The Doppler shift is a frequency change of a repetitive effect, as measured by a receiver, due to the motion of the wave emitter, to the motion of the wave receiver, or both. A demonstration of the effect is provided by the sound of a car's horn that changes from a higher pitch to a lower pitch when a car drives past. Most derivations of the Doppler formula assume sinusoidal waves. We want to call the attention of readers to a derivation that uses a graphical representation based on a graph of position versus time. We will adopt a particle model of machine gun bullets fired at a constant period, thus getting a derivation that just requires the knowledge of kinematics. Furthermore, we will explain how to introduce the time dilation in order to get the formula for the relativistic Doppler shift.