https://file.aiquickdraw.com/mathbot/file/1726196484033nea8i5n9.jpg

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.

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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.

Use the graph of y = f(x) shown below to graph the function g, where g(x) = 3f(x).

Explore how to apply vertical scaling to a function using the transformation g(x) = 3f(x). Learn how to graphically represent a vertical stretch and understand key concepts like functions and graph transformations. Suitable for students in Grades 9-12.

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