https://file.aiquickdraw.com/mathbot/file/1727884750876wtagzwqj.jpg

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.

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 various transformations to a function's graph, including horizontal compressions, stretches, and reflections across axes. Suitable for high school students studying graph transformations in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation