Certainly! Let's analyze the function [tex]\( g(x) = f(x + 8) + 4 \)[/tex] and compare it to the original function [tex]\( f(x) \)[/tex].
### Step-by-Step Explanation:
1. Expression Analysis:
- The function [tex]\( g(x) \)[/tex] is given as [tex]\( g(x) = f(x + 8) + 4 \)[/tex].
2. Horizontal Shift:
- The term [tex]\( f(x + 8) \)[/tex] inside the function indicates a horizontal shift.
- Generally in transformations, [tex]\( f(x + h) \)[/tex] represents a horizontal shift of the function [tex]\( f(x) \)[/tex]. If [tex]\( h \)[/tex] is positive, the shift is to the left, and if [tex]\( h \)[/tex] is negative, the shift is to the right.
- Here, [tex]\( x + 8 \)[/tex] means [tex]\( h = 8 \)[/tex], hence [tex]\( f(x + 8) \)[/tex] represents a horizontal shift to the left by 8 units.
3. Vertical Shift:
- The term [tex]\( + 4 \)[/tex] outside the function indicates a vertical shift.
- In transformations, [tex]\( f(x) + k \)[/tex] represents a vertical shift of the function [tex]\( f(x) \)[/tex]. If [tex]\( k \)[/tex] is positive, the shift is upwards, and if [tex]\( k \)[/tex] is negative, the shift is downwards.
- Here [tex]\( + 4 \)[/tex] means the graph of [tex]\( f(x) \)[/tex] is shifted vertically upwards by 4 units.
### Combined Effect:
- The transformation [tex]\( g(x) = f(x + 8) + 4 \)[/tex] therefore involves two transformations combined:
- A horizontal shift to the left by 8 units.
- A vertical shift upwards by 4 units.
### Conclusion:
- The graph of [tex]\( g(x) \)[/tex] compared to the graph of [tex]\( f(x) \)[/tex] is:
- Horizontally shifted to the left by 8 units.
- Vertically shifted upwards by 4 units.
Thus, the best statement that compares the graph of [tex]\( g(x) \)[/tex] with the graph of [tex]\( f(x) \)[/tex] is:
"The graph of [tex]\( g(x) \)[/tex] is horizontally shifted to the left by 8 units and vertically shifted upwards by 4 units compared to the graph of [tex]\( f(x) \)[/tex]."
