The problem involves graph transformations based on the given graph of $f(x)$. Let's analyze the graph and transformations:

1. The graph of $f(x)$: The graph provided shows a piecewise linear function, which starts at $(0, 0)$, goes up to $(1, 1)$, down to $(2, 0)$, and then up to $(3, 1)$.

Now, let's apply the transformations to graph each function:

(a) $y = f(2x)$

This represents a horizontal compression by a factor of 2. Every point on the original graph is compressed horizontally, meaning we divide the $x$-coordinates by 2.

• Point $(0, 0)$ remains at $(0, 0)$.
• Point $(1, 1)$ becomes $\left(\frac{1}{2}, 1\right)$.
• Point $(2, 0)$ becomes $(1, 0)$.
• Point $(3, 1)$ becomes $\left(\frac{3}{2}, 1\right)$.

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

This represents a horizontal stretch by a factor of 2. Every point on the original graph is stretched horizontally, meaning we multiply the $x$-coordinates by 2.

• Point $(0, 0)$ remains at $(0, 0)$.
• Point $(1, 1)$ becomes $(2, 1)$.
• Point $(2, 0)$ becomes $(4, 0)$.
• Point $(3, 1)$ becomes $(6, 1)$.

(c) $y = f(-x)$

This is a reflection across the $y$-axis. Every point on the original graph has its $x$-coordinate negated.

• Point $(0, 0)$ remains at $(0, 0)$.
• Point $(1, 1)$ becomes $(-1, 1)$.
• Point $(2, 0)$ becomes $(-2, 0)$.
• Point $(3, 1)$ becomes $(-3, 1)$.

(d) $y = -f(-x)$

This involves two transformations: a reflection across the $y$-axis and a reflection across the $x$-axis. Therefore, every point’s $x$-coordinate is negated, and the $y$-coordinate is also negated.

• Point $(0, 0)$ remains at $(0, 0)$.
• Point $(1, 1)$ becomes $(-1, -1)$.
• Point $(2, 0)$ becomes $(-2, 0)$.
• Point $(3, 1)$ becomes $(-3, -1)$.

If you need help sketching these transformations or any further clarifications, feel free to ask!

Here are five follow-up questions to deepen your understanding:

1. How do horizontal compressions and stretches affect the graph of a function?
2. What is the general rule for reflecting a function across the $x$-axis and $y$-axis?
3. How does the transformation $y = f(-x)$ change the graph's symmetry?
4. Can a graph transformation result in no visible changes? Why or why not?
5. How would vertical transformations affect the graph, like $y = 2f(x)$ or $y = \frac{1}{2}f(x)$?

Tip: When graphing transformations, always start by identifying how the transformation affects the $x$- and $y$-coordinates separately to guide your adjustments.