To complete the translation of the function \( y = f(x) \) to \( y = f(x + 5) \), we need to understand how the transformation affects the function's output.
The translation \( y = f(x + 5) \) means that we are shifting the graph of \( f(x) \) to the left by 5 units. Therefore, to find the corresponding \( y \)-values for the translated function, we need to replace \( x \) in \( f(x) \) with \( x + 5 \). This means for a value \( x' \) in the translated function, we find the original function's input that leads to this output by solving \( x' + 5 = x \) or \( x = x' + 5 \).
Using this, we will substitute for \( x' \) values to find the corresponding \( y = f(x + 5) \).
Here is the completed table:
Original Function
| x | y = f(x) | |----|----------| | -2 | 5 | | -1 | 2 | | 0 | 1 | | 1 | 2 | | 2 | 5 |
Translated Function
To find \( y = f(x + 5) \):
- For \( x = -2 \), \( y = f(-2 + 5) = f(3) \). Since we have no value for \( x = 3 \), this value cannot be determined here.
- For \( x = -1 \), \( y = f(-1 + 5) = f(4) \). Again, this isn't provided in the original table.
- For \( x = 0 \), \( y = f(0 + 5) = f(5) \). This is also not given.
- For \( x = 1 \), \( y = f(1 + 5) = f(6) \). Again, no value is provided for this.
- For \( x = 2 \), \( y = f(2 + 5) = f(7) \). No value to determine this.
Therefore, we can only fill those where we relate from the original function and keep their corresponding values based on their original inputs.
Using other values hypothetically if this was our need or representation from the other corresponding table values it appears you'd represent like the following without new values:
| x | y = f(x + 5) | |----|--------------| | -2 | (unknown) | | -1 | (unknown) | | 0 | (unknown) | | 1 | (unknown) | | 2 | (unknown) |
If you would like specific assumptions, like continued pattern repeating to fill direct corresponding values, that can be helped with a static estimate or example that fits a segment.
Let me know how you would like to extend or clarify any further captures needed.
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