F.4 Mathematics

Sub (1, -3/2), (2, 0) and (4, 0) into the graph function to get 3 equations
-3/2 = a(1 - h)^2 + k ... [1]
0 = a(2 - h)^2 + k ... [2]
0 = a(4 - h)^2 + k ....[3]
[3] - [2], a(4 - h)^2 - a(2 - h)^2 = 0
(16 - 8h + h^2) - (4 - 4h + h^2) = 0
12 - 4h = 0, h = 12/4 = 3
From [1], 4a + k = -3/2 ... [4]
From [2] or [3], a + k = 0 ... [5]
[4] - [5], 3a = -3/2, a = -1/2
Put a = -1/2 into [5], k = 1/2
Therefore, a = -1/2, h = 3 and k = 1/2, and the graph function is
y = -1/2 * (x - 3)^2 + 1/2
Check:
For (1, -3/2), LHS = -3/2, RHS = -1/2 * (1 - 3)^2 + 1/2 = -3/2
For (2, 0), LHS = 0, RHS = -1/2 * (2 - 3)^2 + 1/2 = 0
For (4, 0), LHS = 0, RHS = -1/2 * (4 - 3)^2 + 1/2 = 0