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 need to understand the effects of each term in the transformed function.
1. Horizontal Shift:
- The term [tex]\((x-3)\)[/tex] inside the squared function indicates a horizontal shift.
- In general, [tex]\( f(x-h) \)[/tex] represents a horizontal shift to the right by [tex]\( h \)[/tex] units.
- Here, [tex]\((x-3)\)[/tex] suggests that the graph shifts 3 units to the right.
2. Vertical Shift:
- The [tex]\(-1\)[/tex] outside the squared function indicates a vertical shift.
- In general, [tex]\( f(x) - k \)[/tex] represents a vertical shift downward by [tex]\( k \)[/tex] units.
- Here, [tex]\(-1\)[/tex] suggests that the graph shifts 1 unit down.
Combining both transformations:
- The graph of [tex]\( f(x) = x^2 \)[/tex] shifts 3 units to the right and 1 unit down to get the graph of [tex]\( f(x) = (x-3)^2 - 1 \)[/tex].
Thus, the best description of the transformation is right 3 units, down 1 unit.
