Learn one curve, and you've learned a family.

This is $y = (x+1)(x-2)^2$. Nudge its equation and the whole graph slides, stretches or squashes — every time in a way you can predict. But first, read the curve itself.

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The brackets hand you the roots. $(x+1)$ is a single factor, so at $x = -1$ the curve cuts straight through the axis.

$(x-2)^2$ is squared: it never changes sign, so at $x = 2$ the curve comes down, touches the axis, and turns back up.

Keep the original as a dashed ghost and follow just two points: the maximum $M(0,\,4)$ and the touching point $T(2,\,0)$. Every transformation comes down to one question — where do they go?

Outside the bracket, acting on $y$: everything lifts by 2. $M$ goes to $(0,\,6)$, $T$ to $(2,\,2)$. Changes outside do exactly what they say.

Now the $+2$ is inside, acting on $x$ before $f$ sees it — and the curve moves left 2, not right. $M$ to $(-2,\,4)$, $T$ to $(0,\,0)$.

Multiply outside and every height doubles: $M$ climbs to $(0,\,8)$. But $T$ stays put — twice zero is still zero. A vertical stretch pins every point on the $x$-axis.

Multiply inside and it reverses once more: $x$-coordinates halve, squashing the curve towards the $y$-axis. $T$ slides in to $(1,\,0)$; $M$, sitting on the $y$-axis, doesn't move at all.

Two edits at once — inside $-2$ moves it right 2, outside $+1$ lifts it 1. Track the points and you can't go wrong: $M$ to $(2,\,5)$, $T$ to $(4,\,1)$.

Four moves. One rule: inside reverses.

Cubics and quartics, reciprocal graphs, where graphs meet, and twelve graded exam questions with full worked solutions are waiting.

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