\[ f(x)=-x^{2}+8 x+1 \] (a) Express $f$ in standard form. \[ f(x)= \]

Step 1 ：The given function is \(f(x) = -x^2 + 8x + 1\).

Step 2 ：The standard form of a quadratic function is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

Step 3 ：To express the given function in standard form, we need to complete the square.

Step 4 ：To complete the square, we need to find a value that when added and subtracted inside the square makes it a perfect square trinomial. This value is \(\left(\frac{b}{2a}\right)^2\), where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\). In this case, \(a = -1\) and \(b = 8\), so the value is \(\left(\frac{8}{2(-1)}\right)^2 = (-4)^2 = 16\).

Step 5 ：So, we rewrite the function as \(f(x) = -(x^2 - 8x + 16) + 16 + 1 = -(x-4)^2 + 17\).

Step 6 ：The expanded form of the function \(f(x) = -(x-4)^2 + 17\) is indeed \(-x^2 + 8x + 1\), which is the same as the original function. Therefore, the standard form of the function is correct.

Step 7 ：\(\boxed{f(x) = -(x-4)^2 + 17}\) is the standard form of the function.