The problem asks to transform the graph of $y = g(x)$ to $y = g(2x)$. To do this, you need to understand how the transformation $y = g(2x)$ affects the graph.

Transformation:

The equation $y = g(2x)$ represents a horizontal compression by a factor of 2. This means that every x-coordinate of the original graph $g(x)$ is halved, pulling the graph towards the y-axis.

Steps:

1. Identify key points: Choose some important points on the original graph, particularly the vertices and intercepts.
2. Apply the transformation: For each point $(x, y)$ on the graph of $g(x)$, the corresponding point on the graph of $g(2x)$ will be $\left( \frac{x}{2}, y \right)$.
3. Sketch the new graph: Once the points have been transformed, redraw the graph based on these points.

Example with key points:

Suppose the original graph has the following points:

• $(-4, 0)$
• $(-2, -4)$
• $(2, 4)$

After applying the transformation $x \to \frac{x}{2}$, the new points will be:

• $(-2, 0)$
• $(-1, -4)$
• $(1, 4)$

These points should then be used to plot the graph of $y = g(2x)$.

Let me know if you would like a more detailed visual sketch or have any questions.

5 related questions:

1. What happens if we change the function to $y = g( \frac{x}{2} )$?
2. How does the transformation $y = g(-x)$ affect the graph?
3. What is the impact of applying $y = 2g(x)$ to the graph?
4. Can you describe how a vertical shift affects the graph of a function?
5. How do you recognize symmetry in a graph?

Tip: Always track how the input variable $x$ is affected by transformations such as stretching, compressing, reflecting, or shifting to correctly redraw graphs.