To create a table for the reflection of the function \( f(x) = (x - 1)^2 \) over the x-axis, we first need to evaluate \( f(x) \) for the given \( x \) values and then take the negative values of those outputs, since reflecting a function over the x-axis involves negating the function's y-values.
Step 1: Calculate \( f(x) \)
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When \( x = 0 \): \[ f(0) = (0 - 1)^2 = 1^2 = 1 \]
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When \( x = 1 \): \[ f(1) = (1 - 1)^2 = 0^2 = 0 \]
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When \( x = 2 \): \[ f(2) = (2 - 1)^2 = 1^2 = 1 \]
Step 2: Calculate \( -f(x) \)
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For \( x = 0 \): \[ -f(0) = -1 \]
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For \( x = 1 \): \[ -f(1) = -0 = 0 \]
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For \( x = 2 \): \[ -f(2) = -1 \]
Final Table:
| x | 0 | 1 | 2 | |---|------|-----|------| | -f(x) | -1 | 0 | -1 |
This table shows the values of \( -f(x) \) for the given \( x \)-values, representing the reflection of the function \( f(x) = (x - 1)^2 \) over the x-axis.