5.
$$f ( x ) = x ^ { 2 } - 8 x + 19$$
- Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
The curve \(C\) with equation \(y = \mathrm { f } ( x )\) crosses the \(y\)-axis at the point \(P\) and has a minimum point at the point \(Q\).
- Sketch the graph of \(C\) showing the coordinates of point \(P\) and the coordinates of point \(Q\).
- Find the distance \(P Q\), writing your answer as a simplified surd.