To determine how the graph of the function [tex]\( y = f(x-3) + 2 \)[/tex] can be obtained from the graph of the function [tex]\( y = f(x) \)[/tex], we need to consider the transformations involved in the equation.
1. Horizontal Shift:
- The term [tex]\( (x-3) \)[/tex] inside the function argument indicates a horizontal shift.
- Specifically, [tex]\( x-3 \)[/tex] represents a shift to the right by 3 units. This is because substituting [tex]\( x \)[/tex] with [tex]\( x-3 \)[/tex] means that every point on the graph of [tex]\( y = f(x) \)[/tex] is moved 3 units to the right to produce the graph of [tex]\( y = f(x-3) \)[/tex].
2. Vertical Shift:
- The term [tex]\( +2 \)[/tex] added to the function indicates a vertical shift.
- Specifically, [tex]\( +2 \)[/tex] means shifting the graph upward by 2 units. This means that every point on the graph of [tex]\( y = f(x-3) \)[/tex] is moved 2 units upward to produce the graph of [tex]\( y = f(x-3) + 2 \)[/tex].
Summarizing the effects of these transformations:
- The graph of [tex]\( y = f(x) \)[/tex] is first shifted 3 units to the right.
- Then the resulting graph is shifted 2 units upward.
Therefore, the graph of the function [tex]\( y = f(x-3) + 2 \)[/tex] can be obtained from the graph of [tex]\( y = f(x) \)[/tex] by:
Shifting it 3 units to the right and 2 units upward.
This corresponds to the correct description of the transformation:
shift 3 units to the right and 2 units upward.
