What is Variation table?

Example 3.10

Consider the function graphed below on $\intcc{-2}{4}$.

Reading the graph: $f$ increases from $f(-2) = -1.5$ up to its maximum $f(1) = 3$, then decreases down to $f(4) = -1.5$. Its variation table is

Example 3.11 (Proving a variation)

Show that $f(x) = 3x + 1$ is strictly increasing on $\R$. Take any $u < v$ and compare the images:

because $v - u > 0$. So $f(u) < f(v)$: the function is strictly increasing. The same computation with a negative slope, e.g. $g(x) = -2x + 5$, gives $g(v) - g(u) = -2(v-u) < 0$: strictly decreasing.