1. **State the problem:** We need to graph the piecewise function: $$f(x) = \begin{cases} 7 - \frac{1}{2}x & \text{if } x \leq -4 \\ -3 & \text{if } -4 < x \leq 5 \\ (x - 5)^2 & \text{if } x > 5 \end{cases}$$ 2. **Understand each piece:** - For $x \leq -4$, the function is linear: $7 - \frac{1}{2}x$. - For $-4 < x \leq 5$, the function is constant: $-3$. - For $x > 5$, the function is quadratic: $(x - 5)^2$. 3. **Evaluate key points:** - At $x = -4$, linear piece: $7 - \frac{1}{2}(-4) = 7 + 2 = 9$. - At $x = -4$, constant piece starts at $-3$ (open circle since $x > -4$). - At $x = 5$, constant piece value is $-3$. - At $x = 5$, quadratic piece starts at $(5 - 5)^2 = 0$ (open circle since $x > 5$). 4. **Graph behavior:** - For $x \leq -4$, plot the line $y = 7 - \frac{1}{2}x$ ending at point $(-4,9)$ (closed circle). - For $-4 < x \leq 5$, plot the horizontal line $y = -3$ from just right of $x = -4$ to $x = 5$ (closed circle at $x=5$). - For $x > 5$, plot the parabola $y = (x - 5)^2$ starting just right of $x=5$ at $y=0$. 5. **Summary:** The graph consists of three parts connected by open or closed circles at the boundaries to indicate inclusion or exclusion of points. Final answer: The piecewise function is graphed as described above with the three segments and their respective domains.