Reciprocal and reciprocal-square graphs

For \(y=\dfrac a x\) and \(y=\dfrac a{x^2}\), \(a\ne0\) and \(x\ne0\). Both have two separate branches. The axes are asymptotes: the curves approach them without meeting them. Never join the branches across \(x=0\).

Example

For \(y=\dfrac1x\), inputs \(-2,-1,1,2\) give outputs \(-\dfrac12,-1,1,\dfrac12\). The two branches have opposite signs.

For \(y=\dfrac1{x^2}\), the same inputs give \(\dfrac14,1,1,\dfrac14\). Both branches are above the horizontal axis and the graph is symmetric about the vertical axis.

A negative coefficient reflects either graph in the horizontal axis. A zero denominator is undefined, not a point at the origin.