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The given problem asks to compare the function \( g(x) = \frac{2(3^x)}{3^7} \) with the function \( f(x) = 3^x \) in terms of horizontal translation and vertical stretch.

Let's break this down:

### Step 1: Simplify \( g(x) \)
The function \( g(x) = \frac{2(3^x)}{3^7} \) can be rewritten as:
\[
g(x) = 2 \cdot \frac{3^x}{3^7}
\]
Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), this simplifies to:
\[
g(x) = 2 \cdot 3^{x-7}
\]

Now we can clearly see that \( g(x) = 2 \cdot 3^{x-7} \).

### Step 2: Analyze Horizontal Translation
The function \( f(x) = 3^x \) has been modified in \( g(x) \) to \( 3^{x-7} \). This represents a **horizontal translation** to the right by 7 units, because subtracting from the \( x \)-variable shifts the graph to the right.

### Step 3: Analyze Vertical Stretch
There is a multiplication by 2 in \( g(x) = 2 \cdot 3^{x-7} \). This represents a **vertical stretch** by a factor of 2, since multiplying the entire function by a constant greater than 1 stretches the graph vertically.

### Final Answer:
- **Horizontal translation:** 7 units to the right.
- **Vertical stretch:** By a factor of 2.

Would you like any further clarification or details on this?

Here are 5 related questions to deepen your understanding:
1. What is the effect of a horizontal translation to the left on the graph of \( f(x) = 3^x \)?
2. How would a vertical compression by a factor of \( \frac{1}{2} \) change the graph of \( f(x) = 3^x \)?
3. What happens if the exponent in \( f(x) = 3^x \) is modified to \( 3^{-x} \)?
4. Can you describe how the graph of \( g(x) = \frac{2(3^x)}{3^7} \) compares to \( h(x) = 2(3^x) \)?
5. How would adding a constant outside the function \( g(x) = 2(3^x) \) shift the graph?

**Tip:** When analyzing transformations of functions, always focus on changes in the exponent for horizontal shifts and multipliers for vertical changes.

Compare g(x) = (2 * 3^x) / 3^7 with f(x) = 3^x in terms of horizontal translation and vertical stretch.

Compare the exponential functions g(x) = (2 * 3^x) / 3^7 and f(x) = 3^x to identify a horizontal translation of 7 units to the right and a vertical stretch by a factor of 2. This solution explains the transformation using exponent rules, ideal for high school students.

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