To describe how the graph of $y = f(2x - 2)$ is related to the graph of $y = f(x)$, we need to break down the transformation by considering the effects of the stretch, reflection, and translation on the graph of $y = f(x)$.

Step 1: Factor the inside of the function

The function $y = f(2x - 2)$ can be rewritten to make transformations more obvious by factoring the argument of the function as follows:

$y = f(2(x - 1))$

Now, we can see the transformations clearly.

Step 2: Identify the transformations

1. Horizontal Compression (Stretch):

   • The factor of $2$ inside the function affects the horizontal scaling of the graph.
   • Specifically, the graph is compressed horizontally by a factor of 2. This means that every $x$-coordinate of the points on the graph of $y = f(x)$ is divided by 2 (or the graph appears to be stretched towards the $y$-axis).

2. Horizontal Translation (Shift):

   • The term $(x - 1)$ indicates a horizontal shift to the right by 1 unit.
   • This is because $x - 1$ means we are moving every point on the graph of $f(x)$ 1 unit to the right.

Summary of Transformations:

• Horizontal Compression by a factor of 2 (the graph gets narrower).
• Shift right by 1 unit.

There is no reflection or vertical translation present in this transformation.

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Would you like a detailed explanation or have any questions about these transformations?

Here are 5 related questions to expand on the concept:

1. How would the graph of $y = f(x - 3)$ be related to the graph of $y = f(x)$?
2. What happens if the function is $y = f(-x)$? How does this affect the graph?
3. How would the graph change if the transformation was $y = f(\frac{1}{2}x)$?
4. What effect does a vertical stretch or compression have on a function?
5. What happens if the function is $y = f(2x) + 3$?

Tip: Always consider horizontal transformations (inside the function) before vertical transformations (outside the function).