Example 6 Given the function $f(x)=3x+4$, find the analytical expression of the function $y=g(x)$ according to the following conditions.
(1) The graph of $y=g(x)$ is obtained by shifting the graph of $y=f(x)$ 1 unit to the left and then 2 units downward;
(2) The graph of $y=g(x)$ is symmetrical to the graph of $y=f(x)$ about the $y$-axis;
(3) The graph of $y=g(x)$ is symmetrical to the graph of $y=f(x)$ about the line $y=1$;
(4) The graph of $y=g(x)$ is symmetrical to the graph of $y=f(x)$ about the line $y=-x$;
(5) The graph of $y=g(x)$ is centrally symmetrical to the graph of $y=f(x)$ about the point $P(a, b)$.
Let $P(x, y)$ be any point on the new graph obtained after the transformation of the function $f(x)=3 x+4$, which is transformed from the point $P^{\prime}\left(x^{\prime}, y^{\prime}\right)$ on the original graph.
(1) $\left\{\begin{array}{l}x=x^{\prime}-1, \\ y=y^{\prime}-2,\end{array}\right.$ Therefore, $y+2=3(x+1)+4$, which means $g(x)=3 x+5$.
(2) $\left\{\begin{array}{l}x=-x^{\prime}, \\ y=y^{\prime},\end{array}\right.$ Therefore, $y=-3 x+4$, which means $g(x)=-3 x+4$.
(3) $\left\{\begin{array}{l}x=x^{\prime}, \\ \frac{1}{2}\left(y+y^{\prime}\right)=1,\end{array}\right.$ Therefore, $2-y=3 x+4$, which means $g(x)=-3 x-2$.
(4) $\left\{\begin{array}{l}x=-y^{\prime}, \\ y=-x^{\prime},\end{array}\right.$ Therefore, $-x=3(-y)+4$, which means $g(x)=\frac{1}{3}(x+4)$.
(5) $\left\{\begin{array}{l}\frac{1}{2}\left(x+x^{\prime}\right)=a, \\ \frac{1}{2}\left(y+y^{\prime}\right)=b,\end{array}\right.$ Therefore, $2 b-y=3(2 a-x)+4$, which means
$$
g(x)=3 x+2 b-6 a-4 .
$$

Solution Guidance The substitution method is the basic approach to solving problems involving translation, reflection, scaling, and even rotation transformations of function graphs.