To determine the transformation from the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( f(x) = (x-3)^2 - 1 \)[/tex], we follow these steps:
1. Horizontal Shift:
- The term [tex]\((x-3)\)[/tex] inside the square function indicates a horizontal shift.
- Specifically, [tex]\(x-3\)[/tex] means we shift the graph to the right by 3 units.
2. Vertical Shift:
- The [tex]\(-1\)[/tex] outside of the square function affects the vertical position of the graph.
- A [tex]\(-1\)[/tex] indicates we move the graph down by 1 unit.
Combining these shifts:
- The graph of [tex]\( f(x) = x^2 \)[/tex] shifts right 3 units and down 1 unit to get the graph of [tex]\( f(x) = (x-3)^2 - 1 \)[/tex].
Therefore, the best description of the transformation is:
- Right 3 units, down 1 unit.
Hence, the correct answer is: right 3 units, down 1 unit.
