The graph shows a piecewise function, $f(x)$, made up of three distinct line segments. To find the equation of each segment, we will determine the slope and y-intercept for each part.

Identifying the intervals and finding the equations:

1. First segment (Leftmost horizontal line):

   • This segment is horizontal and exists from $x = -\infty$ to $x = -8$.
   • The line is constant at $y = -10$ (no slope), so the equation is: $f(x) = -10 \quad \text{for} \quad x \leq -8$

2. Second segment (Middle line):

   • This segment starts at $(-8, -10)$ and ends at $(1, 5)$.
   • To find the slope ($m$): $m = \frac{5 - (-10)}{1 - (-8)} = \frac{15}{9} = \frac{5}{3}$
   • Using the point-slope form $y - y_1 = m(x - x_1)$ with $(x_1, y_1) = (-8, -10)$: $y + 10 = \frac{5}{3}(x + 8)$ Expanding: $y = \frac{5}{3}x + \frac{5}{3}(8) - 10 = \frac{5}{3}x + \frac{40}{3} - 10 = \frac{5}{3}x + \frac{40}{3} - \frac{30}{3} = \frac{5}{3}x + \frac{10}{3}$ So, the equation for this segment is: $f(x) = \frac{5}{3}x + \frac{10}{3} \quad \text{for} \quad -8 < x \leq 1$

3. Third segment (Rightmost line):

   • This segment starts at $(1, 5)$ and continues upwards.
   • Using points $(1, 5)$ and $(7, 13)$ to find the slope: $m = \frac{13 - 5}{7 - 1} = \frac{8}{6} = \frac{4}{3}$
   • Using point-slope form $y - y_1 = m(x - x_1)$ with $(x_1, y_1) = (1, 5)$: $y - 5 = \frac{4}{3}(x - 1)$ Expanding: $y = \frac{4}{3}x - \frac{4}{3} + 5 = \frac{4}{3}x + \frac{11}{3}$ So, the equation for this segment is: $f(x) = \frac{4}{3}x + \frac{11}{3} \quad \text{for} \quad x > 1$

Piecewise Function: