1. **State the problem:** We need to express the graph as a piecewise function $f(x)$ based on the two line segments described. 2. **Identify the points and slopes:** - First segment passes through $(-7,8)$ and $(0,-5)$ (closed circle at $x=0$). - Second segment passes through $(0,-5)$ (open circle) and $(4,6)$. 3. **Calculate the slope of the first segment:** $$m_1 = \frac{-5 - 8}{0 - (-7)} = \frac{-13}{7}$$ 4. **Write the equation of the first line segment:** Using point-slope form with point $(-7,8)$: $$y - 8 = m_1(x + 7)$$ $$y - 8 = \frac{-13}{7}(x + 7)$$ $$y = \frac{-13}{7}x - 13 + 8 = \frac{-13}{7}x - 5$$ 5. **Calculate the slope of the second segment:** $$m_2 = \frac{6 - (-5)}{4 - 0} = \frac{11}{4}$$ 6. **Write the equation of the second line segment:** Using point-slope form with point $(0,-5)$: $$y - (-5) = m_2(x - 0)$$ $$y + 5 = \frac{11}{4}x$$ $$y = \frac{11}{4}x - 5$$ 7. **Define the piecewise function with domain restrictions:** - First segment includes $x$ from $-7$ to $0$ (closed at $0$): $$f(x) = \frac{-13}{7}x - 5, \quad -7 \leq x \leq 0$$ - Second segment starts just after $0$ (open circle at $0$) to $4$: $$f(x) = \frac{11}{4}x - 5, \quad 0 < x \leq 4$$ **Final answer:** $$ f(x) = \begin{cases} \frac{-13}{7}x - 5, & -7 \leq x \leq 0 \\ \frac{11}{4}x - 5, & 0 < x \leq 4 \end{cases} $$