https://r2.erweima.ai/mathbot/image/1725745745168xsg7lrvq.jpg

The image shows a graph of a function \( f(x) \) and a set of transformations of the function, with instructions to sketch the transformations. The transformations are:

(a) \( y = f(2x) \)  
(b) \( y = f\left(\frac{1}{2}x\right) \)  
(c) \( y = f(-x) \)  
(d) \( y = -f(-x) \)

### Step-by-step explanation of each transformation:

1. **\( y = f(2x) \)** (horizontal compression):  
   This transformation compresses the graph horizontally by a factor of 2. Every \( x \)-coordinate is divided by 2, meaning that for each point on the original graph, you plot the corresponding point at half the \( x \)-value.

2. **\( y = f\left(\frac{1}{2}x\right) \)** (horizontal stretch):  
   This transformation stretches the graph horizontally by a factor of 2. Every \( x \)-coordinate is multiplied by 2, which means the graph appears wider. Points on the graph are moved to twice their original \( x \)-coordinate.

3. **\( y = f(-x) \)** (reflection over the y-axis):  
   This reflects the graph over the \( y \)-axis. Each point \( (x, y) \) on the original graph is transformed to \( (-x, y) \).

4. **\( y = -f(-x) \)** (reflection over both axes):  
   This reflects the graph over both the \( x \)-axis and \( y \)-axis. Each point \( (x, y) \) is transformed to \( (-x, -y) \), effectively rotating the graph 180 degrees about the origin.

If you'd like, I can generate visual representations of these transformations based on the provided graph.

Let me know if you'd like more details on any of the transformations, or have any questions.

---

### Relative Questions:
1. How does vertical scaling affect the graph of a function?
2. What happens when a function is shifted horizontally or vertically?
3. What are the effects of combining multiple transformations on a function?
4. How do reflections of a function affect its properties (e.g., symmetry)?
5. What are the differences between horizontal and vertical transformations?

### Tip:
When applying transformations, focus on one type at a time (e.g., horizontal vs. vertical) to avoid confusion, and remember that horizontal transformations affect the \( x \)-axis while vertical transformations affect the \( y \)-axis.

Grafik f diberikan di bawah ini. Gunakanlah untuk mendeskripsikan fungsi-fungsi berikut: (a) y = f(2x), (b) y = f(1/2x), (c) y = f(-x), (d) y = -f(-x).

Learn how to apply function transformations like horizontal compression, horizontal stretch, and reflections over the y-axis and both axes using the graph of f(x). This problem covers key graphing concepts and is suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation