To show that \( h(x) \) is the reflection about the origin of the function \( f(x) \), we need to determine the values of \( h(x) \) based on the values of \( f(x) \).
A reflection about the origin means that for every point \( (x, f(x)) \) on the graph of \( f(x) \), there is a corresponding point \( (-x, -f(x)) \) on the graph of \( h(x) \). Thus, we can find \( h(x) \) by taking the negative of the corresponding \( f(x) \) values at the negative of the same \( x \) values.
Here are the given values of \( f(x) \) and the corresponding values of \( h(x) \):
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For \( f(0) = 0 \):
- \( h(0) = -f(0) = -0 = 0 \)
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For \( f(1) = 1 \):
- \( h(-1) = -f(1) = -1 \)
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For \( f(2) = 8 \):
- \( h(-2) = -f(2) = -8 \)
Thus, the completed table will look like this:
\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline 0 & 0 \ 1 & 1 \ 2 & 8 \ \hline \end{array} \]
\[ \begin{array}{|c|c|} \hline x & h(x) \ \hline 0 & 0 \ -1 & -1 \ -2 & -8 \ \hline \end{array} \]
So, the completed table shows that \( h(x) \) is indeed the reflection about the origin of \( f(x) \).