Special Points on Reciprocal Graphs

An axis minimum becomes a positive vertical asymptote.

An axis maximum becomes a negative vertical asymptote.

Approach from above becomes approach from below, and vice versa.

An isolated zero of \(f\) gives a vertical asymptote of \(1/f\) when \(f(x)\to0\) through nonzero values as \(x\) approaches that input. Always keep the original domain restrictions; an excluded input does not become an intercept.

The first two pairs use \(a=0\): \(f(x)=x^2\) and \(f(x)=-x^2\), both on \(\mathbb R\). Their reciprocals are \(1/x^2\) and \(-1/x^2\), with \(x\ne0\). The final pair uses \(f(x)=1+\dfrac1{x+3.2}\), \(x\ne-3.2\), and \(\dfrac1{f(x)}=\dfrac{x+3.2}{x+4.2}\), \(x\ne-3.2,-4.2\). Both horizontal asymptotes are \(y=1\). The original excluded input \(-3.2\) remains a hole at \((-3.2,0)\) in the reciprocal.