4.

To reflect a point \((x, f(x))\) about the origin, we transform it to the point \((-x, -f(x))\).

Starting with the given points:

\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -4 & 8 \ -3 & 2 \ -2 & 0 \ -1 & 2 \ 0 & 8 \ \hline \end{array} \]

We need to find \( h(x) \) for corresponding positive \( x \) values:

1. For \( x = 4 \) (which reflects to \( -4 \)): \( h(4) = -f(-4) = -8 \)

2. For \( x = 3 \) (which reflects to \( -3 \)): \( h(3) = -f(-3) = -2 \)

3. For \( x = 2 \) (which reflects to \( -2 \)): \( h(2) = -f(-2) = -0 = 0 \)

4. For \( x = 1 \) (which reflects to \( -1 \)): \( h(1) = -f(-1) = -2 \)

5. For \( x = 0 \): Since reflecting at zero keeps it the same, \( h(0) = -f(0) = -8 \)

Now we can populate the table:

\[ \begin{array}{|c|c|} \hline x & h(x) \ \hline 4 & -8 \ 3 & -2 \ 2 & 0 \ 1 & -2 \ 0 & -8 \ \hline \end{array} \]

Thus, the completed table showing the reflection about the origin is:

\[ \begin{array}{|c|c|} \hline x & h(x) \ \hline 4 & -8 \ 3 & -2 \ 2 & 0 \ 1 & -2 \ 0 & -8 \ \hline \end{array} \]