(Diagram and setup from UCSMP Precaluclus and Discrete Mathematics, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Are the curves in the locus some specific (named) type of curve? Is there a simple way to describe the curves (equations)?
The question need not be limited to the specific polynomial given--a similar sort of locus is generated by the zeros of nearly any quartic polynomial as the constant term is varied.