The transformation $f(x) \mapsto f(x+8) - 7$ involves two steps applied to the graph of the original function $f(x)$:

1. Horizontal Shift: The term $f(x + 8)$ indicates a shift of the graph of $f(x)$ 8 units to the left. This is because adding 8 inside the function argument $x$ moves the graph in the opposite direction along the x-axis.

2. Vertical Shift: The term $-7$ indicates a shift of the graph 7 units down. This is a direct translation along the y-axis.

In summary, the transformation $f(x) \mapsto f(x+8) - 7$ shifts the graph of $f(x)$ 8 units to the left and 7 units down.

Would you like further details or have any questions on this transformation?

Related Questions:

1. What would the transformation $f(x) \mapsto f(x - 3) + 5$ do to the graph?
2. How does the graph change with the transformation $f(x) \mapsto -f(x)$?
3. What effect does multiplying $f(x)$ by a constant have on its graph?
4. How would the graph of $f(x) \mapsto f(2x)$ be affected?
5. How does the transformation $f(x) \mapsto f(x) + c$ generally affect the graph of $f(x)$?

Tip: When dealing with transformations, always remember that changes inside the function argument $f(x + a)$ affect horizontal shifts, and changes outside, like $f(x) + b$, affect vertical shifts.