Given the function \(f(x) = x^2 + ax + b\) with roots \(x = 1\) and \(x = 9\), we can express it as:

\(f(x) = (x - 1)(x - 9)\)

Expanding this, we get:

\(f(x) = x^2 - 10x + 9\)

Thus, \(a = -10\) and \(b = 9\).

To find the vertex, use the vertex formula \(x = \frac{-b}{2a}\) or symmetry:

\(x = \frac{1 + 9}{2} = 5\)

Substitute \(x = 5\) into \(f(x)\):

\(f(5) = 5^2 - 10 \times 5 + 9 = -16\)

Thus, the vertex is \((5, -16)\).