Concavity and convexity

Besides “rising” or “falling”, the graph of a function can curve “upwards” or “downwards”: this is the distinction between convexity and concavity.

Definition — Convex and concave

A function $f$ is convex on an interval $I$ if, taking two points $A$ and $B$ of its graph in $I$, the segment $AB$ lies above or on the graph (“belly up”, $\cup$). It is concave if instead the segment $AB$ lies below or on the graph (“belly down”, $\cap$).

Remark

Parabolas with $a > 0$ are convex on the whole of $\mathbb{R}$; those with $a < 0$ are concave. The function $y = |x|$ is convex on $\mathbb{R}$. The homographic function $y = 1/x$ is convex on $(0;+\infty)$ and concave on $(-\infty;0)$. The change of behaviour is an inflection point, a concept we shall return to in Year Four with derivatives.

Links

Topics: Functions and properties
Concepts: Concavity · Convexity · Inflection point
Functions: Homographic function · Parabola · Absolute value
Skills: Interpreting a graph