Question 5 of 10 How does the graph of $$ f(x)=\frac{1}{2}(3)^{x+2}-1 $$ relate to its parent function? $$ \square $$ A. The parent function has been translated to the left. $$ \square $$ B. The parent function has been stretched. $$ \square $$ C. The parent function has been translated down. $$ \square $$ D. The parent function has been compressed.

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To analyze how the graph of the function $$ f(x) = \frac{1}{2}(3)^{x+2} - 1 $$ relates to its parent function, we need to identify the transformations applied to the parent function $$ g(x) = 3^x $$.

1. **Horizontal Translation**: The term $$ x + 2 $$ indicates a horizontal translation. Specifically, $$ x + 2 $$ means the graph is shifted to the left by 2 units.

2. **Vertical Stretch/Compression**: The coefficient $$ \frac{1}{2} $$ in front of $$ (3)^{x+2} $$ indicates a vertical compression by a factor of $$ \frac{1}{2} $$. This means that the output values of the function are halved compared to the parent function.

3. **Vertical Translation**: The term $$ -1 $$ at the end indicates a vertical translation downward by 1 unit.

Now, let's evaluate the options:

- **A. The parent function has been translated to the left.** (True, due to $$ x + 2 $$)
- **B. The parent function has been stretched.** (False, it has been compressed)
- **C. The parent function has been translated down.** (True, due to $$ -1 $$)
- **D. The parent function has been compressed.** (True, due to $$ \frac{1}{2} $$)

Since the question asks how the graph relates to its parent function, the most significant transformations are the horizontal translation to the left and the vertical compression.

However, if we are to choose the most relevant transformation based on the options provided, the correct answer would be:

**A. The parent function has been translated to the left.**

This is because the horizontal translation is a more direct transformation of the input variable $$ x $$ compared to the vertical changes.

If you need to select only one option, choose **A**. If multiple selections are allowed, you could also consider **C** and **D**.
The graph has been translated to the left.

Question 5 of 10
How does the graph of relate to its parent function?
A. The parent function has been translated to the left.
B. The parent function has been stretched.
C. The parent function has been translated down.
D. The parent function has been compressed.

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Question 5 of 10 How does the graph of relate to its parent function? A. The parent function has been translated to the left. B. The parent function has been stretched. C. The parent function has been translated down. D. The parent function has been compressed.