To understand how the function [tex]\( g(x) = x^2 \)[/tex] is transformed to obtain the function [tex]\( h(x) = g(x) - 5 \)[/tex], let's break down the transformation step-by-step:
1. Original Function: The original function is [tex]\( g(x) = x^2 \)[/tex].
2. Transformation Applied: The transformation applied to [tex]\( g(x) \)[/tex] is subtracting 5 from it, resulting in the new function:
[tex]\[
h(x) = g(x) - 5
\][/tex]
3. Substitute [tex]\( g(x) \)[/tex] into [tex]\( h(x) \)[/tex]: Since [tex]\( g(x) = x^2 \)[/tex], we substitute [tex]\( x^2 \)[/tex] into the equation for [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = x^2 - 5
\][/tex]
4. Graphical Interpretation of the Transformation:
- The transformation [tex]\( h(x) = x^2 - 5 \)[/tex] is a vertical shift of the original function [tex]\( g(x) = x^2 \)[/tex].
- Specifically, subtracting 5 from the function shifts the graph downward by 5 units. This is because for every value of [tex]\( x \)[/tex], the output of [tex]\( h(x) \)[/tex] will be 5 units less than the output of [tex]\( g(x) \)[/tex].
Given these steps, we can conclude that the graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] shifted vertically downward by 5 units.
Thus, the correct statement is:
D. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted down 5 units.
