To describe 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 analyze the changes made to the function.
1. Horizontal Shift:
- The expression [tex]\( (x - 3) \)[/tex] inside the parentheses indicates a horizontal shift.
- If the term is [tex]\( (x - h) \)[/tex], the graph of the function shifts to the right by [tex]\( h \)[/tex] units. Here, [tex]\( h = 3 \)[/tex], so the graph shifts to the right by 3 units.
2. Vertical Shift:
- The expression [tex]\(-1\)[/tex] outside the parentheses indicates a vertical shift.
- If the term is [tex]\(- k\)[/tex] added or subtracted from the function, the graph shifts downward by [tex]\( k \)[/tex] units. Here, [tex]\( k = 1 \)[/tex], so the graph shifts downward by 1 unit.
Putting these two transformations together:
- The graph shifts 3 units to the right.
- The graph shifts 1 unit downward.
Hence, the best description of the transformation is:
right 3 units, down 1 unit
