The table of variations

What is the purpose of the variation table?

The variation table is a visual summary of a function’s behaviour across its domain. For each interval, it uses arrows to indicate whether the function is increasing or decreasing, and it specifies the extreme values (maximum or minimum) reached.

How to construct it

1. Identify the intervals where the function changes direction (these points are called local extrema).
2. On the first row, list the values of x that delimit these intervals, in ascending order.
3. On the second line, draw an arrow pointing upwards (increasing) or downwards (decreasing), and indicate the values of f(x) at the endpoints.

Example: f(x) = x² - 4x + 3

This function is decreasing on ]–∞; 2] and then increasing on [2; +∞[. Its minimum is f(2) = 4 – 8 + 3 = –1.

x	-∞	...	2	...	+∞
f(x)	+∞ (downward arrow)		-1		+∞ (upward arrow)

We can thus see directly that f has a minimum of -1, reached at x = 2.

Connection to the graph

Each upward arrow corresponds to a section of the curve that ‘rises’ when read from left to right, and each downward arrow corresponds to a section that ‘falls’. The point where the arrow changes direction corresponds to a vertex (trough or peak) of the curve.

Pitfall to avoid

Do not confuse the minimum of the function (a value of f(x)) with the point at which it is reached (a value of x). We write: “the minimum of f is -1, reached at x = 2”, and not “the minimum is 2”.