To write the piecewise function for the given graph, let's analyze the two line segments on the graph:

Segment 1 (for $x < -3$):

• This line passes through the points $(-6, 6)$ and $(-3, 3)$.
• To find the equation of the line, we first compute the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 6}{-3 - (-6)} = \frac{-3}{3} = -1$ So, the slope is $-1$.
• Now, using the point-slope form of the equation $y - y_1 = m(x - x_1)$, we can substitute one of the points, say $(-3, 3)$: $y - 3 = -1(x + 3)$ Simplifying: $y = -x$ Therefore, the equation for this segment is $y = -x$ for $x < -3$.

Segment 2 (for $x \geq -3$):

• This line passes through the points $(-3, -5)$ and $(3, 1)$.
• Again, we compute the slope $m$: $m = \frac{1 - (-5)}{3 - (-3)} = \frac{1 + 5}{3 + 3} = \frac{6}{6} = 1$ So, the slope is $1$.
• Using the point-slope form with the point $(3, 1)$: $y - 1 = 1(x - 3)$ Simplifying: $y = x - 2$ Therefore, the equation for this segment is $y = x - 2$ for $x \geq -3$.

Final Piecewise Function: