Quick questions on Reciprocal functions - TCE Mathematics Specialised (Tasmania)

The single most important detail when sketching a reciprocal is the direction of the vertical asymptote at each zero of $f$. Read the sign of $f$ on each side of the zero. If $f$ changes from positive to negative through the zero (a simple crossing with negative gradient), then $\dfrac{1}{f}$ goes from $+\infty$ down to $-\infty$ across the asymptote. If $f$ only touches the axis (a repeated root, so $f$ does not change sign), the reciprocal shoots to the same infinity on both sides.