Turn up the volume on a song and its sound wave stretches taller; nudge a character across a video game and its whole path slides sideways. Animators, sound engineers and app designers reshape curves like this constantly — moving, stretching and flipping one basic shape instead of rebuilding it from scratch.

Learning graphs one point at a time would take forever. The clever shortcut is this: once you know the shape of a single curve — say y = x^2 or y = \sin x — you can predict the graph of a whole family of related curves just by shifting, stretching, or reflecting the one you already know.

One known shape turns into hundreds. You never re-plot a point; you just move the picture around. A transformation is a rule that takes the graph of y = f(x) and rearranges it in a predictable way.

The six standard moves

Every transformation you meet is one of these six, applied to y = f(x):

• y = f(x) + a — shift up by a.
• y = f(x + a) — shift left by a (the counter-intuitive one — a plus moves it left).
• y = a\,f(x) — stretch vertically by factor a (taller if a > 1).
• y = f(ax) — stretch horizontally by factor \tfrac{1}{a} (so a > 1 actually squashes it).
• y = -f(x) — reflect in the x-axis (flip upside down).
• y = f(-x) — reflect in the y-axis (flip left-to-right).

There is one rule that ties them all together, and it is worth carving into your memory: a change outside the bracket affects y and does the obvious thing; a change inside the bracket affects x and does the opposite of what you expect.

Shift it yourself

The faint dashed curve is the base parabola y = x^2. Pull the sliders to move it: h slides it right (inside the bracket) and k slides it up (outside). Together they make y = (x - h)^2 + k. Watch how the inside h and the outside k behave differently.

Worked examples

1. Describe how y = (x - 3)^2 + 2 transforms y = x^2.

The (x - 3) is inside the bracket, so it acts on x — and inside is backward, so the minus 3 shifts the curve right by 3. The +2 is outside, so it does the obvious thing: up by 2. Net move: right 3, up 2. The vertex slides from (0, 0) to (3, 2).

2. Describe how y = -x^2 transforms y = x^2.

The minus sits outside, multiplying the whole function, so it is y = -f(x): a reflection in the x-axis. The upward smile becomes a downward frown; every y-value flips sign.

3. Combine two moves: sketch y = 2(x + 1)^2 from y = x^2.

Take them one at a time. The (x + 1) is inside → backward → shift left by 1. The \times 2 is outside → obvious → vertical stretch by factor 2 (twice as steep). So the parabola moves one step left and is pulled twice as tall. Its vertex ends up at (-1, 0).

Beyond parabolas: the wave family

The best thing about transformations is that they don't care which curve you start from — the same six moves work on any known shape. Take the wave y = \sin x, which rolls up and down between -1 and 1. From that single wave you can read off a whole orchestra of related graphs:

• y = 3\sin x — outside, obvious → the wave gets three times taller (a bigger amplitude, a louder sound).
• y = \sin(2x) — inside, backward → the wave is squashed to half the width, so it wiggles twice as fast (a higher pitch).
• y = \sin x + 2 — outside → the whole wave lifts up by 2.

You never learned any of these graphs from scratch — you transformed the one wave you already knew. That is the entire trick, reused forever.

See it explained