To solve this question, we analyze how transformations affect the given graph $f(x)$, as follows:

Transformations Explained

1. $f(x - 3)$ (Option A):

   • Horizontal shift to the right by 3 units.
   • Each $x$-coordinate increases by 3, but $y$-values remain unchanged.

2. $f(x) - 2$ (Option B):

   • Vertical shift downward by 2 units.
   • Each $y$-coordinate decreases by 2, but $x$-values remain the same.

3. $-f(x)$ (Option C):

   • Reflection across the $x$-axis.
   • The $y$-coordinates are multiplied by -1, flipping the graph vertically.

4. $f(x + 1)$ (Option D):

   • Horizontal shift to the left by 1 unit.
   • Each $x$-coordinate decreases by 1, while $y$-values stay the same.

5. $f(-x)$ (Option E):

   • Reflection across the $y$-axis.
   • The $x$-coordinates are negated, flipping the graph horizontally.

6. $|f(x)|$ (Option F):

   • All negative $y$-values become positive, reflecting any part of the graph below the $x$-axis upward.

Step-by-Step Summary of Each Option

1. Option A: $f(x - 3)$:

   • Shift the graph 3 units to the right.

2. Option B: $f(x) - 2$:

   • Shift the graph 2 units downward.

3. Option C: $-f(x)$:

   • Reflect the graph vertically across the $x$-axis.

4. Option D: $f(x + 1)$:

   • Shift the graph 1 unit to the left.

5. Option E: $f(-x)$:

   • Reflect the graph horizontally across the $y$-axis.

6. Option F: $|f(x)|$:

   • Reflect all parts of the graph below the $x$-axis upward so that $y \geq 0$.

Graph Summary

If you plot each of these transformations, the resulting graphs would look like:

• $f(x-3)$ shifted right.
• $f(x)-2$ shifted down.
• $-f(x)$ flipped vertically.
• $f(x+1)$ shifted left.
• $f(-x)$ flipped horizontally.
• $|f(x)|$, where negative $y$-values are reflected upward.

Do you want detailed graphs of these transformations? Let me know!

Related Questions

1. How do you identify horizontal and vertical shifts in a function?
2. What happens when a graph is reflected across the $x$-axis?
3. How do you handle $|f(x)|$ for a given function graph?
4. What is the impact of $f(-x)$ on the graph of $f(x)$?
5. How do you graph $f(x \pm c)$ transformations step-by-step?

Tip: Always analyze transformations in the order: reflections, stretches/compressions, and then shifts.