Function Transformations: Introduction

Calculus Readiness · Axiom Academy

One parent shape, a whole family of graphs — slide it, flip it, stretch it, and read the new equation right off the picture. Learn one curve, get a thousand for free Plotting points for every new function is slow, and you never have to. Almost every graph you meet is just a familiar parent — here the parabola y = x^2 — that has been moved, flipped, or scaled. Recognize the parent and the four moves, and you can picture the whole graph from the equation alone. Watch the parent y = x^2 run through its family: it lifts up by k , slides right by h , flips over the x -axis, then stretches by a factor a — the equation re-typesetting at every beat. Same shape throughout; only its place, orientation, and size change. Every graph in the family is the same parent in a new place, pose, or size — that is what a transformation is. Slide it: shifts move the whole graph Grab the vertex, or drive the two sliders. Adding k outside the square lifts the graph; subtracting h inside slides it right. The faint parent stays put so you can see that the new graph g(x) = (x - h)^2 + k is just the parent picked up and set down — its vertex lands exactly at . Outside the square acts on y and behaves as you'd expect; inside acts on x and runs backwards — (x-h) shifts right . Stretch it, flip it: one factor changes the size

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