To address the problem of comparing the graphs of [tex]\( g(x) = f(x) + 2 \)[/tex] with [tex]\( f(x) \)[/tex]:
1. Identify the Transformation:
The equation [tex]\( g(x) = f(x) + 2 \)[/tex] indicates that we are adding a constant, 2, to the function [tex]\( f(x) \)[/tex]. This addition of a constant affects the graph of the function.
2. Understand the Effect of Adding a Constant:
When we add a positive constant to a function [tex]\( f(x) \)[/tex], it results in a vertical shift of the graph of the function. Specifically:
- Adding a positive constant [tex]\( ( + 2 ) \)[/tex] to [tex]\( f(x) \)[/tex] shifts the graph of the function upwards.
3. Determine the Direction of the Shift:
The term [tex]\( +2 \)[/tex] signifies the amount of shift:
- If the constant was negative ([tex]\( f(x) - 2 \)[/tex]), the graph would shift downwards.
- Since [tex]\( f(x) + 2 \)[/tex] is positive, the graph shifts upwards by 2 units.
4. Compare [tex]\( g(x) \)[/tex] to [tex]\( f(x) \)[/tex]:
Thus, the graph of [tex]\( g(x) \)[/tex] ([tex]\( g(x) = f(x) + 2 \)[/tex]) is the graph of [tex]\( f(x) \)[/tex] moved 2 units up in the vertical direction.
5. Evaluate All Given Options:
- A. The graph of [tex]\( g(x) \)[/tex] is shifted 2 units up. (Correct)
- B. The graph of [tex]\( g(x) \)[/tex] is shifted 2 units down. (Incorrect)
- C. The graph of [tex]\( g(x) \)[/tex] is shifted 2 units to the right. (Incorrect)
- D. The graph of [tex]\( g(x) \)[/tex] is vertically stretched by a factor of 2. (Incorrect)
Conclusion:
The best comparison is provided by option A: The graph of [tex]\( g(x) \)[/tex] is shifted 2 units up.
Therefore, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
