Let's analyze the piecewise function shown in the graph step-by-step.

1. First Interval $-6 \leq x \leq -1$:

   • This part of the graph shows a straight line starting at the point $(-6, -2)$ and ending at the point $(-1, 4)$.
   • To find the equation of this line, we use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{-1 - (-6)} = \frac{4 + 2}{-1 + 6} = \frac{6}{5} = \frac{6}{5}.$
   • Using the point-slope form $y - y_1 = m(x - x_1)$, with one of the points $(-6, -2)$: $y + 2 = \frac{6}{5}(x + 6).$
   • Simplifying this equation to the slope-intercept form $y = mx + b$: $y = \frac{6}{5}x + \frac{36}{5} - 2 = \frac{6}{5}x + \frac{36}{5} - \frac{10}{5} = \frac{6}{5}x + \frac{26}{5}.$
   • So, the equation for this interval is: $f(x) = \frac{6}{5}x + \frac{26}{5}.$

2. Second Interval $-1 < x \leq 3$:

   • This part of the graph is a horizontal line between the points $(-1, 4)$ and $(3, 4)$.
   • Since it's a horizontal line, the equation is simply: $f(x) = 4.$

3. Third Interval $3 < x \leq 6$:

   • This part of the graph shows a straight line starting at the point $(3, 4)$ and ending at the point $(6, 1)$.
   • To find the equation of this line: $m = \frac{1 - 4}{6 - 3} = \frac{-3}{3} = -1.$
   • Using the point-slope form with the point $(3, 4)$: $y - 4 = -1(x - 3).$
   • Simplifying to the slope-intercept form: $y = -x + 3 + 4 = -x + 7.$
   • So, the equation for this interval is: $f(x) = -x + 7.$

Final Piecewise Function

The complete description of the piecewise function is: