A plotted point is (X,Y), not necessarily (x,y). Once Y=mX+c is found, replace each axis symbol by its stated expression to recover the original law. Find m,c from two transformed points, substitute the axis definitions, then solve only as far as requested. When undoing y2, keep both signs unless positivity is given; logarithms require positive arguments. If Y=y2,X=x3 and Y=2X+1, then y2=2x3+1. At x=0, y=+-1; taking only the positive root loses one permitted value. Check: Y= e2y means y= 12ln Y with Y>0; Y=y3 has a unique real cube root, including when Y<0.

A plotted point is \((X,Y)\), not necessarily \((x,y)\). Once \(Y=mX+c\) is found, replace each axis symbol by its stated expression to recover the original law.

Find \(m,c\) from two transformed points, substitute the axis definitions, then solve only as far as requested. When undoing \(y^2\), keep both signs unless positivity is given; logarithms require positive arguments.

If \(Y=y^2,X=x^3\) and \(Y=2X+1\), then \(y^2=2x^3+1\). At \(x=0\), \(y=\pm1\); taking only the positive root loses one permitted value.

Check: \(Y=\mathrm e^{2y}\) means \(y=\tfrac12\ln Y\) with \(Y>0\); \(Y=y^3\) has a unique real cube root, including when \(Y<0\).

Recovering nonlinear laws from transformed axes

Recovering nonlinear laws from transformed axes