a) Vertex at (0.5, -12.5), roots at (-2, 0) and (p, 0).
Axis of symmetry: \( x = 0.5 \).
Distance from vertex to \( x = -2 \): \( 0.5 – (-2) = 2.5 \).
Other root symmetric about \( x = 0.5 \): \( p = 0.5 + 2.5 = 3 \).
Result: \( p = 3 \). [2]

b) Function: \( f(x) = ax^2 + bx + c \).
Use roots: \( f(-2) = 0 \): \( 4a – 2b + c = 0 \) (1).
Use root \( p = 3 \): \( f(3) = 0 \): \( 9a + 3b + c = 0 \) (2).
Use vertex: \( f(0.5) = -12.5 \): \( \frac{1}{4}a + \frac{1}{2}b + c = -12.5 \) (3).
Subtract (1) from (2): \( 5a + 5b = 0 \implies a + b = 0 \implies b = -a \).
Substitute into (1): \( 4a – 2(-a) + c = 6a + c = 0 \implies c = -6a \).
Substitute into (3): \( \frac{1}{4}a + \frac{1}{2}(-a) – 6a = -12.5 \implies -\frac{1}{4}a – 6a = -12.5 \implies -\frac{25}{4}a = -12.5 \).
Solve: \( a = 2 \).
Then: \( b = -2 \), \( c = -6 \cdot 2 = -12 \).
Results: (i) \( a = 2 \), (ii) \( b = -2 \), (iii) \( c = -12 \). [3]

c) Axis of symmetry: \( x = -\frac{b}{2a} \).
From (b): \( a = 2 \), \( b = -2 \).
Calculate: \( -\frac{-2}{2 \cdot 2} = \frac{2}{4} = 0.5 \).
Result: \( x = 0.5 \). [2]