To find the reflection of the function \( f(x) = (x - 1)^2 \) over the x-axis, we need to negate the output of the function. That is, the reflected function \( g(x) \) will be \( g(x) = -f(x) \).
Let's calculate \( f(x) \) for the given x-values and then determine the reflected values \( g(x) \):
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For \( x = 0 \): \[ f(0) = (0 - 1)^2 = (-1)^2 = 1 \] Therefore, \[ g(0) = -f(0) = -1 \]
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For \( x = 1 \): \[ f(1) = (1 - 1)^2 = 0^2 = 0 \] Therefore, \[ g(1) = -f(1) = -0 = 0 \]
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For \( x = 2 \): \[ f(2) = (2 - 1)^2 = 1^2 = 1 \] Therefore, \[ g(2) = -f(2) = -1 \]
Now we can fill in the table:
| \( x \) | \( -f(x) \) | |:------:|:------:| | 0 | -1 | | 1 | 0 | | 2 | -1 |
Thus, the completed table for the reflection over the x-axis is:
| \( x \) | \( -f(x) \) | |:------:|:------:| | 0 | -1 | | 1 | 0 | | 2 | -1 |
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