To determine which table represents the function \( g(x) = -f(-x) \) resulting in a reflection about the origin of the function graphed, we first must analyze how to derive \( g(x) \) based on the points from the parabola given by \( f(x) = 2(x+22) \).
In order to reflect a point about the origin, if we have a point \( (x, f(x)) \), the reflected point will be \( (-x, -f(x)) \).
Let's assume we have the following points from the graph of \( f(x) \):
- For \( x = 1 \), \( f(1) = 2 \), so the reflected point is \( (-1, -2) \).
- For \( x = 2 \), \( f(2) = 0 \), so the reflected point is \( (-2, 0) \).
- For \( x = 3 \), \( f(3) = 2 \), so the reflected point is \( (-3, -2) \).
Now, organizing these reflected points for \( g(x) = -f(-x) \):
- When \( x = 1 \), \( g(1) = -f(-1) \) leads to \( (-1, -2) \).
- When \( x = 2 \), \( g(2) = -f(-2) \) leads to \( (-2, 0) \).
- When \( x = 3 \), \( g(3) = -f(-3) \) leads to \( (-3, -2) \).
Now, we can write these points down in terms of \( g(x) \):
- For \( x = -1 \), \( g(-1) = 2 \) (from the reflected point)
- For \( x = -2 \), \( g(-2) = 0 \) (from the reflected point)
- For \( x = -3 \), \( g(-3) = 2 \) (from the reflected point)
Based on the provided choices, the corresponding table for the function \( g(x) = -f(-x) \) should have:
- \( g(1) = -2 \)
- \( g(2) = 0 \)
- \( g(3) = -2 \)
The correct table that reflects these points would be:
\[ \begin{array}{c|c} x & g(x) \ \hline 1 & -2 \ 2 & 0 \ 3 & -2 \ \end{array} \]
This matches the table shown in your provided options:
- \( x \) \( g(x) \)
- 1 -2
- 2 0
- 3 -2
Therefore, that would be the answer.
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