Let's go over the transformations listed and match them with the corresponding graphs:

1. $2f(x)$: This transformation vertically stretches the graph by a factor of 2, multiplying all the $y$-values by 2.

   • Look for the graph where the vertical scale is stretched but the $x$-coordinates remain the same.
   • Graph A shows this vertical stretch, so $2f(x)$ corresponds to A.

2. $f(x - 2)$: This transformation shifts the graph horizontally to the right by 2 units.

   • Look for the graph where all points have been moved 2 units to the right along the $x$-axis.
   • Graph E shows this horizontal shift, so $f(x - 2)$ corresponds to E.

3. $f(x) - 2$: This transformation shifts the graph downward by 2 units.

   • Look for the graph where the entire graph is shifted down by 2 units along the $y$-axis.
   • Graph D shows this vertical shift downward, so $f(x) - 2$ corresponds to D.

4. $-f(x)$: This transformation reflects the graph over the $x$-axis (flips vertically).

   • Look for the graph where the $y$-values are inverted, making it a reflection over the $x$-axis.
   • Graph C is the reflection, so $-f(x)$ corresponds to C.

5. $f(-x)$: This transformation reflects the graph over the $y$-axis (flips horizontally).

   • Look for the graph where the $x$-coordinates are inverted, reflecting the graph over the $y$-axis.
   • Graph B shows this horizontal reflection, so $f(-x)$ corresponds to B.

Summary:

• $2f(x)$: A
• $f(x - 2)$: E
• $f(x) - 2$: D
• $-f(x)$: C
• $f(-x)$: B

Would you like any further explanation on these transformations?

Here are five related questions to deepen your understanding:

1. What would $\frac{1}{2}f(x)$ do to the graph of $f(x)$?
2. How does $f(x + 2)$ differ from $f(x - 2)$?
3. How can you combine multiple transformations (e.g., vertical stretch and shift) in a single function?
4. What effect does $f(x) + 3$ have on the graph of $f(x)$?
5. Can you describe what $f(2x)$ would look like compared to $f(x)$?

Tip: Always apply horizontal transformations (shifts and stretches) before vertical ones when analyzing composite transformations.