Transformations Summary

Pre-Calculus · Axiom Academy

SUMMARY Transformations of Functions One parent graph, four moves. Everything in this unit is a shift, a stretch, or a flip applied to a function you already know. Every transformed graph comes from a parent function reshaped by the same four moves: vertical shift, horizontal shift, stretch/compression, and reflection. Outside the function changes the output (vertical, "what you'd expect"); inside changes the input (horizontal, and it runs opposite to the sign). packages all four at once — read a , b , h , k straight off the equation. Stretches and reflections change a graph's shape ; shifts only change its position . When several moves combine, order matters : do the inside ( b , then h ) before the outside ( a , then k ). Core Concept Translations (Shifts) A shift slides the whole graph without changing its shape. Add outside to move up/down; subtract inside to move right/left. Vertical: +k up, -k down — behaves exactly as written. Horizontal: (x-h) shifts right h , (x+h) shifts left — opposite the sign. Core Concept Stretches & Compressions Scaling pulls the graph toward or away from an axis, changing its shape. The vertical factor is direct; the horizontal factor is reciprocal. Vertical: stretches taller, compresses flatter. Horizontal: compresses (squeeze by 1/b ), stretches wider. A reflection is a flip across an axis — the sign tells you which one. It lands the parent graph as a mirror image. Over the x -axis: negate the output, -f(x) (the case).

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