Suppose we want to make a quadratic polynomial y = f(x) = a0 + a1x + a2x
2
that passes through three specified points p1 = (p1, q1), p2 = (p2, q2), p3 = (p3, q3). Consider the
equation
0 = det




1 x x2 y
1 p1 p
2
1
q1
1 p2 p
2
2
q2
1 p3 p
2
3
q3




.
The determinant above implicitly gives an equation y = f(x) (it’s easy to solve for y since no y2, y3,etc terms appear).
(a) Write out the matrix above, using p1 = (0, 0), p2 = (1, 1), p3 = (3, 5) for the constants pi, qi, but leaving x and y as variables. Solve the equation det(A) = 0 to get y = a quadratic polynomial in x. Check directly that the parabola passes through p1, p2, p3.