1. **State the problem:** We need to graph the piecewise function $$f(x) = \begin{cases} 7 + 0.5x & \text{if } 0 \leq x \leq 8 \\ -5 + 2x & \text{if } x > 8 \end{cases}$$ 2. **Understand the pieces:** - For $0 \leq x \leq 8$, the function is linear with slope $0.5$ and intercept $7$. - For $x > 8$, the function is linear with slope $2$ and intercept $-5$. 3. **Calculate key points:** - At $x=0$, $f(0) = 7 + 0.5 \times 0 = 7$ (closed point). - At $x=8$, $f(8) = 7 + 0.5 \times 8 = 7 + 4 = 11$ (open circle because the second piece takes over immediately after 8). - For $x > 8$, the function starts just after $x=8$: $$f(8^+) = -5 + 2 \times 8 = -5 + 16 = 11$$ So the second piece starts at $(8,11)$ but with an open circle at $x=8$ for the first piece and a ray starting just after $8$. 4. **Graph shape:** - From $x=0$ to $x=8$, a line segment with slope $0.5$ starting at $(0,7)$ and ending at $(8,11)$ with an open circle. - From $x>8$, a ray starting just after $(8,11)$ with slope $2$ going upwards. 5. **Conclusion:** The graph matches the description of option A: piecewise linear with a filled point at $(0,7)$, line segment increasing with slope $0.5$ from $0$ to $8$, open circle at $(8,11)$, then a steeper ray with slope $2$ starting just after $8$. Final answer: **Graph A** matches the function.