Reciprocal graphs and asymptotes

For \(y=a/x+b\), with \(a\ne0\), x cannot be zero, giving vertical asymptote \(x=0\). As the magnitude of x grows, \(a/x\) approaches zero, so the horizontal asymptote is \(y=b\). Draw the two branches separately.

Example

For \(y=2/x+1\), the x-intercept solves \(2/x+1=0\), so it is \((-2,0)\). There is no y-intercept.

For \(y=a/x^2\), the two branches have the same sign and the y-axis is a line of symmetry. For \(y=a/x\), opposite inputs give opposite outputs. A dashed asymptote is a guide, not part of the curve.