Example 2. Plot the graph of the function $y=2-\frac{3 x+1}{2 x-2}$.
Solution. Let's represent the given function as $y=-\frac{2}{x-1}+\frac{1}{2}$. We will construct the graph step by step.

1) Let $\Gamma$ be the graph of the function $y=\frac{1}{x}$ - a hyperbola with equal branches (Fig. 5.13, a).
2) $\Gamma_{1}: y=\frac{1}{x-1}$ - a shift of the hyperbola 1 unit to the right along the $O x$ axis (Fig. $5.13, b$ ).
3) $\Gamma_{2}: y=-\frac{1}{x-1}$ - reflection of $\Gamma$ about the $O x$ axis (Fig. 5.13, c).
4) $\Gamma_{3}: y=-\frac{2}{x-1}$ - stretching $\Gamma_{1}$ by a factor of 2 from the $O x$ axis (Fig. 5.13, d).
5) $\Gamma_{4}: y=-\frac{2}{x-1}+\frac{1}{2}$ - a shift of $\Gamma_{3}$ upwards by $\frac{1}{2}$ (Fig. 5.13, e).