Answer :
To understand the transformation of the function [tex]\( g(x) = x^2 \)[/tex] to [tex]\( h(x) = (x+7)^2 \)[/tex], we need to analyze the form of the new function [tex]\( h(x) \)[/tex].
### Step-by-Step Analysis
1. Understanding the Original Function:
The original function is [tex]\( g(x) = x^2 \)[/tex]. This is a standard parabola whose vertex is at the origin [tex]\((0, 0)\)[/tex].
2. Applying the Transformation:
The new function is given by [tex]\( h(x) = (x + 7)^2 \)[/tex].
3. Impact of the Transformation:
- When a constant is added inside the argument of the function [tex]\( f \in h(x) = f(x + c) \)[/tex], it results in a horizontal shift.
- Specifically, if [tex]\( h(x) = (x + c)^2 \)[/tex], the graph of the function shifts horizontally.
- Here [tex]\( c = 7 \)[/tex]. Since it is [tex]\( x + 7 \)[/tex], it indicates a shift to the left.
4. Direction of the Shift:
- Adding a positive number inside the parentheses (i.e., [tex]\( x + 7 \)[/tex]) actually shifts the graph to the left by the absolute value of that number.
Therefore, the graph of [tex]\( h(x) = (x + 7)^2 \)[/tex] is obtained by shifting the graph of [tex]\( g(x) = x^2 \)[/tex] horizontally to the left by 7 units.
### Conclusion
The correct description of the transformation is:
B. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted left 7 units.
### Step-by-Step Analysis
1. Understanding the Original Function:
The original function is [tex]\( g(x) = x^2 \)[/tex]. This is a standard parabola whose vertex is at the origin [tex]\((0, 0)\)[/tex].
2. Applying the Transformation:
The new function is given by [tex]\( h(x) = (x + 7)^2 \)[/tex].
3. Impact of the Transformation:
- When a constant is added inside the argument of the function [tex]\( f \in h(x) = f(x + c) \)[/tex], it results in a horizontal shift.
- Specifically, if [tex]\( h(x) = (x + c)^2 \)[/tex], the graph of the function shifts horizontally.
- Here [tex]\( c = 7 \)[/tex]. Since it is [tex]\( x + 7 \)[/tex], it indicates a shift to the left.
4. Direction of the Shift:
- Adding a positive number inside the parentheses (i.e., [tex]\( x + 7 \)[/tex]) actually shifts the graph to the left by the absolute value of that number.
Therefore, the graph of [tex]\( h(x) = (x + 7)^2 \)[/tex] is obtained by shifting the graph of [tex]\( g(x) = x^2 \)[/tex] horizontally to the left by 7 units.
### Conclusion
The correct description of the transformation is:
B. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted left 7 units.
