Combined input transformations

For \(y=\mathrm{f}(ax+b)\), \(a\ne0\), track the original input \(u\) by \(ax+b=u\). A point \((u,v)\) moves to \(\left(\dfrac{u-b}{a},v\right)\).

Equivalently, first scale horizontally by \(\dfrac1{|a|}\), reflect in the \(y\)-axis if \(a<0\), then translate horizontally by \(-\dfrac ba\).

Example

For \(y=\mathrm{f}(2x-6)\), compress horizontally by scale factor \(\dfrac12\), then move right by \(3\). The point \((4,7)\) moves to \((5,7)\).

For \(y=Af(ax+b)+B\), the full point mapping is \(\left(\dfrac{u-b}{a},Av+B\right)\). Apply the same mapping to domain endpoints and relevant asymptotes.