To create a table for the reflection over the x-axis of the function \( f(x) = (x-1)^2 \), we first need to understand that reflecting a function over the x-axis involves changing the sign of the output value \( f(x) \). Therefore, the reflected function can be written as \( g(x) = -f(x) \), which means:
\[ g(x) = -(x-1)^2 \]
Next, we'll create a table of values for both the original function \( f(x) \) and its reflection \( g(x) \).
Table of Values
| \( x \) | \( f(x) = (x-1)^2 \) | \( g(x) = -(x-1)^2 \) | |---------|-----------------------|------------------------| | -2 | 9 | -9 | | -1 | 4 | -4 | | 0 | 1 | -1 | | 1 | 0 | 0 | | 2 | 1 | -1 | | 3 | 4 | -4 | | 4 | 9 | -9 |
Explanation:
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For \( x = -2 \):
- \( f(-2) = (-2 - 1)^2 = 9 \)
- \( g(-2) = -9 \)
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For \( x = -1 \):
- \( f(-1) = (-1 - 1)^2 = 4 \)
- \( g(-1) = -4 \)
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For \( x = 0 \):
- \( f(0) = (0 - 1)^2 = 1 \)
- \( g(0) = -1 \)
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For \( x = 1 \):
- \( f(1) = (1 - 1)^2 = 0 \)
- \( g(1) = 0 \)
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For \( x = 2 \):
- \( f(2) = (2 - 1)^2 = 1 \)
- \( g(2) = -1 \)
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For \( x = 3 \):
- \( f(3) = (3 - 1)^2 = 4 \)
- \( g(3) = -4 \)
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For \( x = 4 \):
- \( f(4) = (4 - 1)^2 = 9 \)
- \( g(4) = -9 \)
This completes the table showing the values for the function and its reflection over the x-axis.