https://file.aiquickdraw.com/mathbot/file/17274039897968lbx16kf.jpg

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!

The graph of f is given. Use it to graph the following functions: (a) y = f(2x), (b) y = f(1/2x), (c) y = f(-x), (d) y = -f(-x).

Learn how to apply function transformations such as horizontal compression, stretch, and reflections on a given graph of f(x). This exercise focuses on graphing f(2x), f(1/2x), f(-x), and -f(-x), exploring the effects of these transformations step by step. Ideal for high school students.

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