x)^{2}=y^{2}+x^{2}-2 y \cdot x \cdot \frac{1}{2}$.
Thus, $y^{2}-x y+2 x-1=0(0 \leqslant y \leqslant 1)$.
Using a graphing calculator, we can obtain its approximate graph (as shown in Figure 7).
Let $t=2-y$, then
$$
\begin{aligned}
x= & \frac{1-y^{2}}{2-y} \\
= & \frac{1-(2-t)^{2}}{t} \\
= & -\left(t+\frac{3}{t}\right)+ \\
& 4(1 \leqslant t \leqslant 2) .
\end{aligned}
$$

Therefore, the range of $x$ is $[0,4$ -
$$
2 \sqrt{3} \text {. }
$$