The problem now involves the function transformation $g(x) = 3f(x)$.

This transformation involves a vertical stretch by a factor of 3. Specifically, every y-coordinate of the original graph of $f(x)$ will be multiplied by 3. This means that each point on the graph will be stretched away from the x-axis by a factor of 3, while the x-coordinates remain unchanged.

Step-by-step process:

1. For each point on the original graph of $f(x)$, multiply the y-coordinate by 3.
   • For example, if a point on $f(x)$ is $(x, y)$, the corresponding point on $g(x)$ will be $(x, 3y)$.
2. Leave the x-coordinates the same.
3. Sketch the new graph after applying this vertical stretch.

Now, let's analyze the options:

• Option A: This graph shows the correct vertical stretch, where the y-values have been multiplied by 3, causing the graph to stretch away from the x-axis.
• Options B, C, and D do not reflect the correct vertical scaling; they either appear compressed or have incorrect points.

Thus, Option A is the correct graph.

Let me know if you'd like more detailed information or have any questions!

Here are 5 related questions for further exploration:

1. What happens to the graph if the function is $g(x) = \frac{1}{2}f(x)$?
2. How does a negative vertical stretch, such as $g(x) = -3f(x)$, affect the graph?
3. What is the general effect of multiplying a function by a constant on the graph?
4. Can you explain the difference between vertical and horizontal stretches?
5. How would you graph $g(x) = 2f(x) + 3$, which involves both a vertical stretch and a vertical shift?

Tip: A vertical stretch $g(x) = a \cdot f(x)$ stretches or compresses the graph by a factor of $a$ in the y-direction. If $|a| > 1$, the graph stretches; if $0 < |a| < 1$, it compresses.