https://file.aiquickdraw.com/mathbot/file/1725442233125endlmtzt.jpg

The image you've provided shows a piecewise linear function on a graph with \( y \) on the vertical axis and \( x \) on the horizontal axis. The function appears to be a series of connected linear segments with certain critical points where the slope changes.

Here’s a breakdown of the graph:
1. The graph starts at \( (0, 0) \) and increases linearly to \( (10, 27) \).
2. It stays constant at \( y = 27 \) from \( x = 10 \) to \( x = 21 \).
3. The graph then decreases linearly from \( (21, 27) \) to \( (31, 0) \).
4. Finally, it remains at \( y = 0 \) from \( x = 31 \) to \( x = 42 \).

### Analysis of the Graph:
- **First Segment (0 to 10):** This is a line with a positive slope, increasing from \( (0,0) \) to \( (10, 27) \). The slope \( m_1 \) can be calculated as:
  \[
  m_1 = \frac{27 - 0}{10 - 0} = \frac{27}{10} = 2.7
  \]
  So, the equation for this segment is \( y = 2.7x \).

- **Second Segment (10 to 21):** This is a horizontal line where \( y \) is constant at 27. The equation for this segment is \( y = 27 \).

- **Third Segment (21 to 31):** This is a line with a negative slope, decreasing from \( (21, 27) \) to \( (31, 0) \). The slope \( m_2 \) is:
  \[
  m_2 = \frac{0 - 27}{31 - 21} = \frac{-27}{10} = -2.7
  \]
  So, the equation for this segment is \( y = -2.7(x - 21) + 27 \).

- **Fourth Segment (31 to 42):** This is a horizontal line where \( y \) is constant at 0. The equation for this segment is \( y = 0 \).

### Specific Questions:

1. **How does the function behave between \( x = 0 \) and \( x = 10 \)?**
2. **What is the slope of the graph between \( x = 21 \) and \( x = 31 \)?**
3. **At what points does the function reach its maximum value?**
4. **What are the equations of the segments where the function is constant?**
5. **How would the graph change if the slope of the first segment were doubled?**

**Tip:** When analyzing piecewise functions, breaking them into segments and analyzing each segment individually helps to better understand the overall behavior of the function.

Analyze the piecewise linear function shown in the graph.

Explore the analysis of a piecewise linear function graph, focusing on calculating the slopes of individual segments and understanding the behavior of the function. Suitable for high school students (Grades 9-12).

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