We look at a familiar one-parameter family of quadratic functions on the complex
plane. After restricting the parameter to be real, we explore when the critical points
of the functions and their iterates are real and when they are not real. We prove
that when the parameter is greater than or equal to 2, all critical points are
real. When the parameter is between 0 and 2, critical points for the original
function are real but there is an iterate with nonreal critical points. When the
parameter is equal to 0, all critical points are 0. When the parameter is less
than 0, the critical points are all 0 or are nonreal. Finally, we compare the
locations of these critical points to the contour plots of the real parts of the
functions for different values of the parameter and for different iterates of the
function.
