To reflect a function \( f(x) \) about the origin, we need to apply the transformation \( g(x) = -f(-x) \). This means that we will take each point \( (x, f(x)) \) on the graph of \( f(x) \), negate the x-coordinate (replace \( x \) with \( -x \)), and then negate the resulting y-coordinate (multiply \( f(-x) \) by \( -1 \)).
First, let's find a few key points from the original function \( f(x) = 2(x + 22) \).
- To find the points, we can choose some x-values:
- For \( x = -22 \): \( f(-22) = 2(-22 + 22) = 2(0) = 0 \) ➔ Point: \( (-22, 0) \)
- For \( x = -21 \): \( f(-21) = 2(-21 + 22) = 2(1) = 2 \) ➔ Point: \( (-21, 2) \)
- For \( x = -20 \): \( f(-20) = 2(-20 + 22) = 2(2) = 4 \) ➔ Point: \( (-20, 4) \)
Now, let's reflect these points about the origin to find corresponding points on the function \( g(x) = -f(-x) \):
- \( (x, f(x)) \) gives:
- \( (-22, 0) \) reflects to \( (22, 0) \)
- \( (-21, 2) \) reflects to \( (21, -2) \)
- \( (-20, 4) \) reflects to \( (20, -4) \)
Thus, the corresponding points on \( g(x) = -f(-x) \) will be:
- \( (22, 0) \)
- \( (21, -2) \)
- \( (20, -4) \)
Now we can summarize these into a table: | x | f(x) | g(x) | |-------|--------|--------| | -22 | 0 | 0 | | -21 | 2 | -2 | | -20 | 4 | -4 | | 20 | 0 | 0 | | 21 | -2 | -2 | | 22 | -4 | -4 |
This table isn't strictly necessary unless testing or comparing functions, but it demonstrates the transformation from \( f \) to \( g \) through point reflection about the origin.
Thus, the resulting table of coordinates for \( g(x) \) indicating a reflection about the origin from \( f(x) \) would ideally show: | x | g(x) | |-------|--------| | 20 | -4 | | 21 | -2 | | 22 | 0 |
Select this option or create similar observations if the function needs visual reinforcement.
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