To analyze the function \( f(x) = (x + 2)^2 + 1 \) and create a table of values for \( f(x) \) evaluated at \( x = 0, -1, \) and \( -2 \), we first calculate the function values for these \( x \) values.
Step 1: Calculate \( f(x) \) for the given \( x \) 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: Create the table of values for \( f(x) \)
Now we can summarize these results in a table:
| x | f(x) | |-----|------| | 0 | 5 | | -1 | 2 | | -2 | 1 |
Step 3: Calculate \( -f(-x) \) for the same \( x \) values
Next, we will compute \( f(-x) \) for \( x = 0, -1, \) and \( -2 \):
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For \( x = 0 \): \[ f(-0) = f(0) = 5 \implies -f(-0) = -5 \]
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For \( x = -1 \): \[ f(-(-1)) = f(1) = (1 + 2)^2 + 1 = (3)^2 + 1 = 9 + 1 = 10 \implies -f(-(-1)) = -10 \]
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For \( x = -2 \): \[ f(-(-2)) = f(2) = (2 + 2)^2 + 1 = (4)^2 + 1 = 16 + 1 = 17 \implies -f(-(-2)) = -17 \]
Step 4: Create the table of values for \( -f(-x) \)
Here is the summary of these calculations:
| x | -f(-x) | |-----|--------| | 0 | -5 | | -1 | -10 | | -2 | -17 |
Final Summary
Here are the final tables summarizing \( f(x) \) and \( -f(-x) \):
Table of \( f(x) \)
| x | f(x) | |-----|------| | 0 | 5 | | -1 | 2 | | -2 | 1 |
Table of \( -f(-x) \)
| x | -f(-x) | |-----|--------| | 0 | -5 | | -1 | -10 | | -2 | -17 |
Using these tables, you can analyze the symmetry of the function and its reflection properties!