Answer:
p = -4
q = 9
Explanation:
The vertex form of a parabola is:
y = a (x − h)² + k
Given that the leading coefficient is 1 and the vertex is (2,5):
y = (x − 2)² + 5
Distributing:
y = x² − 4x + 4 + 5
y = x² − 4x + 9
Alternatively, using calculus, if y = x² + px + q, then the derivative is:
dy/dx = 2x + p
The turning point occurs when the derivative is 0:
0 = 2x + p
0 = 2(2) + p
p = -4
Plugging in and solving for q:
y = x² − 4x + q
5 = (2)² − 4(2) + q
5 = 4 − 8 + q
q = 9