We teach conics as a family of curves and many use physical models to show how different cross-sections of double napped cones yield each of the four conic sections. Unfortunately, I haven't seen any sources show the same result from an algebraic perspective.
This construction graphs [math]ax^2+cy^2+dx+ey-1=0[/math] and allows users to determine the parameter conditions for a, c, d, and e under which each of the conics appears.
NOTE: This is construction is equivalent to the seemingly more general [math]Ax^2+Cy^2+Dx+Ey+F=0[/math], but diving both sides by -F gives an equation equivalent to the one graphed here.
