The question asks to manipulate the graph of the function $f(x)$ in three different ways:

1. $f(-x)$: This represents a reflection of the graph of $f(x)$ across the y-axis. To draw this, flip the graph horizontally. For example, a point $(a, b)$ on the original graph will become $(-a, b)$ on the new graph.

2. $-f(x)$: This represents a reflection of the graph of $f(x)$ across the x-axis. To draw this, flip the graph vertically. A point $(a, b)$ on the original graph will become $(a, -b)$ on the new graph.

3. $-f(-x)$: This combines both transformations: first reflect the graph horizontally across the y-axis (giving $f(-x)$), and then reflect that result vertically across the x-axis (giving $-f(-x)$).

Let's now break it down:

Original graph of $f(x)$:

• There is a downward curve from $(-3, 4)$ to $(0, 0)$.
• A small line segment from $(0, 0)$ to $(1, 1)$.
• An open circle at $(1, 2)$ followed by a horizontal line from $(1, 3)$ to $(2, 3)$.
• An upward line starting from an open circle at $(2, 3)$ and passing through $(3, 4)$.

a) $f(-x)$:

• Reflect the graph across the y-axis. For example:
  • $(-3, 4)$ becomes $(3, 4)$,
  • $(0, 0)$ stays the same,
  • $(1, 1)$ becomes $(-1, 1)$,
  • The rest follows similarly.

b) $-f(x)$:

• Reflect the graph across the x-axis. For example:
  • $(-3, 4)$ becomes $(-3, -4)$,
  • $(0, 0)$ stays the same,
  • $(1, 1)$ becomes $(1, -1)$,
  • And so on.

c) $-f(-x)$:

• First reflect the graph across the y-axis (as in part a), and then reflect the result across the x-axis. For example:
  • $(-3, 4)$ becomes $(3, 4)$ (after reflecting across y-axis), and then $(3, -4)$ after reflecting across x-axis.
  • Continue similarly for the rest of the graph.

Do you want me to create the graphs for each transformation?

Follow-up Questions:

1. What general effect does $f(-x)$ have on a graph?
2. How does the transformation $-f(x)$ change the shape of a function graphically?
3. Can you combine multiple transformations like $-f(-x)$? How?
4. How can the domain and range change when applying transformations like these?
5. What are the coordinates of key points after applying $-f(x)$ to your graph?

Tip: When transforming functions, always reflect points individually to ensure accuracy.