Vertex form of a parabola, the normal at its y-intercept, and the length of the chord cut off by the normal

Part of the graph of a quadratic function \(\mathrm{f}\) is shown in the diagram below. The minimum point of the curve is \((1,-4)\), and the curve meets the \(y\)-axis at the point \(P\).

1. State the equation of the line of symmetry of the curve.[1]

The function can be expressed as \(\mathrm{f}(x)=\frac12(x-h)^2+k\), where \(h\) and \(k\) are integers.

2. Write down \(h\) and \(k\).[2]
3. Find the \(y\)-coordinate of \(P\).[2]

The line \(L\) is the normal to the curve at \(P\), as shown in the diagram below.

4. Find an equation of \(L\) in the form \(y=mx+c\).[4]

The normal \(L\) meets the curve again at the point \(Q\).

5. Calculate the exact length of \(PQ\).[8]