Sketch the reciprocal of a function, relating zeros to asymptotes and turning points to turning points

What this dot point is asking

SCSA wants you to transform a known graph into the graph of its reciprocal using these structural rules, without computing a new formula point by point.

Zeros become asymptotes

Wherever $f(x) = 0$, the reciprocal $\tfrac{1}{f(x)}$ is undefined and blows up, so each zero of $f$ gives a vertical asymptote of the reciprocal. The sign of $f$ just either side of the zero tells you whether the reciprocal branch goes to $+\infty$ or $-\infty$.

Large becomes small, small becomes large

Sign and fixed points

The reciprocal has the same sign as $f$, since $\tfrac{1}{f}$ is positive exactly when $f$ is positive. The graphs cross where $f(x) = \tfrac{1}{f(x)}$, that is where $f(x) = \pm 1$; these points are shared by both curves.

Turning points swap type

Where $f$ has a local maximum (and is positive there), $\tfrac{1}{f}$ has a local minimum at the same $x$, because dividing one into a peak gives a trough. Likewise a positive local minimum of $f$ becomes a local maximum of the reciprocal. Where $f$ is negative the roles invert accordingly.

Sketching routine

Mark the zeros of $f$ as asymptotes of the reciprocal, mark the $f = \pm 1$ crossing points, send the reciprocal toward $y = 0$ where $f$ is large, and turn the peaks into troughs. Then join smoothly.