  The purpose of this note is give some evidence in support of conjectures of
Poonen, and Morton and Silverman, on the periods of rational numbers under the
iteration of quadratic polynomials. In particular, Poonen conjectured that
there are at most 9 periodic points defined over the rational numbers for any
map in the family x^2 + c for c rational. We verify this conjecture for c
values up to height 10^8. For quadratic number fields, we provide evidence that
the upper bound on the exact period of Q-rational periodic point is 6.
