In this problem, you're asked to match different transformations of the function $f(x)$ to their corresponding graphs.

Let's break down the transformations:

1. $y = \frac{1}{2} f(x - 1)$:

   • Horizontal shift: The graph shifts right by 1 unit (because of $x - 1$).
   • Vertical shrink: The graph is compressed vertically by a factor of $\frac{1}{2}$.

2. $y = f\left( \frac{1}{2} x \right)$:

   • Horizontal stretch: The graph is stretched horizontally by a factor of 2 (since $\frac{1}{2} x$ is inside the function).

3. $y = f(-x)$:

   • Reflection over the y-axis.

4. $y = f(x - 2)$:

   • Horizontal shift: The graph shifts right by 2 units (because of $x - 2$).

5. $y = 2f(x)$:

   • Vertical stretch: The graph is stretched vertically by a factor of 2.

6. $y = f(2x)$:

   • Horizontal shrink: The graph is compressed horizontally by a factor of $\frac{1}{2}$ (because of $2x$).

Now, let's match these transformations to the given graphs:

1. For $y = \frac{1}{2} f(x - 1)$:

   • The graph should be shifted right by 1 and compressed vertically.

2. For $y = f\left( \frac{1}{2} x \right)$:

   • This graph should be stretched horizontally by a factor of 2.

3. For $y = f(-x)$:

   • This graph is a reflection of the original over the y-axis.

4. For $y = f(x - 2)$:

   • The graph should be shifted right by 2.

5. For $y = 2f(x)$:

   • This graph should be vertically stretched by a factor of 2.

6. For $y = f(2x)$:

   • The graph should be horizontally compressed.

Would you like help applying this to specific graphs, or would you like additional clarification? Let me know!

Here are five questions to extend your understanding:

1. How does a vertical stretch or shrink affect the graph of a function?
2. What is the difference between a horizontal stretch and a horizontal shrink?
3. How does reflecting a function over the y-axis change the appearance of the graph?
4. What happens to a function graphically when the transformation is inside the parentheses, affecting $x$?
5. How do shifts (horizontal and vertical) affect the graph of a function?

Tip: Always remember that transformations inside the function (affecting $x$) modify the graph horizontally, while transformations outside the function (affecting $f(x)$) modify it vertically.