发挥 fāhuī to display

to display; to exhibit; to bring out implicit or innate qualities; to express (a thought or moral); to develop (an idea); to elaborate (on a theme)

How to remember 发挥: 发 send out + 挥 to direct = to display.

Characters

Pairs with 作用 and 水平: 发挥作用 (fāhuī zuòyòng) - play a role/have an effect, 正常发挥 (zhèngcháng fāhuī) - perform up to one's usual level.
Central in sports and exam talk: commentators and students both say 发挥得好/不好 (fāhuī de hǎo/bù hǎo) about how well someone performed.
Not "invent" or "show off an object" - it means bringing out ability or potential that already exists, distinct from 表现 (biǎoxiàn), which covers outward behavior/demonstration more generally.