To express the function graphed as a piecewise function, we need to analyze the different line segments and their corresponding equations. The graph shows two distinct linear segments:

1. First Segment (from $x = -7$ to $x = -2$):

   • This segment is a line with a slope passing through points (-7, -5) and (-2, -2).
   • Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, the slope $m$ is: $m = \frac{-2 - (-5)}{-2 - (-7)} = \frac{-2 + 5}{-2 + 7} = \frac{3}{5}$
   • Now, using the point-slope form $y - y_1 = m(x - x_1)$ and the point (-7, -5), we find the equation of this line: $y + 5 = \frac{3}{5}(x + 7)$ Simplifying: $y = \frac{3}{5}(x + 7) - 5$

2. Second Segment (from $x = -2$ to $x = 4$):

   • This segment is a line with slope through points (-2, 2) and (4, 8).
   • Again, using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, the slope $m$ is: $m = \frac{8 - 2}{4 - (-2)} = \frac{6}{6} = 1$
   • Now using the point-slope form $y - y_1 = m(x - x_1)$ with the point (-2, 2), the equation is: $y - 2 = 1(x + 2)$ Simplifying: $y = x + 2$

3. Third Segment (from $x = 4$ to $x = 7$):

   • This is a line segment passing through points (4, 8) and (7, 10).
   • The slope is: $m = \frac{10 - 8}{7 - 4} = \frac{2}{3}$
   • Using point-slope form with point (4, 8): $y - 8 = \frac{2}{3}(x - 4)$ Simplifying: $y = \frac{2}{3}(x - 4) + 8$

Piecewise Function

The piecewise function that describes the graph is: