6.1.2 Exam: Semester 2 Exam Question 7 of 40 How does the graph of $$ f(x)=-3^{2 x}-4 $$ differ from the graph of $$ g(x)=-3^{2 x} $$ ? A. The graph of $$ f(x) $$ is shifted four units down from the graph of $$ g(x) $$. B. The graph of $$ f(x) $$ is shifted four units to the left of the graph of $$ g(x) $$. The graph of $$ f(x) $$ is shifted four units up from the graph of $$ g(x) $$ D. The graph of $$ f(x) $$ is shifted four units to the right of the graph of $$ g(x) $$.

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UpStudy AI · Accepted Answer

To determine how the graph of $$ f(x) = -3^{2x} - 4 $$ differs from the graph of $$ g(x) = -3^{2x} $$, let's analyze the transformations involved.

### Understanding the Functions

1. **Original Function ($$ g(x) $$):**
   $$
   g(x) = -3^{2x}
   $$
   - This is an exponential function with a base of 3 raised to the power of $$ 2x $$, and the entire expression is negated.

2. **Transformed Function ($$ f(x) $$):**
   $$
   f(x) = -3^{2x} - 4
   $$
   - This function is the same as $$ g(x) $$ but with an additional $$-4$$ at the end.

### Identifying the Transformation

The difference between $$ f(x) $$ and $$ g(x) $$ is the $$-4$$ term in $$ f(x) $$. This represents a **vertical shift** of the graph.

- **Vertical Shift:** Adding or subtracting a constant to the function $$ g(x) $$ shifts the graph **up or down**.
  - **$$ g(x) + 4 $$:** Shifts the graph **4 units up**.
  - **$$ g(x) - 4 $$:** Shifts the graph **4 units down**.

Since $$ f(x) = g(x) - 4 $$, the graph of $$ f(x) $$ is shifted **4 units down** from the graph of $$ g(x) $$.

### Conclusion

**Answer A** correctly describes this transformation:

**A. The graph of $$ f(x) $$ is shifted four units down from the graph of $$ g(x) $$.**

Answer: A
The graph of $$ f(x) = -3^{2x} - 4 $$ is shifted four units down from the graph of $$ g(x) = -3^{2x} $$.

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