To complete the table for the translated function \( y = f(x + 5) \), we need to determine how the input values will change based on the translation. The translation \( y = f(x + 5) \) means that we take the original input \( x \) and subtract 5 to find the corresponding output of \( f \).
Let's find the new \( x \) values for each of the original \( y = f(x) \) pairs:
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For \( y = 5 \) (originally when \( x = -2 \)): \[ x + 5 = -2 \implies x = -2 - 5 = -7 \]
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For \( y = 2 \) (originally when \( x = -1 \)): \[ x + 5 = -1 \implies x = -1 - 5 = -6 \]
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For \( y = 1 \) (originally when \( x = 0 \)): \[ x + 5 = 0 \implies x = 0 - 5 = -5 \]
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For \( y = 2 \) (originally when \( x = 1 \)): \[ x + 5 = 1 \implies x = 1 - 5 = -4 \]
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For \( y = 5 \) (originally when \( x = 2 \)): \[ x + 5 = 2 \implies x = 2 - 5 = -3 \]
Now we can complete the translated function table:
| Translated function | x | y = f(x + 5) | |---------------------|-----|---------------| | | -7 | 5 | | | -6 | 2 | | | -5 | 1 | | | -4 | 2 | | | -3 | 5 |
So the completed table for the translated function \( y = f(x + 5) \) looks like this:
| Original function | x | y = f(x) | |---------------------|-----|----------| | | -2 | 5 | | | -1 | 2 | | | 0 | 1 | | | 1 | 2 | | | 2 | 5 |
| Translated function | x | y = f(x + 5) | |---------------------|-----|---------------| | | -7 | 5 | | | -6 | 2 | | | -5 | 1 | | | -4 | 2 | | | -3 | 5 |