1. **State the problem:** We need to express the graphed function as a piecewise function based on the given points and intervals. 2. **Analyze the graph:** - From $x=-1$ to $x=2$, the graph is a line starting at $(-1,-4)$ with a solid dot and ending at $(2,4)$ with an open circle. - From $x=3$ to beyond $x=6$, the graph starts at $(3,3)$ with an open circle and slopes downward passing through $(6,0)$ and continues. 3. **Find the equation for the first piece:** - The line passes through $(-1,-4)$ and approaches $(2,4)$. - Slope $m=\frac{4 - (-4)}{2 - (-1)}=\frac{8}{3}$. - Equation using point-slope form: $$y - (-4) = \frac{8}{3}(x - (-1))$$ $$y + 4 = \frac{8}{3}(x + 1)$$ $$y = \frac{8}{3}x + \frac{8}{3} - 4 = \frac{8}{3}x + \frac{8}{3} - \frac{12}{3} = \frac{8}{3}x - \frac{4}{3}$$ 4. **Domain for first piece:** $-1 \leq x < 2$ (solid dot at $-1$, open circle at $2$). 5. **Find the equation for the second piece:** - The line passes through $(3,3)$ and $(6,0)$. - Slope $m=\frac{0 - 3}{6 - 3} = \frac{-3}{3} = -1$. - Equation using point-slope form: $$y - 3 = -1(x - 3)$$ $$y - 3 = -x + 3$$ $$y = -x + 6$$ 6. **Domain for second piece:** $3 < x$ (open circle at $3$, continues beyond $6$). 7. **Write the piecewise function:** $$f(x) = \begin{cases} \frac{8}{3}x - \frac{4}{3} & \text{for } -1 \leq x < 2 \\ -x + 6 & \text{for } x > 3 \end{cases}$$ Note: The function is not defined between $2$ and $3$ based on the graph. **Final answer:** $$f(x) = \begin{cases} \frac{8}{3}x - \frac{4}{3} & -1 \leq x < 2 \\ -x + 6 & x > 3 \end{cases}$$