Graphing by Transformations

Graphing by transformations means reading a formula as edits to a parent function — horizontal moves from inside, vertical moves from outside — and applying stretches and reflections before shifts.

Graph g(x) = 2(x − 1)² + 3 from the parent f(x) = x². Inside first: (x − 1) slides the graph right 1. Outside next, in order: the 2 stretches outputs vertically, then the +3 lifts everything 3. Track one point through the pipeline: the parent's (1, 1) moves right to (2, 1), stretches to (2, 2), lifts to (2, 5) — and the check confirms it: g(2) = 2(1)² + 3 = 5. The vertex rides from (0, 0) to (1, 3), and g(1) = 3 agrees. One tracked point plus the vertex catches nearly every ordering mistake.