Slide it. Stretch it. Flip it.
Change one number in a function and the whole graph moves. Learn transformations by doing — turn the knobs, watch the graph respond, and build the intuition that static textbook figures can't.
Start playing ↓ already know the basics? jump to the function library →A like y = x2 + 1 takes any input x and produces exactly one output y. Every run of the machine is plotted as a point at . Drag the slider and watch it work.
The graph is the complete record of the function: every input paired with its output, plotted as one curve. Every move on this page is an edit to the machine — change what comes out (the well-behaved knobs) or change what goes in (the sneaky one).
Change the constant at the end and it's applied after everything else: every output rises or falls by exactly that much. The graph doesn't tilt or bend — it translates straight up or down. The arrows track individual points making the trip.
Outside the function, numbers do exactly what they say: +3 means up 3, −3 means down 3. Savor it — the next knob isn't so polite.
Now edit what goes in, before the function ever sees it: y = 2 − (x − 3)²4. Subtracting 3 shifts the whole graph… to the right?! Almost everyone guesses left. Here's the click: x − 3 is a delay. The function receives every input as 3 smaller than it really is, so whatever used to happen at 0 now happens at 3.
Inside the parentheses, signs work backwards. f(x − 3) shifts right; f(x + 3) shifts left. Outside the function (the green knob), numbers behave as written.
Still feels like a trick? Two pens draw simultaneously, left to right. The blue pen repeats the white pen's every move — three steps behind.
Slide a line sideways and the result looks identical to sliding it down — a line has no landmark, so both descriptions are true at once. The dome's peak makes the horizontal shift unambiguous.
The coefficient out front multiplies every output. At a = 2, every height doubles — a vertical stretch that reads as narrower. At a = 0.5, heights halve — a compression that reads as wider. Push it negative and every output changes sign: the parabola reflects through the x-axis.
y = a·f(b·(x − h)) + k — the complete formula, and a shelf of nine functions to aim it at. You already own a, h, and k. The new knob, b, multiplies x before anything else happens — it's an inside knob, so it works backwards too: b = 2 squeezes the graph to half its width. Some of these functions come with dashed — watch them follow your knobs. Pick a card, or grab the curve and drag it.
| You write | The graph… | Watch out |
|---|---|---|
| y = f(x) + 3 | translates up 3 | behaves as written |
| y = f(x) − 3 | translates down 3 | |
| y = f(x − 3) | shifts right 3 | backwards! inside the parentheses, − shifts right, + shifts left |
| y = f(x + 3) | shifts left 3 | |
| y = 2·f(x) | vertical stretch ×2 (reads narrower) | heights scale; points on the x-axis don't move |
| y = 12·f(x) | compression to half (reads wider) | |
| y = −f(x) | reflects through the x-axis | a stretch by −1: every output changes sign |
| y = f(2·x) | squeezes to half the width | backwards again! it's inside the function — bigger b, narrower graph |
| y = f(12·x) | stretches twice as wide |
A dashed target appears. Tune your knobs until your curve lands exactly on it. Six rounds, from one knob to all three — no formulas to recall, just the feel you've built.
A function is a rule that assigns each input exactly one output. y = x/2 + 1 says: whatever x comes in, the output is half of it, plus 1. Same input, same output, every single time — that reliability is what makes a function drawable as a curve.
More: function (mathematics) →(x, y) is an address. Start at the origin (0, 0). x says how far right to go (negative means left); y says how far up (negative means down). The point lives where you stop.
More: coordinate system →An asymptote is a line a curve gets forever closer to but never touches. The graph can come within a hair of it — closer than any distance you name — yet never arrives. Asymptotes act like invisible walls and floors, and they move when you turn the h and k knobs.
More: asymptote →Sine is just the mathematician's name for the perfect wave — a curve that rises, falls, and repeats in identical cycles forever. It comes from trigonometry (you'll meet it properly there), but for today you can read sin(x) as "the function whose graph is a wave that comes and goes."
More: sine wave →Why does minus mean right? Ask: where does the new graph produce the output the old one made at x = 0? Set the inside to zero: x − 3 = 0, so x = 3. The output that lived at 0 now lives at 3 — and the same relocation happens to every point. In general, f(x − h) moves every output h units to the right.
More: transformations of functions →