Six, the axis of symmetry of the function $y=(m-1) x^{2}+2 m x+3 m-2$ is the line $\mathrm{x}=2$, find the value of $\mathrm{m}$, and draw the graph of this function (10 points)
Six, solve $\because$ the axis of symmetry of the parabola $y=a x^{2}+b x+c$ is $x=-\frac{b}{2 a}, \therefore-\frac{2 m}{2(m-1)}=2$, solving yields $m=\frac{2}{3}$.
Substituting $m=\frac{2}{3}$, we get the analytical expression of the function as $y=-\frac{1}{3} x^{2}+\frac{4}{3} x$.
Draw the graph of $y=-\frac{1}{3} x^{2}+\frac{4}{3} x$ (as shown in Figure 5).
$$
y=-\frac{1}{3} x^{2}+\frac{4}{3} x=-\frac{1}{3}(x-2)^{2}+\frac{4}{3}
$$
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline $\mathbf{x}$ & $\cdots$ & 0 & 1 & 2 & 3 & 4 & $\cdots$ \\
\hline $\mathbf{y}$ & $\cdots$ & 0 & 1 & $\frac{4}{3}$ & 1 & 0 & $\cdots$ \\
\hline
\end{tabular}