We have investigated the optical output of the free-electron laser for infrared experiments (FELIX) when it is driven by an electron beam with a ramped energy. We show that the applied slow ramp on the electron beam energy leads to a frequency chirp on each picosecond optical pulse. Typical values for the chirp are 0.2% frequency sweep across a 1.5-ps-long optical pulse. The optical pulses were analyzed with a double-grating pair and with a second-order autocorrelator. The pulse duration was reduced in the double-grating pair by 20%. A linear dependence of the chirp on the cavity desynchronization was measured.