Assess learners' ability to analyze how coefficients a, b, and c in y = ax^2+bx+c (and its vertex form) determine a parabola's opening direction, width/steepness, vertex coordinates, axis of symmetry, and roots. Items should require using -b/(2a), completing the square to find the vertex, using the discriminant to predict the number and nature of real roots, and reasoning about how changing each coefficient shifts, reflects, or scales the graph. Include tasks that compare functions, predict graph changes from coefficient variation, and extract vertex, axis, and roots from equations and graphs.