Translating a reciprocal

Consider the rational function $y = \dfrac{1}{x - 1} + 2$. It is the reciprocal $y = \dfrac{1}{x}$ translated by $1$ in the $x$-direction and $2$ in the $y$-direction.

The asymptotes $x = 0$ and $y = 0$ of $y = \dfrac{1}{x}$ move along with it, becoming $x = 1$ (vertical) and $y = 2$ (horizontal). The point where the two asymptotes cross, $(1, 2)$, is the center of symmetry. The graph draws the horizontal asymptote $y = 2$ as a line, and the large dot marks the center $(1, 2)$; near the vertical asymptote $x = 1$ the graph shoots up steeply.