To transform \( f(x) = 3x \) into \( -f(3x - 2) + 1 \), we will break it down into steps while filling in the provided table.
We start with \( f(x) = 3x \) and want to calculate \( -f(3x - 2) + 1 \).
Step 1: Evaluate \( f(3x - 2) \) \[ f(3x - 2) = 3(3x - 2) = 9x - 6 \]
Step 2: Apply the negative \[ -f(3x - 2) = -(9x - 6) = -9x + 6 \]
Step 3: Add 1 \[ -f(3x - 2) + 1 = -9x + 6 + 1 = -9x + 7 \]
Now, we'll fill out the table with the steps of transformation.
Table
| x | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 | |-------|--------------|-------------------|----------------|---------------------|----------------| | 1 | \( 3(3x - 2) \) | \( -9x + 6 \) | \( -9x + 7 \) | | | | 2 | \( 3(3x - 2) \) | \( -9x + 6 \) | \( -9x + 7 \) | | | | 3 | \( 3(3x - 2) \) | \( -9x + 6 \) | \( -9x + 7 \) | | |
Since none of the options exactly match our calculations or steps, here’s how we can fit the closest options to the steps:
Final Options
- Step 1: \( 3(3x - 2) \) could be represented as \( 3^{(3x - 2)} \).
- Step 2: Represents negative \( 9x + 6 \) could correspond to \( -3(3x - 2) + 1 \) or \( -9x + 6 \) (fitting into the options).
- Step 3: Taking \( -f(3x - 2) + 1 \) would result in \( -9x + 7 \) or correct fits.
To summarize, the correct order of operations based on the transformation is:
- Calculate \( f(3x - 2) = 3(3x - 2) \)
- Apply negative: \( -(3(3x - 2)) + 1 \)
- Final result: calculate \( -(9x - 6) + 1 \).
Given the nature of the options, it seems a more precise interpretation of the options should be determined since not all completely match the pathways created based on the transformations.