The transformation \( f(x) \mapsto f(x + 7) - 7 \) can be analyzed in two parts:
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Horizontal Shift: The expression \( f(x + 7) \) represents a horizontal shift of the graph of \( f(x) \) to the left by 7 units. This happens because for any input \( x \), the function evaluates at \( x + 7 \), which means you're taking the value of \( f \) at a point that is 7 units to the left of \( x \).
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Vertical Shift: The expression \( f(x + 7) - 7 \) indicates that after applying the horizontal shift, the entire graph is then shifted downward by 7 units. The subtraction of 7 results in a vertical translation of every point on the graph of \( f(x + 7) \) by 7 units down.
Combining these two transformations, the overall effect of the transformation \( f(x) \mapsto f(x + 7) - 7 \) is to shift the graph of \( f(x) \) left by 7 units and then down by 7 units.
In summary, the transformation results in:
- A shift left by 7 units,
- Followed by a shift down by 7 units.
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