Graph transformations

For $y=A f(B(x-h))+k$, a point $(u,v)$ on the original graph becomes $(h+u/B,k+Av)$ when $B\ne0$. This point mapping gives the transformation order without guessing from the formula. Absolute values reflect selected portions; reciprocals turn zeros into potential singularities.

Horizontal translation and point mapping

For $a>0$:

Equation	Translation	Point mapping
$y=f(x-a)$	a units in the positive x-direction	$(p,q)\mapsto(p+a,q)$
$y=f(x+a)$	a units in the negative x-direction	$(p,q)\mapsto(p-a,q)$

The introductory examples change $x^2$ into $(x+1)^2$ by moving left one unit and into $(x-2)^2$ by moving right two. Solve for a known point’s new input to verify direction.