To analyze the graphs of \( f(x) \) and \( g(x) \) based on the given tables:
Table for \( f(x) \): \[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -2 & -6 \ 0 & -4 \ 2 & -2 \ 4 & 0 \ \hline \end{array} \]
Table for \( g(x) \): \[ \begin{array}{|c|c|} \hline x & g(x) \ \hline -2 & 6 \ 0 & 4 \ 2 & 2 \ 4 & 0 \ \hline \end{array} \]
Examining the Options
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Reflection over the x-axis: To be a reflection over the x-axis, \( g(x) \) should equal \( -f(x) \).
- For \( x = -2\): \( g(-2) = 6 \) and \( -f(-2) = 6 \) (check).
- For \( x = 0\): \( g(0) = 4 \) and \( -f(0) = 4 \) (check).
- For \( x = 2\): \( g(2) = 2 \) and \( -f(2) = 2 \) (check).
- For \( x = 4\): \( g(4) = 0 \) and \( -f(4) = 0 \) (check).
Since all conditions hold, \( g(x) = -f(x) \), meaning that one graph is a reflection of the other over the x-axis.
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Reflection over the y-axis: To be a reflection over the y-axis, we need \( g(x) \) to equal \( f(-x) \).
- For \( x = -2\): \( g(-2) = 6 \) and \( f(2) = -2 \) (not equal).
- For the other values, the condition does not hold either. Therefore, this is not true.
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Reflection about the origin: To reflect about the origin, \( g(x) \) should equal \( -f(-x) \).
- For \( x = -2\): \( g(-2) = 6 \) and \( -f(2) = 2 \) (not equal).
- Similarly, this does not hold for other values either. Thus, this is also not true.
Conclusion
The only true description from the options provided is:
Option #1: One graph is a reflection of the other over the x-axis.