The gradient function of \(y = ax^{2} + bx + c\) is \(8x + 4\).

Explanation

\(y = ax^{2} + bx + c\)

Gradient = \(\frac{\mathrm d y}{\mathrm d x} = 2ax + b = 8x + 4\)

Equating, we have

\(2a = 8 \implies a = 4\)

\(b = 4\)

For minimum value, gradient = 0

\(8x + 4 = 0 \implies x = -\frac{1}{2}\)

At \(x = -\frac{1}{2}, y = 1\)

\(1 = 4(-\frac{1}{2})^{2} + 4(-\frac{1}{2}) + c\)

\(1 = 1 - 2 + c\)

\(1 = c - 1 \implies c = 2\)

\(a, b, c = 4, 4, 2\).