Can I write the parametric equations for the graphs in the following case: on the x-axis, I want to plot a real number 'b'. On the y-axis, I want to plot the roots (all real roots) for x of the equation (7+b^{2})x^{3}+(6-b)x^{2}+9x-6=0. e.g. when b=1, I plot 1 on the x-axis and x=0.46124674 (the real root of the equation 8x^3+5x^2+9x-6=0) on the y-axis. (I chose the coefficients on the cubic pretty much arbitrarily but I want to see how it's done.)
I suggest that you plot a graph of the curves involved in that example and post it. Maybe that will inspire someone to comment on the general situation. I'm guessing that your are interested in the general question of whether the roots of a polynomial equation can be expressed as functions of the coefficients of the equation.