Assess students' ability to analyze how coefficients a, b, and c in f(x)=ax^2+bx+c determine the vertex (including use of -b/2a or completing the square), axis of symmetry, discriminant and resulting nature/number of roots (using the quadratic formula), y-intercept, and overall graph shape (opening direction and width). Include tasks requiring computation, interpretation of parameter changes, justification of root behavior from the discriminant, and sketching or describing resulting parabola transformations.