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.