Example 1 Study the relationship between the graphs of three functions $f(x)=\frac{1}{x}, g(x)=\frac{1}{x+3}, h(x)=\frac{x+4}{x+3}$, and provide the monotonic intervals and the center of symmetry of the graph of the function $y=h(x)$.
Let $P\left(a, \frac{1}{a}\right)$ be any point on the graph of the function $y=f(x)$, then the corresponding point $Q\left(a-3, \frac{1}{a}\right)$ is a point on the graph of the function $y=g(x)$. Shifting point $P$ 3 units to the left results in point $Q$, so shifting the graph of the function $y=f(x)$ 3 units to the left results in the graph of the function $y=g(x)$.
$$
h(x)=\frac{x+4}{x+3}=\frac{(x+3)+1}{x+3}=1+\frac{1}{x+3} .
$$

Let $R\left(a, \frac{1}{a+3}\right)$ be any point on the graph of the function $y=g(x)$, then the corresponding point $S\left(a, 1+\frac{1}{a+3}\right)$ is a point on the graph of the function $y=h(x)$. Shifting point $R$ 1 unit upwards results in point $S$, so shifting the graph of the function $y=g(x)$ 1 unit upwards results in the graph of the function $y=h(x)$.

In summary, shifting the graph of the function $y=f(x)$ 3 units to the left results in the graph of the function $y=g(x)$, and shifting the graph of the function $y=g(x)$ 1 unit upwards results in the graph of the function $y=h(x)$.

Since the center of symmetry of the graph of the function $y=f(x)$ is the point $O(0,0)$, after shifting the graph 3 units to the left and 1 unit upwards, the center of symmetry $O(0,0)$ becomes the point $O'(-3,1)$. Therefore, the center of symmetry of the graph of the function $y=h(x)$ is the point $O'(-3,1)$; the intervals of monotonic decrease for the function $y=f(x)$ are $(-\infty, 0)$ and $(0,+\infty)$. After shifting the graph 3 units to the left and 1 unit upwards, the corresponding intervals become $(-\infty,-3)$ and $(-3,+\infty)$. Therefore, the intervals of monotonic decrease for the function $y=h(x)$ are $(-\infty,-3)$ and $(-3,+\infty)$.

Key Points for Solving the Problem: Do similar functions have a connection in their graphs? How does the graph of the function $y=f(x)$ transform into the graph of the function $y=f(x+a)+b$? Through the study of this problem, understand and master the methods and approaches for research!