To determine the effects of the transformations on the graph of the function [tex]\( f \)[/tex] to obtain the graph of the function [tex]\( g \)[/tex] where [tex]\( g(x) = f(x - 3) + 4 \)[/tex], let's break down the transformations step by step:
1. Horizontal Shifting:
- The term [tex]\( x - 3 \)[/tex] inside the function [tex]\( f \)[/tex] indicates a horizontal shift.
- When we have [tex]\( f(x - a) \)[/tex], it means the graph of [tex]\( f \)[/tex] is shifted to the right by [tex]\( a \)[/tex] units.
- Here, [tex]\( a = 3 \)[/tex], so the graph of [tex]\( f \)[/tex] is shifted 3 units to the right.
2. Vertical Shifting:
- The "+4" outside the function [tex]\( f \)[/tex] indicates a vertical shift.
- When we add a constant [tex]\( b \)[/tex] to the function [tex]\( f \)[/tex], i.e., [tex]\( f(x) + b \)[/tex], it means the graph of [tex]\( f \)[/tex] is shifted up by [tex]\( b \)[/tex] units.
- Here, [tex]\( b = 4 \)[/tex], so the graph of [tex]\( f \)[/tex] is shifted 4 units up.
Therefore, combining these two transformations, the graph of the function [tex]\( f \)[/tex] is shifted 3 units to the right and 4 units up to obtain the graph of [tex]\( g \)[/tex].
So, the correct statement is:
C. The graph of function [tex]\( f \)[/tex] is shifted right 3 units and up 4 units.
