Let's begin by understanding the transformation between the functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex].
1. We start with the function [tex]\( g(x) = x^2 \)[/tex].
2. The function [tex]\( h(x) \)[/tex] is defined as [tex]\( h(x) = g(x) - 5 \)[/tex].
To understand how [tex]\( h(x) \)[/tex] transforms from [tex]\( g(x) \)[/tex], we substitute [tex]\( g(x) \)[/tex] into the equation for [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = g(x) - 5 \][/tex]
[tex]\[ h(x) = x^2 - 5 \][/tex]
This new function [tex]\( h(x) \)[/tex] represents a transformation of the original function [tex]\( g(x) = x^2 \)[/tex].
Let's break down what the transformation [tex]\( h(x) = x^2 - 5 \)[/tex] means:
- The term [tex]\( - 5 \)[/tex] indicates a vertical shift of the graph.
- Specifically, subtracting 5 from the [tex]\( y \)[/tex]-values (outputs) of [tex]\( g(x) \)[/tex] means we are shifting the entire graph of [tex]\( g(x) \)[/tex] downward by 5 units.
So, the graph of [tex]\( h(x) = x^2 - 5 \)[/tex] can be described as taking the graph of [tex]\( g(x) = x^2 \)[/tex] and shifting it vertically downward by 5 units.
Let's revisit the given statements and match them with the above transformation analysis:
A. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted down 5 units.
B. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted left 5 units.
C. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted up 5 units.
D. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted right 5 units.
The correct statement is:
A. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted down 5 units.
Thus, the correct answer is [tex]\( \boxed{1} \)[/tex] which corresponds to Choice [tex]\( \text{A} \)[/tex].
