When you flex something, you bend or contract it. To appear super tough, you might flex your archery bow while flexing your muscles.
Things curve and bend when they flex, whether they're human joints or objects made of a bendable material like plastic or soft wood. Another way to flex is to tense or contract a muscle: "The weightlifter flexes her biceps." Experts suspect that flexible came first, from the Latin root flexibilis, "flexible, pliant, or yielding," and figuratively, "tractable, or easily influenced."