To create a piecewise function from the given graph, we need to identify the distinct linear segments of the graph and their respective equations.

The graph consists of several line segments, which we will analyze step-by-step:

1. From $(-10, -5)$ to $(0, 0)$:

   • This segment is a straight line.
   • The slope $m$ is calculated as follows: $m = \frac{0 - (-5)}{0 - (-10)} = \frac{5}{10} = \frac{1}{2}$
   • The equation of the line in slope-intercept form $y = mx + b$: $y = \frac{1}{2}x + b$
   • To find $b$, use the point $(0, 0)$: $0 = \frac{1}{2}(0) + b \implies b = 0$
   • Thus, the equation is: $y = \frac{1}{2}x \quad \text{for} \quad -10 \leq x \leq 0$

2. From $(0, 0)$ to $(5, 0)$:

   • This segment is a horizontal line.
   • The equation of the line: $y = 0 \quad \text{for} \quad 0 < x \leq 5$

3. From $(5, 0)$ to $(7.5, 2.5)$:

   • This segment is a straight line.
   • The slope $m$ is calculated as follows: $m = \frac{2.5 - 0}{7.5 - 5} = \frac{2.5}{2.5} = 1$
   • The equation of the line in slope-intercept form $y = mx + b$: $y = 1x + b$
   • To find $b$, use the point $(5, 0)$: $0 = 1(5) + b \implies b = -5$
   • Thus, the equation is: $y = x - 5 \quad \text{for} \quad 5 < x \leq 7.5$

4. From $(7.5, 2.5)$ to $(10, 5)$:

   • This segment is a straight line.
   • The slope $m$ is calculated as follows: [ m = \frac{5 - 2.5}{10 - 7.5} = \frac{2.5}{