To complete the table for the function transformation \( f(x) = 3^x \) reflecting the transformation to \(-f(3x - 2) + 1\), we will analyze the transformations step by step.
-
Step 1: Evaluate \( f(x) = 3^x \) for \( x = 1, 2, 3 \):
- \( f(1) = 3^1 = 3 \)
- \( f(2) = 3^2 = 9 \)
- \( f(3) = 3^3 = 27 \)
-
Step 2: Evaluate \( f(3x - 2) \): For each \( x \) value:
- For \( x = 1 \): \( f(3(1) - 2) = f(1) = 3 \)
- For \( x = 2 \): \( f(3(2) - 2) = f(4) = 3^4 = 81 \)
- For \( x = 3 \): \( f(3(3) - 2) = f(7) = 3^7 = 2187 \)
-
Step 3: Evaluate \(-f(3x - 2)\):
- For \( x = 1 \): \(-f(3(1) - 2) = -3\)
- For \( x = 2 \): \(-f(3(2) - 2) = -81\)
- For \( x = 3 \): \(-f(3(3) - 2) = -2187\)
-
Step 4: Evaluate \(-f(3x - 2) + 1\):
- For \( x = 1 \): \(-3 + 1 = -2\)
- For \( x = 2 \): \(-81 + 1 = -80\)
- For \( x = 3 \): \(-2187 + 1 = -2186\)
-
Step 5: Complete the table: The final values will be filled in as per the transformations applied.
Here is what the completed table looks like:
| Step 1 | Step 2 | Step 3 | Step 4 | Step 5 | |--------|--------|--------|--------|------------------| | 3 | 3 | -3 | -2 | -2 | | 9 | 81 | -81 | -80 | -80 | | 27 | 2187 | -2187 | -2186 | -2186 |
So, the final column (Step 5) is the outcome of the transformation \( -f(3x - 2) + 1 \) for the given \( x \) values.