To determine how the graph of the function [tex]\(g(x) = f(x) + 6\)[/tex] differs from the graph of the function [tex]\(f(x)\)[/tex], we need to analyze the impact of adding 6 to the function [tex]\(f(x)\)[/tex].
Given:
[tex]\[ f(x) = 2^x \][/tex]
[tex]\[ g(x) = f(x) + 6 \][/tex]
Substituting [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x + 6 \][/tex]
The expression [tex]\( g(x) = 2^x + 6 \)[/tex] indicates that for every value of [tex]\( x \)[/tex], the value of [tex]\( g(x) \)[/tex] is the value of [tex]\( f(x) \)[/tex] plus 6.
Graphically, adding a constant to a function [tex]\( f(x) \)[/tex] results in a vertical shift of the graph. Here's a step-by-step explanation:
1. Understanding the Vertical Shift: Adding a positive constant (in this case, 6) to [tex]\( f(x) \)[/tex] shifts the entire graph of [tex]\( f(x) \)[/tex] vertically upwards by that constant.
2. Graph Analysis:
- The original graph of [tex]\( f(x) = 2^x \)[/tex] remains unchanged in shape.
- By adding 6 to [tex]\( f(x) \)[/tex], every y-coordinate of [tex]\( f(x) \)[/tex] increases by 6 units, effectively shifting the entire graph vertically upward.
3. Confirming the Shift: Since 6 is added to every output value of [tex]\( f(x) \)[/tex], the entire graph of [tex]\( g(x) \)[/tex] is elevated by 6 units on the y-axis.
Based on this analysis, the correct transformation is a vertical shift upwards by 6 units. Therefore, the correct answer is:
D. The graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] shifted 6 units up.
