To determine the correct transformation from the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( g(x) = (x - 3)^2 - 1 \)[/tex], we need to understand the changes made to the function in its equation.
Consider the function [tex]\( f(x) = x^2 \)[/tex]:
1. Horizontal Shift:
- The term [tex]\((x - 3)\)[/tex] inside the squared function affects the horizontal position of the graph.
- The subtraction of 3 indicates a shift to the right by 3 units. In general, [tex]\( f(x - c) \)[/tex] represents a shift to the right by [tex]\( c \)[/tex] units when [tex]\( c \)[/tex] is positive.
2. Vertical Shift:
- The [tex]\(-1\)[/tex] outside the squared function affects the vertical position of the graph.
- The subtraction of 1 moves the graph down by 1 unit. In general, [tex]\( f(x) - k \)[/tex] represents a downward shift by [tex]\( k \)[/tex] units when [tex]\( k \)[/tex] is positive.
Combining these observations, we can see that the graph of [tex]\( g(x) = (x - 3)^2 - 1 \)[/tex] is obtained by shifting the graph of [tex]\( f(x) = x^2 \)[/tex] to the right by 3 units and then down by 1 unit.
The best description of the transformation is:
right 3 units, down 1 unit.
