Question

Linear Algebra mixed with Calculus

Let T: P^2-->R^3 be the linear transformation given by T(p(x)) = , B={1,x, x^2} the standard basis for P^2 and B' = {e1, e2, e3} be the standard basis for R^3. Use the inverse of the matrix [T]_B,B' to find a quadratic polynomial passing through (-3, 75) and (5, 99) and whose tangent line at x=2 has a slope of 13.