Question
What does the transformation f(x)↦f(x–2)+4 do to the graph of f(x)?
Answers
GPT-4o mini
The transformation \( f(x) \mapsto f(x - 2) + 4 \) consists of two distinct operations applied to the function \( f(x) \):
1. **Horizontal Shift:** The term \( f(x - 2) \) indicates a horizontal shift of the graph of \( f(x) \) to the right by 2 units. This is because replacing \( x \) with \( x - 2 \) effectively shifts every point on the graph 2 units to the right.
2. **Vertical Shift:** The addition of 4, as in \( f(x - 2) + 4 \), indicates a vertical shift of the graph upward by 4 units. This means that after the horizontal shift has been made, every point on the graph is moved up 4 units.
Combining these two transformations, the overall effect on the graph of \( f(x) \) is:
- Shift the graph 2 units to the right.
- Shift the graph up 4 units.
Thus, the final transformation results in the graph of \( f(x) \) being shifted to the right by 2 units and then moved upward by 4 units.
1. **Horizontal Shift:** The term \( f(x - 2) \) indicates a horizontal shift of the graph of \( f(x) \) to the right by 2 units. This is because replacing \( x \) with \( x - 2 \) effectively shifts every point on the graph 2 units to the right.
2. **Vertical Shift:** The addition of 4, as in \( f(x - 2) + 4 \), indicates a vertical shift of the graph upward by 4 units. This means that after the horizontal shift has been made, every point on the graph is moved up 4 units.
Combining these two transformations, the overall effect on the graph of \( f(x) \) is:
- Shift the graph 2 units to the right.
- Shift the graph up 4 units.
Thus, the final transformation results in the graph of \( f(x) \) being shifted to the right by 2 units and then moved upward by 4 units.