To compare the graph of [tex]\( g(x) = f(x + 3) + 4 \)[/tex] with the graph of [tex]\( f(x) \)[/tex], we need to analyze how the transformations affect the graph.
1. Horizontal Shift:
- When we replace [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex] in [tex]\( f(x) \)[/tex], the graph of the function shifts horizontally.
- Specifically, [tex]\( f(x + 3) \)[/tex] shifts the graph of [tex]\( f(x) \)[/tex] to the left by 3 units. This is because replacing [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex] makes every input to [tex]\( f \)[/tex] happen 3 units earlier than it normally would.
2. Vertical Shift:
- Adding 4 to [tex]\( f(x + 3) \)[/tex], i.e., [tex]\( f(x + 3) + 4 \)[/tex], shifts the graph vertically.
- Adding a positive constant to the function [tex]\( f(x) \)[/tex], shifts the graph up by that constant value. Hence, [tex]\( f(x + 3) + 4 \)[/tex] shifts the graph of [tex]\( f(x + 3) \)[/tex] up by 4 units.
Combining these two transformations:
- The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted left by 3 units.
- After this horizontal shift, the graph is further shifted upwards by 4 units.
This conclusion matches the statement:
- A. The graph of [tex]\( g(x) \)[/tex] is shifted 3 units left and 4 units up.
Thus, the correct comparison between the graphs of [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] is statement A.
