Definition, graph and graphical representation of the square function
Variations and graphical representation
The parabola
The graph of f(x) = x² on a coordinate plane is a curve called a parabola, with its vertex at the origin O(0, 0). It is symmetric about the y-axis, as f(-x) = f(x) for all real numbers x.
Table of values
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| x² | 4 | 1 | 0 | 1 | 4 |
Direction of variation
The square function is not monotonic over the entire real line: its behaviour changes at 0.
- On the interval ]–∞; 0], f is decreasing: if a ≤ b ≤ 0, then f(a) ≥ f(b).
- On the interval [0; +∞[, f is increasing: if 0 ≤ a ≤ b, then f(a) ≤ f(b).
We summarise this in a table of variations: f decreases from +∞ to 0 as x varies from -∞ to 0, then increases from 0 to +∞ as x varies from 0 to +∞.
Minimum of the function
The minimum of f on ℝ is 0, attained only at x = 0. The square function has no maximum: its values can become as large as one wishes.
Pitfall to avoid
Never say that “the square function is increasing on ℝ”: this is false. For example, f(-5) = 25 and f(-1) = 1, so f(-5) > f(-1) even though -5 < -1. You must always specify the interval over which you are discussing the variation.