1. **State the problem:** We need to graph the piecewise function: $$f(x) = \begin{cases} x - 3 & x < -1 \\ -x & -1 \leq x \leq 2 \\ -x - 1 & x > 2 \end{cases}$$ 2. **Understand each piece:** - For $x < -1$, the function is $f(x) = x - 3$, a line with slope 1 and y-intercept -3. - For $-1 \leq x \leq 2$, the function is $f(x) = -x$, a line with slope -1 passing through the origin. - For $x > 2$, the function is $f(x) = -x - 1$, a line with slope -1 and y-intercept -1. 3. **Evaluate boundary points:** - At $x = -1$, from the first piece: $f(-1) = -1 - 3 = -4$ (not included since $x < -1$). - At $x = -1$, from the second piece: $f(-1) = -(-1) = 1$ (included). - At $x = 2$, from the second piece: $f(2) = -2 = -2$ (included). - At $x = 2$, from the third piece: $f(2) = -2 - 1 = -3$ (not included since $x > 2$). 4. **Plot points and lines:** - For $x < -1$, plot the line $y = x - 3$ ending just before $x = -1$ with an open circle at $(-1, -4)$. - For $-1 \leq x \leq 2$, plot the line $y = -x$ from $x = -1$ to $x = 2$ including points $(-1, 1)$ and $(2, -2)$. - For $x > 2$, plot the line $y = -x - 1$ starting just after $x = 2$ with an open circle at $(2, -3)$. 5. **Summary:** The graph consists of three line segments with different slopes and intercepts, connected with open or closed points at the boundaries to reflect inclusivity. Final answer: The piecewise function is graphed as described above with the specified line segments and boundary points.