To help you match the given functions with their corresponding graphs, let's go through the transformations and compare them with the graphs provided.

Here’s a breakdown of each function transformation:

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

   • $f(x-1)$: This is a horizontal shift 1 unit to the right.
   • $\frac{1}{2}$: This compresses the graph vertically by a factor of $\frac{1}{2}$.

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

   • $f\left(\frac{1}{2}x\right)$: This is a horizontal stretch by a factor of 2 (since $\frac{1}{2}$ causes a widening along the x-axis).

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

   • This reflects the graph across the y-axis (a horizontal reflection).

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

   • This is a horizontal shift 2 units to the right.

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

   • This stretches the graph vertically by a factor of 2 (makes it twice as tall).

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

   • This is a horizontal compression by a factor of 2 (the graph gets narrower along the x-axis).

Now, comparing these with the graphs provided (A-F):

• Graph A: Appears to have a horizontal stretch, which would correspond to the function $y = f\left(\frac{1}{2}x\right)$ (Option 2).
• Graph B: Looks like it has been horizontally compressed, so it likely corresponds to $y = f(2x)$ (Option 6).
• Graph C: Seems like a horizontal reflection (flipped across the y-axis), which corresponds to $y = f(-x)$ (Option 3).
• Graph D: This one looks vertically stretched, which would match $y = 2f(x)$ (Option 5).
• Graph E: Appears to be shifted 1 unit to the right and vertically compressed, matching $y = \frac{1}{2} f(x-1)$ (Option 1).
• Graph F: Appears to be shifted 2 units to the right, so it corresponds to $y = f(x-2)$ (Option 4).

Summary of Matches:

1. $y = \frac{1}{2}f(x-1)$ → E
2. $y = f\left(\frac{1}{2}x\right)$ → A
3. $y = f(-x)$ → C
4. $y = f(x-2)$ → F
5. $y = 2f(x)$ → D
6. $y = f(2x)$ → B

Would you like further details or explanations on any of these transformations?

Related Questions:

1. How does a vertical compression affect the overall shape of a function?
2. What is the impact of horizontal stretches on periodic functions?
3. How does the reflection $y = f(-x)$ compare to $y = -f(x)$?
4. Why does $f(x-2)$ shift the graph to the right instead of left?
5. Can a function transformation have both horizontal and vertical changes simultaneously?

Tip:

When identifying transformations, focus on how the function changes with respect to the x-axis (horizontal shifts, stretches) and y-axis (vertical shifts, stretches).