Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the $x$-values at which they occur.
\[
f(x)=1-2 x-4 x^{2} ;[-4,4]
\]

Step 1 ：The function given is \(f(x)=1-2 x-4 x^{2}\) and we are asked to find the absolute maximum and minimum values of the function over the interval \([-4,4]\).

Step 2 ：The function is a quadratic function, and its graph is a parabola. The maximum or minimum value of the function occurs at the vertex of the parabola.

Step 3 ：The x-coordinate of the vertex can be found using the formula \(-b/2a\) where \(a\) and \(b\) are the coefficients of \(x^2\) and \(x\) respectively. In this case, \(a=-4\) and \(b=-2\). So, the x-coordinate of the vertex is \(-(-2)/2*(-4)=-1/4\).

Step 4 ：We also need to check the values of the function at the endpoints of the interval, which are \(-4\) and \(4\).

Step 5 ：By substituting these values into the function, we find that the function values at the endpoints are \(-55\) and \(-71\) respectively.

Step 6 ：The maximum value of the function over the interval \([-4,4]\) is \(1.25\) and it occurs at \(x=-0.25\). The minimum value of the function over the interval \([-4,4]\) is \(-71\) and it occurs at \(x=4\).

Step 7 ：Final Answer: The absolute maximum value of the function over the interval \([-4,4]\) is \(\boxed{1.25}\) at \(x=\boxed{-0.25}\) and the absolute minimum value is \(\boxed{-71}\) at \(x=\boxed{4}\).