What does the transformation f(x)↦f(x+7)–7 do to the graph of f(x)?

Answer 1

The transformation $f(x)\mapsto f(x+7)-7$ can be analyzed in two parts:

1. 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 $x$.

2. 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.