Transformation Playground

Pre-Calculus · Axiom Academy

Shift it, stretch it, flip it: learn the four moves that turn one parent graph into infinitely many. One parent graph, reshaped at will You don't graph every function from scratch. Almost all of them are a single parent function in disguise — the same basic shape, slid, stretched, or flipped into place. Learn a handful of moves and you can sketch a whole family by eye, and read an equation straight off its graph. Watch the parent parabola y = x^2 get molded: lifted up, stretched taller, slid to the right, then flipped upside down. The same curve each time — only its position and proportions change, and the equation changes right along with it. Same parabola, four different homes. Every move below is one of these. Grab the vertex and drag it — or use the sliders. Lifting the whole curve adds to the output: y = x^2 + k . Sliding it sideways changes the input: y = (x-h)^2 . That sideways one feels backwards, so watch the sign as you move. Subtracting inside, (x-h)^2 , slides it right by h — the counter-intuitive one. Multiplying the whole function scales its height: y = a\,x^2 . A factor above 1 makes it steeper and narrower; between 0 and 1, gentler and wider. Flip the sign and the parabola turns upside down. Through it all, the vertex at the origin never moves — every other point does. A negative outside , y = -a\,x^2 , mirrors it across the x-axis — the vertex stays put.

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