Question

What does the transformation f(x)↦f(x+5)–2 do to the graph of f(x)?

Answer 1

The transformation \( f(x) \mapsto f(x+5) - 2 \) involves two operations on the graph of the function \( f(x) \).

1. **Horizontal Shift**: The term \( f(x + 5) \) indicates a horizontal shift of the graph. Specifically, replacing \( x \) with \( x + 5 \) moves the graph of \( f(x) \) to the left by 5 units. This is because for any given value of \( y \) in the original function, \( f(x) \) must now attain the same value at \( x - 5 \) instead of at \( x \).

2. **Vertical Shift**: The term \( -2 \) affects the vertical position of the graph. After shifting left by 5 units, the subtraction of 2 shifts the entire graph downward by 2 units. Consequently, every point on the graph is moved down by 2.

In summary, the transformation \( f(x) \mapsto f(x+5) - 2 \) results in the graph of \( f(x) \) being shifted 5 units to the left and 2 units down.