Let's analyze each part of the problem in detail.

Part (a): Graph Transformation of $y = f\left(\frac{1}{2}x\right)$

You are given a graph of the function $y = f(x)$ and asked to sketch the graph of $y = f\left(\frac{1}{2}x\right)$.

• The transformation $y = f\left(\frac{1}{2}x\right)$ is a horizontal compression by a factor of 2. This means that the graph of $f(x)$ will be squished towards the y-axis, making it twice as narrow horizontally.
• Every point on the graph of $y = f(x)$ will move closer to the y-axis, with the x-coordinates halved.

Part (b): Graph Transformation of $y = 2g(x)$

You are given the graph of $y = g(x)$ and asked to sketch the graph of $y = 2g(x)$.

• The transformation $y = 2g(x)$ is a vertical stretch by a factor of 2. This means that the graph will be stretched away from the x-axis, making it taller.
• Every y-coordinate on the graph of $y = g(x)$ will be multiplied by 2, while the x-coordinates will remain unchanged.

If you want, I can guide you through sketching these transformations step by step or discuss the graphical changes in more detail. Would you like more details, or do you have any questions so far?

Here are 5 relative questions based on this topic:

1. How does the graph of $y = f(kx)$ change for different values of $k$?
2. What happens to the graph of $y = f(x)$ when you apply a vertical stretch by a factor $a$?
3. What is the effect of a horizontal translation on the graph of $f(x)$?
4. How would the graph of $g(x)$ change if the transformation was $y = \frac{1}{2}g(x)$?
5. Can you describe how the transformation $y = -f(x)$ affects the original graph?

Tip: Always focus on whether the transformation is acting on the input (x) or output (y) to understand if it impacts the graph horizontally or vertically.