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: