Let \(0< k <2\). Below is sketched a graph of \(y=f_k(x)\) where \(f_k(x) = x(x-k)(x-2)\). Let \(A(k)\) denote the area of the shaded region.

- Without evaluating them, write down an expression for \(A(k)\) in terms of two integrals.

How is the value of an integral affected if a function gives negative values (is below the \(x\)-axis)?

- Explain why \(A(k)\) is a polynomial in \(k\) of degree \(4\) or less. [You are not required to calculate \(A(k)\) explicitly.]

What is the highest of power of \(k\) we can get by integrating \(f\), with the limits given?

- Verify that \(f_k (1+t) = -f_{2-k}(1-t)\) for any \(t\).

- How can the graph of \(y=f_k(x)\) be transformed to the graph of \(y = f_{2-k}(x)\)?

What happens if we use part (iii), replacing \(1+t\) with \(x\)?

Deduce that \(A(k)=A(2-k)\).

Can we compare \(f_{2-k}(x)\) to \(f_k (x)\)? What do the transformations from part iv) do to the area \(A(k)\)?

- Explain why there are constants \(a, b, c\) such that \[A(k) = a(k-1)^4 + b(k-1)^2 + c.\] [You are not required to calculate \(a,b,c\) explicitly.]

Can we show \(A(1+k) = A(1-k)\)? What implications for symmetry does this have?