The image contains a graph representing a piecewise function, and you are asked to complete the equation for $f(x)$.

Here’s the process to determine the equations for each segment:

1. Segment from $x = -6$ to $x = -2$:

This is a linear segment with a decreasing slope. To find the equation:

• The endpoints are $(-6, 0)$ and $(-2, 2)$.
• Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{-2 - (-6)} = \frac{2}{4} = \frac{1}{2}$.
• The point-slope form of a line is $y - y_1 = m(x - x_1)$. Using the point $(-6, 0)$ and slope $\frac{1}{2}$, the equation becomes: $y - 0 = \frac{1}{2}(x - (-6)) \implies y = \frac{1}{2}(x + 6)$

So, for $-6 \leq x \leq -2$, the equation is $f(x) = \frac{1}{2}(x + 6)$.

2. Segment from $x = -2$ to $x = 3$:

This is a constant function at $f(x) = 2$. This means the function is simply $f(x) = 2$ for $-2 < x \leq 3$.

3. Segment from $x = 3$ to $x = 6$:

This is another linear segment, this time decreasing. The endpoints are $(3, 2)$ and $(6, -4)$.

• The slope is $m = \frac{-4 - 2}{6 - 3} = \frac{-6}{3} = -2$.
• Using point-slope form with the point $(3, 2)$: $y - 2 = -2(x - 3) \implies y = -2(x - 3) + 2 \implies y = -2x + 6 + 2 \implies y = -2x + 8$

So, for $3 < x \leq 6$, the equation is $f(x) = -2x + 8$.

Complete Piecewise Function: