To determine how the graph of [tex]\( g(x) = f(x + 8) + 4 \)[/tex] compares with the graph of [tex]\( f(x) \)[/tex], let's break down the transformation step by step.
1. Horizontal Shift:
- The argument of the function [tex]\( f \)[/tex] is [tex]\( x + 8 \)[/tex]. Adding a positive constant inside the function argument results in a horizontal shift to the left.
- Specifically, [tex]\( f(x + 8) \)[/tex] means the function [tex]\( f(x) \)[/tex] is shifted 8 units to the left on the x-axis.
2. Vertical Shift:
- The entire functional value [tex]\( f(x + 8) \)[/tex] is then increased by 4.
- Adding a positive constant outside the function results in a vertical shift upward.
- Thus, [tex]\( f(x + 8) + 4 \)[/tex] means the function is shifted 4 units up on the y-axis.
Combining these transformations:
- The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 8 units to the left and 4 units up.
Thus, the best statement that describes this transformation is:
B. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 8 units to the left and 4 units up.
