Bot GPT 4-o mini Part A: When the function \( f(x) = (x-1)^2 \) is reflected over the x-axis, the new function becomes \( f(x) = -(x-1)^2 \).
Part B: To complete the table for the reflection over the x-axis of the function \( f(x) = (x-1)^2 \), we first calculate the original values of \( f(x) \) for the given \( x \) values and then negate those values for the reflected function.
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Calculate \( f(x) \) for the original function:
- For \( x = 0 \): \( f(0) = (0-1)^2 = 1 \)
- For \( x = 2 \): \( f(2) = (2-1)^2 = 1 \)
- For \( x = 4 \): \( f(4) = (4-1)^2 = 9 \)
- For \( x = 6 \): \( f(6) = (6-1)^2 = 25 \)
- For \( x = 8 \): \( f(8) = (8-1)^2 = 49 \)
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Now reflect over the x-axis by negating these values:
- For \( x = 0 \): \( f(0) = -1 \)
- For \( x = 2 \): \( f(2) = -1 \)
- For \( x = 4 \): \( f(4) = -9 \)
- For \( x = 6 \): \( f(6) = -25 \)
- For \( x = 8 \): \( f(8) = -49 \)
The completed table is as follows:
| x | 0 | 2 | 4 | 6 | 8 | |-----|------|------|------|------|------| | f(x)| -1 | -1 | -9 | -25 | -49 |
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