We consider a multicarrier chirp-based waveform for joint radar and communication (JRC) systems and derive its time discrete periodic ambiguity function (AF). An advantage of the waveform is that it includes a set of waveform parameters (e.g., chirp rate) which together with the transmit sequence, can be selected to flexibly shape the AF to be thumbtack-like, or to be ridge-like, either along the delay axis or the Doppler axis. These shapes are applicable for different use cases, e.g., target detection or timeand frequency synchronization. The results show that better signal detection performance than OFDM and DFT-s-OFDM can be achieved on channels with large Doppler frequency. Furthermore, it is shown how transmit sequences can be selected in order to achieve 0 dB peak-to-average-power-ratio (PAPR) of the waveform.