To reflect on the origin of the function \( f(x) = (x+2)^2 + 1 \), we will first create a table of values for \( f(x) \) evaluated at \( x = 0, -1, \) and \( -2 \). We will then calculate \( -f(-x) \) for those values of \( x \).
Step 1: Calculate \( f(x) \) at specified values
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For \( x = 0 \): \[ f(0) = (0 + 2)^2 + 1 = (2)^2 + 1 = 4 + 1 = 5 \]
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For \( x = -1 \): \[ f(-1) = (-1 + 2)^2 + 1 = (1)^2 + 1 = 1 + 1 = 2 \]
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For \( x = -2 \): \[ f(-2) = (-2 + 2)^2 + 1 = (0)^2 + 1 = 0 + 1 = 1 \]
Step 2: Calculate \( -f(-x) \)
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For \( x = 0 \): \[ -f(-0) = -f(0) = -5 \]
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For \( x = -1 \): \[ -f(-(-1)) = -f(1) \] Calculate \( f(1) \): \[ f(1) = (1 + 2)^2 + 1 = (3)^2 + 1 = 9 + 1 = 10 \] Therefore, \[ -f(1) = -10 \]
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For \( x = -2 \): \[ -f(-(-2)) = -f(2) \] Calculate \( f(2) \): \[ f(2) = (2 + 2)^2 + 1 = (4)^2 + 1 = 16 + 1 = 17 \] Therefore, \[ -f(2) = -17 \]
Step 3: Create the table
Now, we can organize our findings in a table format:
| \( x \) | 0 | -1 | -2 | |-----------|-------|------|------| | \( f(x) \) | 5 | 2 | 1 | | \( -f(-x) \) | -5 | -10 | -17 |
This table summarizes the reflection of the function \( f(x) \) with respect to the origin, showing both \( f(x) \) and its negative reflection \( -f(-x) \).