[Assess students' ability to analyze how coefficients a, b, and c in y = ax^2 + bx + c affect the parabola's graph, including direction and width of opening, vertex coordinates (via completing the square and -b/2a), axis of symmetry, the discriminant's role in determining number and nature of real roots, and how systematic changes to coefficients transform the graph; include tasks requiring vertex calculation, discriminant evaluation, and interpretation of roots.]