To understand how the graph of [tex]\( g(x) = 3^x - 2 \)[/tex] compares to the graph of [tex]\( f(x) = 3^x \)[/tex], we need to analyze the transformation applied to the function [tex]\( f(x) \)[/tex].
The function [tex]\( g(x) \)[/tex] can be considered as a transformation of [tex]\( f(x) \)[/tex]. Here, [tex]\( g(x) = 3^x - 2 \)[/tex].
In this expression, [tex]\( 3^x \)[/tex] is the original function [tex]\( f(x) \)[/tex]. The transformation applied to [tex]\( f(x) \)[/tex] is the subtraction of 2. This transformation shifts the graph of the original function.
When a constant is subtracted from the entire function, it results in a downward vertical translation of the graph by that constant. In this case, subtracting 2 means the graph of [tex]\( g(x) \)[/tex] is moved down by 2 units.
Therefore, the graph of [tex]\( g(x) \)[/tex] is a vertical translation of the graph of [tex]\( f(x) = 3^x \)[/tex] by 2 units down.
Hence, the correct comparison from the given options is:
The graph of [tex]\( g(x) \)[/tex] is a translation of [tex]\( f(x) \)[/tex] 2 units down.
