Sketching Curves Using Stationary Points

A sketch shows the important features, not every calculated point. Find the intercepts, stationary points and direction between them. Keep any breaks in the domain.

For large $|x|$, the cubic term dominates the linear term. The left end goes down and the right end goes up.

Read the vertical label first. A zero of $f'$ gives a stationary input of $f$, not necessarily an intercept of $f$. Where the derivative graph is below its axis, the original function decreases.

The derivative does not give the original curve height. Curves differing only by an added constant have the same derivative. Use a supplied point or the original equation to locate the sketch vertically.

A derivative graph crosses its axis at $x=-2$ and $x=2$. It is positive outside these values and negative between them. The original curve passes through $(-2,16)$ and $(2,-16)$. Classify these points and describe the curve between them.