Select the correct answer.

The function [tex]$g(x) = x^2$[/tex] is transformed to obtain function [tex]$h(x) = g(x) - 5$[/tex]. Which statement describes how the graph of [tex][tex]$h$[/tex][/tex] is different from the graph of [tex]$g$[/tex]?

A. The graph of [tex]$h$[/tex] is the graph of [tex][tex]$g$[/tex][/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][tex]$h$[/tex][/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][tex]$g$[/tex][/tex] horizontally shifted right 5 units.



Answer :

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].