Reciprocal and squared transformations

**Transformation** | **Point mapping** | **Checks** y= 1f(x) | (u,v) to (u, 1v) | Keep u in D_f, exclude v=0; heights +-1 fixed y=[f(x)]2 | (u,v) to (u,v2) | Outputs non-negative; zeros stay fixed For the reciprocal graph, a continuous zero approached through nonzero values gives an infinite branch. A vertical asymptote of f can give a limiting height 0, but its excluded input remains excluded: it does not automatically become an intercept. Squaring moves heights with 0<|v|<1 nearer the axis and heights with |v|>1 farther away; negative heights become positive. Example For f(x)=x-2, the transforms are 1x-2, with asymptotes x=2, y=0, and (x-2)2, with vertex (2,0).