Quick questions on Sketching y = 1/f(x) from the graph of y = f(x)

Let $f(x)$ be a graphed function and $g(x) = \dfrac{1}{f(x)}$ its reciprocal. Reading off the graph of $f$, here is what each feature becomes.

The cleanest first example has no zeroes and no asymptotes to worry about. Take $f(x) = 2^x$, which is positive everywhere and increasing. Its reciprocal is $g(x) = \dfrac{1}{2^x} = 2^{-x}$, and the dictionary predicts the whole shape before any algebra.

Now bring in a zero. Take the line $f(x) = x - 2$, with its single zero at $x = 2$. The dictionary says $g(x) = \dfrac{1}{x - 2}$ has a vertical asymptote at $x = 2$, is never zero, keeps the sign of the line, and meets the line where $f = \pm 1$.

The rule "a maximum of $f$ becomes a minimum of $g$" needs $f$ to keep the same sign on both sides of the turning point. If $f$ has a maximum but crosses zero on the way, the reciprocal has a vertical asymptote in between, and the neat flip is interrupted. So always check the sign of $f$ around the turning point before declaring the reciprocal's turning point: the flip is clean only when no zero of $f$ lies nearby.