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How does the graph of [tex]f(x) = -3^{2x} - 4[/tex] differ from the graph of [tex]g(x) = -3^{2x}[/tex]?

A. The graph of [tex]f(x)[/tex] is shifted four units to the right of the graph of [tex]g(x)[/tex].
B. The graph of [tex]f(x)[/tex] is shifted four units up from the graph of [tex]g(x)[/tex].
C. The graph of [tex]f(x)[/tex] is shifted four units down from the graph of [tex]g(x)[/tex].
D. The graph of [tex]f(x)[/tex] is shifted four units to the left of the graph of [tex]g(x)[/tex].


Sagot :

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].