1. **State the problem:** We need to complete the description of the piecewise function $f(x)$ based on the graph with three segments: - Segment 1: from $(-6,5)$ to $(-1,-4)$ inclusive - Segment 2: horizontal line from $(-1,-4)$ to $(3,-4)$ inclusive - Segment 3: from $(3,-6)$ to $(6,-3)$ with an open circle at $(3,-6)$ (excluded) 2. **Write the piecewise function form:** $$ f(x) = \begin{cases} \text{Segment 1} & -6 \leq x \leq -1 \\ -4 & -1 < x \leq 3 \\ \text{Segment 3} & 3 < x \leq 6 \end{cases} $$ 3. **Find the equation for Segment 1 (line from $(-6,5)$ to $(-1,-4)$):** - Slope $m = \frac{-4 - 5}{-1 - (-6)} = \frac{-9}{5} = -\frac{9}{5}$ - Use point-slope form with point $(-6,5)$: $$ f(x) - 5 = -\frac{9}{5}(x + 6) $$ - Simplify: $$ f(x) = -\frac{9}{5}x - \frac{54}{5} + 5 = -\frac{9}{5}x - \frac{54}{5} + \frac{25}{5} = -\frac{9}{5}x - \frac{29}{5} $$ 4. **Segment 2 is given as a constant:** $$ f(x) = -4 \quad \text{for} \quad -1 < x \leq 3 $$ 5. **Find the equation for Segment 3 (line from $(3,-6)$ to $(6,-3)$):** - Slope $m = \frac{-3 - (-6)}{6 - 3} = \frac{3}{3} = 1$ - Use point-slope form with point $(3,-6)$: $$ f(x) - (-6) = 1(x - 3) $$ - Simplify: $$ f(x) = x - 3 - 6 = x - 9 $$ - Note: The point $(3,-6)$ is excluded (open circle), so domain is $3 < x \leq 6$ 6. **Final piecewise function:** $$ f(x) = \begin{cases} -\frac{9}{5}x - \frac{29}{5} & -6 \leq x \leq -1 \\ -4 & -1 < x \leq 3 \\ x - 9 & 3 < x \leq 6 \end{cases} $$