1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} -2x - 2 & \text{for } x < 1 \\ -4 & \text{for } 1 \leq x < 6 \\ 5x - 35 & \text{for } x \geq 6 \end{cases}$$ We want to understand and graph this function over the domain $-10 \leq x \leq 10$. 2. **Understand the pieces:** - For $x < 1$, the function is linear with slope $-2$ and intercept $-2$. - For $1 \leq x < 6$, the function is constant at $-4$. - For $x \geq 6$, the function is linear with slope $5$ and intercept $-35$. 3. **Evaluate key points and continuity:** - At $x=1$, from the left: $f(1^-) = -2(1) - 2 = -4$. - At $x=1$, from the right: $f(1) = -4$. - So the function is continuous at $x=1$. - At $x=6$, from the left: $f(6^-) = -4$. - At $x=6$, from the right: $f(6) = 5(6) - 35 = 30 - 35 = -5$. - So there is a jump discontinuity at $x=6$. 4. **Summary:** - The graph is a line with slope $-2$ for $x<1$ ending at point $(1,-4)$. - Then a horizontal line at $y=-4$ from $x=1$ to $x=6$. - Then a line with slope $5$ starting at $(6,-5)$ for $x \geq 6$. 5. **Graphing function for Desmos:** $$y = \begin{cases} -2x - 2 & x < 1 \\ -4 & 1 \leq x < 6 \\ 5x - 35 & x \geq 6 \end{cases}$$ This piecewise function can be plotted directly in Desmos. **Final answer:** The piecewise function is defined and continuous at $x=1$, with a jump at $x=6$, and the graph consists of two linear segments and one constant segment as described.