Worked Example: Function Transformations

Calculus Readiness · Axiom Academy

EXAMPLE Graphing a Transformed Parabola Build y = -2(x-3)^2 + 1 from the parent y = x^2 , one transformation at a time Starting from the parent function f(x) = x^2 , graph g(x) = -2(x-3)^2 + 1 . Identify each transformation in order, track where the vertex lands, and state the final vertex. The dashed grey curve is the parent y = x². The solid orange curve is the result, g(x) = −2(x − 3)² + 1 — narrower, opening downward, with its vertex lifted to (3, 1). Nicely done. You rebuilt a parabola from its parent by peeling the formula apart one transformation at a time. Inside vs. outside: changes inside the function — like (x-3) — move the graph horizontally (and the opposite way you'd guess); changes outside stretch, reflect, or shift it vertically. The vertex carries the story: . The stretch and reflection don't move a point already on the axis of symmetry, so only the two shifts relocate the vertex. Read it from vertex form: for g(x) = a(x-h)^2 + k , the vertex is (h,k) and a controls width (|a|>1 narrower ) and direction (a<0 opens down ) . Result: g(x) = -2(x-3)^2 + 1 is a downward, narrower parabola with vertex (3,1) . The same inside/outside reading works for every parent function — lines, roots, absolute values, exponentials — so once you can decode one formula, you can graph them all.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.