To solve this problem, let's break it down.

Given:

• The original graph represents $y = f(x)$.
• You are asked to graph $g(x) = 3f(x)$, which is a vertical scaling of the function $f(x)$ by a factor of 3.

Vertical Scaling:

The transformation $g(x) = 3f(x)$ means that every y-value of the function $f(x)$ will be multiplied by 3. This results in a vertical stretch of the graph.

For example:

• If $f(x) = 1$, then $g(x) = 3 \times 1 = 3$.
• If $f(x) = -1$, then $g(x) = 3 \times (-1) = -3$.

Steps:

1. Take each point on the original graph of $f(x)$.
2. Multiply the y-coordinate of each point by 3 to find the corresponding point on $g(x)$.

Visualizing the Transformation:

Looking at the graph on the left (the original graph of $f(x)$), there are key points such as:

• $(x = 0, f(x) = 0)$ → This will remain $g(0) = 0$.
• $(x = 1, f(x) = 2)$ → This will become $g(1) = 6$.
• $(x = -1, f(x) = -1)$ → This will become $g(-1) = -3$.

Based on these transformations, we need to identify the graph that reflects this vertical stretch.

Correct Graph:

Comparing the transformations, the correct graph should show the y-values stretched, particularly around the points where $f(x)$ had noticeable values (such as $2$ becoming $6$, and $-1$ becoming $-3$).

Answer: The correct graph is option B since it reflects a vertical stretch of the function by a factor of 3.

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How would the graph change if the function was $g(x) = \frac{1}{2}f(x)$?
2. What effect would a negative multiplier have on the graph, e.g., $g(x) = -3f(x)$?
3. How would you reflect a function across the x-axis?
4. What is the impact of horizontal scaling on a graph?
5. How do vertical shifts differ from vertical stretches?

Tip: Whenever multiplying a function by a constant, remember to adjust the y-coordinates only, leaving the x-coordinates unchanged.