Sketching curves with stationary points

A typical question: Sketch the curve $y = x^3 - 6x^2 + 9x$, showing the coordinates of the stationary points and where the curve meets the axes.

Giveaway words: sketch, showing the coordinates, where the curve meets the axes.

A sketch is not an accurate graph. It needs the right shape and the key points labelled, nothing more.

The same four steps every time:

1. Stationary points: find them and their nature (the five steps from the last lesson).
2. Axes: put $x = 0$ to see where it meets the $y$-axis. Solve $y = 0$, by factorising, for the $x$-axis.
3. Ends: use the highest power of $x$ to see where the curve goes on the far left and the far right.
4. Join: mark the points, then draw one smooth curve that turns only at the stationary points. Label the coordinates.

For large $x$ (positive or negative), the highest power wins. Only the sign of the number in front matters: $3x^3$ behaves like $+x^3$.

In symbols, "as $x \to \infty$, $y \to -\infty$" means: on the far right, the curve heads down forever. ($\infty$ is "infinity", and $\to$ is "heads towards".)

Why does the highest power win? Try $y = x^3 - 6x^2$ at $x = 100$: $x^3 = 1\,000\,000$ but $6x^2$ is only $60\,000$. The biggest power swamps the rest.