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 modifications made to the quadratic function.
1. Identify the Horizontal Shift:
The function [tex]\( f(x) = (x-3)^2 - 1 \)[/tex] has the term [tex]\( (x-3) \)[/tex] inside the squared expression. This [tex]\( (x-3) \)[/tex] indicates a horizontal shift of the graph of [tex]\( f(x) = x^2 \)[/tex].
- Translation Rule: If a term [tex]\( (x - h) \)[/tex] appears in the function, it translates the graph horizontally by [tex]\( h \)[/tex] units.
- Here, [tex]\( h = 3 \)[/tex], so the graph shifts 3 units to the right.
2. Identify the Vertical Shift:
The term [tex]\( -1 \)[/tex] outside the squared expression [tex]\( (x-3)^2 \)[/tex] indicates a vertical shift of the graph.
- Translation Rule: If a term [tex]\( k \)[/tex] is added or subtracted outside the squaring function, it translates the graph vertically by [tex]\( k \)[/tex] units.
- Here, [tex]\( k = -1 \)[/tex], so the graph shifts 1 unit down.
Putting these two observations together:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right by 3 units.
- The graph is then shifted down by 1 unit.
Thus, the transformation from the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( f(x) = (x-3)^2 - 1 \)[/tex] is right 3 units, down 1 unit.
Therefore, the best description of this transformation is:
- right 3 units, down 1 unit
