  Given a Hopf algebra $H$ and a counital $2$-cocycle $\mu$ on $H$, Drinfeld
introduced a notion of twist which deforms an $H$-module algebra $A$ into a new
algebra $A_\mu$. We show that when $A$ is a quadratic algebra, and $H$ acts on
$A$ by degree-preserving endomorphisms, then the twist $A_\mu$ is also
quadratic. Furthermore, if $A$ is a Koszul algebra, then $A_\mu$ is a Koszul
algebra. As an application, we prove that the twist of the $q$-quantum plane by
the quasitriangular structure of the quantum enveloping algebra
$U_q(\mathfrak{sl}_2)$ is a quadratic algebra equal to the $q^{-1}$-quantum
plane.
