To compare the graphs of the functions [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] given that [tex]\( g(x) = f(x) + 2 \)[/tex], we need to understand how modifications in the equation affect the graph of the function.
1. The basic function is [tex]\( f(x) \)[/tex], and we are adding a constant value, which is [tex]\( +2 \)[/tex], to the function to obtain [tex]\( g(x) \)[/tex]. This leads to the equation [tex]\( g(x) = f(x) + 2 \)[/tex].
2. Adding a constant value to a function results in a vertical shift in the graph. Specifically, if we add a positive constant [tex]\( c \)[/tex] to [tex]\( f(x) \)[/tex] as in [tex]\( f(x) + c \)[/tex], this shifts the graph of [tex]\( f(x) \)[/tex] upwards by [tex]\( c \)[/tex] units.
3. Here, the constant [tex]\( c \)[/tex] is 2. Therefore, [tex]\( g(x) = f(x) + 2 \)[/tex] represents the graph of [tex]\( f(x) \)[/tex] shifted upwards by 2 units. This means every point [tex]\( (x, y) \)[/tex] on the graph of [tex]\( f(x) \)[/tex] will move to [tex]\( (x, y+2) \)[/tex] on the graph of [tex]\( g(x) \)[/tex].
Thus, the correct statement that compares the graph of [tex]\( g(x) \)[/tex] with the graph of [tex]\( f(x) \)[/tex] is:
D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 2 units up.
