Whenever teaching or discussing parabolas, conic sections and other curves with my students, I always feel dissatisfied with the standard "find vertex, pick points, connect the dots" method to draw a parabola. This also applies to higher degree curves, using local max/min, inflection points, etc. I feel like it is equivalent to guess-and-check, which definitely has it's uses, but I would really like to give them more of a complete and continuous way to create parabolas and any other curves. One thing I was thinking was to have students to essentially perform the mathematics behind Bezier curves by marking control points, drawing in the scaffold, subdividing, etc until they arrived at a reasonable construction. I'm just not sure exactly how to go about doing this with regards to specific curves or if this would even be the best way.
Does anyone have any experience, insight, or resources on constructing and sketching parabolas, conic sections, or other curves that is different from just picking points and connecting the dots?