To analyze how the graph of [tex]\( h(x) \)[/tex] is different from the graph of [tex]\( g(x) \)[/tex], let's start by understanding the transformation involved.
### Step-by-Step Solution:
1. Given Functions:
- The original function is [tex]\( g(x) = x^2 \)[/tex].
- The transformed function is [tex]\( h(x) = g(x + 7) \)[/tex].
2. Understanding the Transformation:
- To find [tex]\( h(x) \)[/tex], we substitute [tex]\( (x + 7) \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( g \)[/tex].
- Therefore, [tex]\( h(x) = g(x + 7) = (x + 7)^2 \)[/tex].
3. Effect of the Transformation:
- The expression [tex]\( g(x + 7) \)[/tex] represents a horizontal shift.
- In general, a transformation [tex]\( g(x + c) \)[/tex] results in shifting the graph of [tex]\( g(x) \)[/tex] horizontally:
- If [tex]\( c \)[/tex] is positive, [tex]\( x \)[/tex] is replaced with [tex]\( (x + c) \)[/tex], shifting the graph [tex]\( c \)[/tex] units to the left.
- If [tex]\( c \)[/tex] is negative, [tex]\( x \)[/tex] is replaced with [tex]\( (x - c) \)[/tex], shifting the graph [tex]\( |c| \)[/tex] units to the right.
4. Applying this to Our Functions:
- Here, [tex]\( g(x + 7) \)[/tex] implies [tex]\( c = 7 \)[/tex], which is positive.
- Therefore, the graph of [tex]\( g(x) \)[/tex] is shifted to the left by 7 units.
5. Conclusion:
- Given this analysis, the correct description of how the graph of [tex]\( h \)[/tex] differs from the graph of [tex]\( g \)[/tex] is that the graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted left 7 units.
### Answer:
The correct statement is:
B. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted left 7 units.
