The graph you've provided shows a piecewise linear function with two distinct parts, and we are asked to express this as a piecewise function.

Let's analyze the graph:

• The first part (left side) shows a downward-sloping line starting at the point $(-6, 0)$ and passing through $(-3, -6)$.
• The second part (right side) shows an upward-sloping line starting from $(0, 2)$ and going through $(4, 6)$, but with an open circle at $(0, 2)$, indicating that the function does not include this point.

Now, let's determine the equations of each part.

Part 1: Line from $(-6, 0)$ to $(-3, -6)$

• This line has a slope $m = \frac{-6 - 0}{-3 - (-6)} = \frac{-6}{3} = -2$.
• The point-slope form of a line is $y - y_1 = m(x - x_1)$. Using the point $(-6, 0)$: $y - 0 = -2(x + 6)$ Simplifying, we get: $y = -2(x + 6) = -2x - 12$ Therefore, this part of the function is $y = -2x - 12$ for $-6 \leq x \leq -3$.

Part 2: Line from $(0, 2)$ to $(4, 6)$

• This line has a slope $m = \frac{6 - 2}{4 - 0} = \frac{4}{4} = 1$.
• Using the point $(0, 2)$ in the point-slope form: $y - 2 = 1(x - 0)$ Simplifying, we get: $y = x + 2$ Therefore, this part of the function is $y = x + 2$ for $0 < x \leq 4$.

Piecewise Function:

Combining both parts, the piecewise function is: