https://file.aiquickdraw.com/mathbot/file/1726621695891i32ii3ba.jpg

The image contains a table showing the equations of transformations applied to functions, including horizontal and vertical stretches (hsf and vsf) and their corresponding mapping notations. Let's break it down:

### 1. First row:
- **Function:** \( y = 3f\left(\frac{x}{2}\right) \)
- **Stretch:** 
  - hsf = 2 (horizontal stretch factor)
  - vsf = 3 (vertical stretch factor)
- **Mapping notation:**
  - \( x \to 2x \)
  - \( y \to 3y \)

### 2. Second row:
- **Function:** \( y = \frac{1}{4}f\left(\frac{x}{3}\right) \)
- **Stretch:** 
  - hsf = 3
  - vsf = \(\frac{1}{4}\)
- **Mapping notation:**
  - \( x \to 3x \)
  - \( y \to \frac{y}{4} \)

### 3. Third row:
- **Function:** \( y = \frac{2}{5}f(4x) \)
- **Stretch:** 
  - hsf = \(\frac{1}{4}\)
  - vsf = \(\frac{2}{5}\)
- **Mapping notation:**
  - \( x \to \frac{x}{4} \)
  - \( y \to \frac{2}{5}y \)

### 4. Fourth row:
- **Function:** \( y = \frac{5}{9}f\left(\frac{7}{3}x\right) \)
- **Stretch:** 
  - hsf = \(\frac{3}{7}\)
  - vsf = \(\frac{5}{9}\)
- **Mapping notation:**
  - \( x \to \frac{3x}{7} \)
  - \( y \to \frac{5y}{9} \)

Would you like further clarification or details on how to apply these transformations?

Here are five related questions to deepen your understanding:

1. How do you determine whether a function is being horizontally stretched or compressed?
2. What effect does a vertical stretch factor greater than 1 have on the graph of a function?
3. How would you describe the difference between a horizontal stretch and a reflection across the y-axis?
4. What happens to the graph of a function when both horizontal and vertical transformations are applied?
5. Can you apply a horizontal stretch and a translation to a function at the same time?

**Tip:** Horizontal transformations affect the x-coordinate inversely; for example, a factor of \(\frac{1}{2}\) horizontally stretches by 2!

Write the equation of the image y = f(x) after each transformation. Include the mapping notation.

Learn how to apply horizontal and vertical stretches to functions and write the equations of the transformed images. This lesson includes function transformations, mapping notation, and worked examples suitable for high school students in Grades 9-12.

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