Questions: How does the graph of f(x)=-3^(2x)-4 differ from the graph of g(x)=-3^(2x)? A. The graph of f(x) is shifted four units to the left of the graph of g(x). B. The graph of f(x) is shifted four units down from the graph of g(x) C. 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).

How does the graph of f(x)=-3^(2x)-4 differ from the graph of g(x)=-3^(2x)?
A. The graph of f(x) is shifted four units to the left of the graph of g(x).
B. The graph of f(x) is shifted four units down from the graph of g(x)
C. 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).

Transcript text: 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 to the left of the graph of $g(x)$. B. The graph of $f(x)$ is shifted four units down from the graph of $g(x)$ C. 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 $g(x)$.

Solution

Solution Steps

Step 1: Identify the Base Function

The base exponential function is given by the equation: $y = -1 \cdot 3^x$. This represents the graph before any transformations are applied.

Step 2: Determine Transformations

^Horizontal Shift^: The graph is shifted 0 units to the left. This is due to the $(x - 0)$ term in the transformed function.
^Vertical Shift^: The graph is shifted 4 units down. This is indicated by the $-4$ at the end of the function.
^Vertical Stretch/Compression and Reflection^: The graph is compressed vertically by a factor of 1 and is with reflection. This is because of the coefficient $-1$ in front of the base function.

Step 3: Apply Transformations Sequentially

After applying the transformations sequentially, the final form of the function is: $y = -1 \cdot 3^{(x - 0)} - 4$. This equation represents how the graph of the base exponential function is transformed.

Final Answer:

The transformed exponential function $y = -1 \cdot 3^{(x - 0)} - 4$ is obtained by applying a horizontal shift of 0 units to the left, a vertical shift of 4 units down, and a vertical transformation that compresseds the graph by a factor of 1 with reflection.

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