To determine how the graph of the function [tex]\( f(x) = -3^{2x} - 4 \)[/tex] differs from the graph of the function [tex]\( g(x) = -3^2 \)[/tex], let's analyze the transformations applied to each function.
1. Start with the function [tex]\( g(x) = -3^2 \)[/tex]:
[tex]\[
g(x) = -3^2 = -9
\][/tex]
Since [tex]\( g(x) \)[/tex] is a constant function, its graph is a horizontal line at [tex]\( y = -9 \)[/tex].
2. Now, consider the function [tex]\( f(x) = -3^{2x} - 4 \)[/tex]:
- The term [tex]\( -3^{2x} \)[/tex] indicates an exponential function.
- Subtraction of 4, i.e., [tex]\( -4 \)[/tex], indicates a vertical shift of the graph downward by 4 units.
Let's compare these two graphs:
- The graph of [tex]\( g(x) = -3^2 = -9 \)[/tex] is purely horizontal (constant function).
- The graph of [tex]\( f(x) = -3^{2x} - 4 \)[/tex] is the graph of [tex]\( -3^{2x} \)[/tex] shifted down by 4 units.
In conclusion, the graph of [tex]\( f(x) \)[/tex] is shifted four units down from the graph of [tex]\( g(x) \)[/tex].
Therefore, the correct answer is:
C. The graph of [tex]\( f(x) \)[/tex] is shifted four units down from the graph of [tex]\( g(x) \)[/tex].
