Order of Transformations

Pre-Calculus · Axiom Academy

EXAMPLE Order of Transformations Apply transformations in the right sequence to graph a transformed parabola. Graph g(x) = -3(2x - 1)^2 + 4 by transforming the parent parabola f(x) = x^2 . We'll follow the transformations through the correct order , tracking the parent's vertex (0, 0) to the vertex of the new graph. First rewrite the inside as , then read the four moves left to right: Parent f(x) = x^2 (light) and the transformed graph g(x) = -3(2x-1)^2 + 4 (orange). The parent vertex (0,0) lands on the new vertex . You graphed g(x) = -3(2x-1)^2 + 4 by walking one point through the transformations in the correct order. Order is fixed: horizontal stretch/compress, then horizontal shift, then vertical stretch/reflect, then vertical shift. Rewrite first: factoring the inside to exposes a compression by and a shift right — not a shift of 1 . Track the vertex: , and the -3 flips the parabola so it opens downward. Why order matters: doing the shift before the compression would move the vertex the wrong distance and give the wrong graph. Inside the parentheses affects x (horizontal, and in the "opposite" way you'd expect); outside affects y (vertical, exactly as written).

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