Assess students' ability to analyze how variations in the coefficients a, b, and c of f(x)=ax^2+bx+c affect a parabola’s orientation and width, the vertex coordinates, axis of symmetry, and the roots (using the discriminant). Include tasks that require calculating and interpreting the vertex and axis, determining root type and values, predicting graph changes after scaling or sign changes, and explaining results both algebraically and with sketches. Appropriate for formative or summative assessment at the high-school level.