To express the function as a piecewise function, let's examine each segment of the graph and write the corresponding linear equations with their respective intervals.

Step-by-Step Solution

1. Identify the line segments and endpoints:

   • The graph shows two line segments:
     • Segment 1: Starts at $x = -6$ and ends at $x = -2$.
     • Segment 2: Starts at $x = -2$ and extends downwards, ending at $x = 6$.

2. Find the equations of each segment:

   • Segment 1 (from $x = -6$ to $x = -2$):

     • Points on the line: $(-6, 4)$ and $(-2, 0)$.
     • Calculate the slope $m$: $m = \frac{0 - 4}{-2 + 6} = \frac{-4}{4} = -1$
     • Using point-slope form with point $(-6, 4)$: $y - 4 = -1(x + 6)$ Simplify to get the equation: $y = -x - 2$
     • This segment is defined for $-6 \leq x \leq -2$.

   • Segment 2 (from $x = -2$ to $x = 6$):

     • Points on the line: $(-2, 0)$ and $(6, -8)$.
     • Calculate the slope $m$: $m = \frac{-8 - 0}{6 + 2} = \frac{-8}{8} = -1$
     • Using point-slope form with point $(-2, 0)$: $y - 0 = -1(x + 2)$ Simplify to get the equation: $y = -x - 2$
     • This segment is defined for $-2 < x \leq 6$.

3. Write the piecewise function:

   -x - 2 & \text{for } -6 \leq x \leq -2, \\ -x - 2 & \text{for } -2 < x \leq 6. \end{cases}$$

This defines the function in piecewise form based on the intervals observed in the graph.

Would you like more details or have any questions?

Related Questions

1. How do you determine the slope of a line given two points?
2. How would the piecewise function change if one of the intervals had a different slope?
3. What is the point-slope form, and why is it useful here?
4. How do open and closed intervals affect piecewise functions?
5. Can this function be represented as a single equation instead of piecewise?

Tip: In piecewise functions, always pay close attention to open and closed endpoints to match the graph's behavior.