1. The problem is to graph the piecewise function: $$f(x) = \begin{cases} -2x - 2 & \text{for } x < 1 \\ -4 & \text{for } 1 \leq x < 6 \\ 5x - 35 & \text{for } x \geq 6 \end{cases}$$ 2. We will find key points for each piece to plot. 3. For the first piece $f(x) = -2x - 2$ when $x < 1$: - At $x = 0$, $f(0) = -2(0) - 2 = -2$ gives point $(0, -2)$. - At $x = 1$, since the domain is $x < 1$, we use an open circle at $x=1$: $$f(1) = -2(1) - 2 = -4$$ so open circle at $(1, -4)$. 4. For the second piece $f(x) = -4$ when $1 \leq x < 6$: - This is a constant function, so plot a horizontal line at $y = -4$ from $x=1$ to $x=6$. - Closed circle at $x=1$ because $1 \leq x$, so point $(1, -4)$ closed circle. - Open circle at $x=6$ because $x < 6$, so point $(6, -4)$ open circle. 5. For the third piece $f(x) = 5x - 35$ when $x \geq 6$: - At $x=6$, $f(6) = 5(6) - 35 = 30 - 35 = -5$ closed circle at $(6, -5)$. - At $x=7$, $f(7) = 5(7) - 35 = 35 - 35 = 0$ point $(7, 0)$. 6. Summary of points to plot: - $(0, -2)$ - Open circle at $(1, -4)$ for first piece - Closed circle at $(1, -4)$ for second piece - Horizontal line from $(1, -4)$ closed circle to $(6, -4)$ open circle - Closed circle at $(6, -5)$ for third piece - Point $(7, 0)$ on third piece These points and circles define the graph segments clearly.