What is Variation table? — Mathematics

Definition 3.9 High School Mathematics · Chapter 3 — Functions

A variation table summarizes on which intervals $f$ increases and decreases, with arrows $\nearrow$ and $\searrow$ , and records the values of $f$ at the turning points.

Examples

Example 3.10

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

A function defined on [-2, 4], increasing then decreasing. — A function defined on $\intcc{-2}{4}$ , increasing then decreasing.

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

$x$	$-2$		$1$		$4$
$f$	$-1.5$	$\nearrow$	$3$	$\searrow$	$-1.5$

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:

f(v) - f(u) = (3v + 1) - (3u + 1) = 3(v - u) > 0,

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.

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Examples

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