The problem asks to graph four transformations of the given function $f(x)$ based on the graph provided. The transformations are as follows:

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

Let’s briefly break down how each transformation affects the graph:

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

• This is a horizontal compression by a factor of 2.
• For every $x$-coordinate on the original graph, the corresponding point will now occur at $\frac{x}{2}$, making the graph "squished" horizontally by half.

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

• This is a horizontal stretch by a factor of 2.
• Every $x$-coordinate on the original graph will be doubled, spreading the graph out horizontally.

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

• This represents a reflection over the y-axis.
• Every point $(x, y)$ on the original graph will be reflected to $(-x, y)$.

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

• This combines a reflection over the y-axis and a reflection over the x-axis, which results in a reflection through the origin.
• The transformation will flip the graph both horizontally and vertically.

Would you like a detailed description of the points on each graph or further explanations for any of these transformations? Let me know if you'd like to see a sketch or have questions about the transformations.

Here are some related questions for further practice:

1. What effect does $y = f(3x)$ have on the graph?
2. How would the graph change if the transformation was $y = -f(x)$?
3. What does the transformation $y = f(x + 2)$ do to the graph?
4. How do vertical and horizontal transformations differ?
5. What is the graphical interpretation of $y = f\left(\frac{1}{3}x\right)$?

Tip: Remember, multiplying inside the function affects the graph horizontally, while multiplying outside the function affects it vertically!