We prove that the dynatomic curves associated with iteration of zd+c in any fixed characteristic (not dividing d) have gonalities tending to infinity. This implies a uniform upper bound on the number of L-rational preperiodic points of zd+c as L varies over extensions of bounded degree over a fixed function field K and c varies over nonconstant elements of L. It also reduces the strong uniform boundedness conjecture for preperiodic points over number fields to the conjecture for periodic points. This is joint work with John R. Doyle.