The graph shows a piecewise function made up of two parts. Let's break it down and express it in a formal piecewise format:

1. First segment (horizontal line):

   • This part is a horizontal line with a constant $y$-value of $-4$, for all $x \leq -3$.
   • This part has a closed circle at $x = -3$, indicating that the function includes the point $(-3, -4)$.
   • So, the first part of the function is: $f(x) = -4, \quad \text{for} \ x \leq -3$

2. Second segment (linear line):

   • The second part is a sloped line starting at the open circle at $(-3, -4)$ and extending upwards.
   • This part extends to $(4, 8)$, so we need to find the equation of this line. Using the points $(-3, -4)$ and $(4, 8)$, we can calculate the slope $m$: $m = \frac{8 - (-4)}{4 - (-3)} = \frac{8 + 4}{4 + 3} = \frac{12}{7}$
   • Now, using the point-slope form $y - y_1 = m(x - x_1)$ with $x_1 = -3$ and $y_1 = -4$: $y - (-4) = \frac{12}{7}(x - (-3))$ Simplifying: $y + 4 = \frac{12}{7}(x + 3)$ $y = \frac{12}{7}(x + 3) - 4$ Expanding: $y = \frac{12}{7}x + \frac{36}{7} - 4 = \frac{12}{7}x + \frac{36}{7} - \frac{28}{7} = \frac{12}{7}x + \frac{8}{7}$
   • Thus, the equation of the second part of the function is: $f(x) = \frac{12}{7}x + \frac{8}{7}, \quad \text{for} \ -3 < x \leq 4$

Full Piecewise Function