Transformations and Reflections

ACT Math · Axiom Academy

LESSON Transformations and Reflections Shift a graph, flip it, stretch it, or squeeze it — every ACT transformation question is really just reading four simple rules off the equation. 1. Horizontal and Vertical Shifts Shifting a graph slides every point the same distance in the same direction — the shape never changes, only its position. The two shift rules look similar but behave in opposite ways depending on whether the shift happens inside or outside the parentheses. horizontal shift — INSIDE the parentheses, OPPOSITE sign vertical shift — OUTSIDE the parentheses, SAME sign If f(x)=x^2 , find the equation of f shifted right 3 and up 2. Right 3 replaces x with (x-3) ; up 2 adds 2 outside. Expanded, that's x^2-6x+11 — same parabola, moved. 2. Reflections — Flipping Across an Axis A reflection flips every point to the opposite side of a line, the same distance away — like a mirror. The ACT tests reflections of both individual points and whole functions , and the three most common mirror lines each flip a coordinate pair differently. reflecting the point (a,b) over each line Worked example — reflecting a point Reflect the point (4,3) over the x -axis, the y -axis, and the line y=x . Worked example — reflecting a function Over the x -axis: reflect f(x)=x^2 . Negate the whole function: -f(x)=-x^2 — same parabola, now opening downward. Over the y -axis: reflect f(x)=x+2 . Negate x everywhere it appears: f(-x)=-x+2 .

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