https://file.aiquickdraw.com/mathbot/file/1728398005102hir0mulh.jpg

The graph shown in the image is made of two distinct linear segments. Based on the points and slopes, we can write the piecewise function for it.

- The first part of the graph goes from point \((-5, 2)\) to the origin \((0, 0)\), which suggests a linear segment for \(x < 0\).
  - The slope \(m\) of this line can be calculated using the two points:  
    \[
    m = \frac{0 - 2}{0 - (-5)} = \frac{-2}{5}
    \]
    The line has a slope of \(-\frac{2}{5}\) and passes through \((0, 0)\), so the equation of the line is:  
    \[
    y = -\frac{2}{5}x
    \]

- The second part starts at \((0, 0)\) and extends to \((2, 1)\), for \(x \geq 0\).  
  - The slope of this line is:  
    \[
    m = \frac{1 - 0}{2 - 0} = \frac{1}{2}
    \]
    This line also passes through \((0, 0)\), and its equation is:
    \[
    y = \frac{1}{2}x
    \]

Thus, the piecewise function can be written as:
\[
f(x) = \begin{cases} 
-\frac{2}{5}x & \text{for } x < 0 \\
\frac{1}{2}x & \text{for } x \geq 0 
\end{cases}
\]

Now, looking at the options in the image, none of them seem to match exactly what we've derived, but the closest option in structure should reflect a change in slope for \(x < 0\) and \(x \geq 0\). You might need to check for approximation, but if I were to pick the closest answer based on structure alone, I would say **option B** looks similar.

Would you like more details or have any questions?

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### Related Questions:
1. How do you calculate the slope of a line between two points?
2. What is the general form of a piecewise function?
3. How does the slope affect the steepness of a line?
4. How would the function change if the lines had different y-intercepts?
5. Can you rewrite the piecewise function if the lines were not continuous?

**Tip**: In a piecewise function, it's important to ensure that each piece covers the correct domain (i.e., the range of \(x\) values where each line is valid).

Write a piecewise function for the graph shown below, and choose the correct answer from the provided options.

Solve a piecewise function problem by analyzing the graph. Learn how to calculate the slope of each segment and write the function for both x < 0 and x ≥ 0. This problem is suitable for Grades 9-11 students and involves understanding linear functions and their representation as piecewise functions.

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