Evaluate students' ability to analyze how coefficients a, b, and c in y = ax^2 + bx + c affect a parabola’s direction and width, determine the vertex and axis of symmetry (using the vertex formula or completing the square), and predict the nature and roots (using the discriminant and factorization). Tasks require calculating vertex and axis, finding or classifying roots, predicting graph changes when each coefficient is varied, and justifying conclusions with algebraic work and sketches.