In this chapter, the authors begin the study of the quadratic family of functions. While these functions look simple enough, their dynamics are amazingly complicated. Indeed, this behavior was only completely understood in the 1990’s. The authors introduce two of the most important types of bifurcations that occur in dynamics. To understand bifurcation behavior, it is often helpful to look at the bifurcation diagram. The authors also describe analytically the period-doubling bifurcation. Thus there are two typical cases for a period-doubling bifurcation. As the parameter changes, a fixed point may change from attracting to repelling and, at the same time, give birth to an attracting 2-cycle. Alternatively, the fixed point may change from repelling to attracting and, at the same time, give birth to a repelling cycle of period 2.