How does the graph of [tex]g(x) = 3^x - 2[/tex] compare to the graph of [tex]f(x) = 3^x[/tex]?

A. The graph of [tex]g(x)[/tex] is a translation of [tex]f(x)[/tex] 2 units left.
B. The graph of [tex]g(x)[/tex] is a translation of [tex]f(x)[/tex] 2 units right.
C. The graph of [tex]g(x)[/tex] is a translation of [tex]f(x)[/tex] 2 units up.
D. The graph of [tex]g(x)[/tex] is a translation of [tex]f(x)[/tex] 2 units down.



Answer :

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.