ACT-Style: Transformation of Functions

ACT Math · Axiom Academy

EXAMPLE ACT-Style: Transformation of Functions Reflect, then shift twice — chaining three transformations in the order the ACT actually tests them The graph of f(x) = x^2 - 4 is reflected over the x -axis, then translated left 4 units and up 6 units. Which equation represents the resulting function h(x) ? Nice work — you chained a reflection with two shifts in the exact order the ACT expects: transform the ORIGINAL function first, then move the result. Reflection first: "reflect over the x -axis" means g(x) = -f(x) — negate the WHOLE function, including any constant already inside it. Skipping this step (or applying it last) changes both the answer's sign pattern and its vertex. Left/right shifts feel backwards: "left h units" replaces x with (x+h) , not (x-h) — the sign flips relative to the direction. Every "shift left" trap answer on the ACT shifts right instead. Up/down shifts are the intuitive ones: "up k units" adds k to the whole function — no sign flip, applied AFTER the horizontal shift. Result: h(x) = -(x+4)^2 + 10 , expanded: h(x) = -x^2 - 8x - 6 . Multi-transformation ACT questions almost always chain a reflection or stretch with one or two shifts — work through them in the stated order, one operation at a time, and track the vertex as you go.

This is the written version of the interactive lesson above. See the full ACT Math course.