Question
What does the transformation f(x)↦f(x–3)–2 do to the graph of f(x)?
Answers
GPT-4o mini
The transformation \( f(x) \mapsto f(x-3) - 2 \) involves two steps applied to the graph of the function \( f(x) \):
1. **Horizontal Shift**: The term \( f(x-3) \) indicates a horizontal shift of the graph of \( f(x) \). Specifically, this transforms the graph to the right by 3 units. This is because you are replacing \( x \) with \( x-3 \), which means that for any given \( y \) value that was originally at \( x \), it now occurs at \( x+3 \).
2. **Vertical Shift**: The term \( -2 \) in \( f(x-3) - 2 \) indicates a vertical shift downwards by 2 units. This means that every point on the graph of \( f(x-3) \) will be shifted down by 2 units.
In summary, the transformation \( f(x) \mapsto f(x-3) - 2 \) shifts the graph of \( f(x) \) 3 units to the right and then 2 units down.
1. **Horizontal Shift**: The term \( f(x-3) \) indicates a horizontal shift of the graph of \( f(x) \). Specifically, this transforms the graph to the right by 3 units. This is because you are replacing \( x \) with \( x-3 \), which means that for any given \( y \) value that was originally at \( x \), it now occurs at \( x+3 \).
2. **Vertical Shift**: The term \( -2 \) in \( f(x-3) - 2 \) indicates a vertical shift downwards by 2 units. This means that every point on the graph of \( f(x-3) \) will be shifted down by 2 units.
In summary, the transformation \( f(x) \mapsto f(x-3) - 2 \) shifts the graph of \( f(x) \) 3 units to the right and then 2 units down.