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 relationship between these two functions.
### Step-by-Step Solution:
1. Identify the Parent Function:
The parent function here is [tex]\( f(x) = 3^x \)[/tex].
2. Understand the Transformation:
The function [tex]\( g(x) \)[/tex] can be expressed as:
[tex]\[
g(x) = 3^x - 2
\][/tex]
This shows that [tex]\( g(x) \)[/tex] is derived from [tex]\( f(x) \)[/tex] with an adjustment.
3. Determine the Nature of the Transformation:
The expression [tex]\( 3^x - 2 \)[/tex] indicates a translation. Specifically, we are subtracting 2 from the output of [tex]\( 3^x \)[/tex].
4. Vertical Translation:
Subtracting 2 from a function [tex]\( f(x) \)[/tex] results in a vertical shift. More precisely, for any value of [tex]\( 3^x \)[/tex]:
[tex]\[
g(x) = 3^x - 2
\][/tex]
This shifts every point on the graph of [tex]\( f(x) = 3^x \)[/tex] down by 2 units.
Therefore, the graph of [tex]\( g(x) = 3^x - 2 \)[/tex] is a vertical translation of the graph of [tex]\( f(x) = 3^x \)[/tex] down by 2 units.
### Conclusion:
The correct description from the given options is:
The graph of [tex]\( g(x) \)[/tex] is a translation of [tex]\( f(x) 2 \)[/tex] units down.
