1. **Problem 1: Find $g(81)$ and $g(-32)$ for a periodic function $y = g(x)$ with period approximately 8.** 2. The function $g$ repeats every 8 units, so to find $g(81)$ and $g(-32)$, we reduce these inputs modulo the period 8. 3. Calculate $81 \mod 8$: $$81 = 8 \times 10 + 1 \implies 81 \mod 8 = 1$$ 4. Calculate $-32 \mod 8$: $$-32 = 8 \times (-4) + 0 \implies -32 \mod 8 = 0$$ 5. From the graph description, at $x=0$, $g(0) = 12$ (peak), and at $x=1$, the value is near the peak but we need to check the graph carefully. 6. Since the graph peaks at $x=0$ and $x=8$ with $y=12$, and dips to $0$ at $x=4$, the value at $x=1$ is close to the peak, so $g(1) \approx 8$ (from options given, closest is 8). 7. At $x=0$, $g(0) = 0$ or $12$? The description says peak at $x=0$ is $12$, but options for $g(-32)$ are (A) 0 (B) 3 (C) 5 (D) 8. Since $g(-32) = g(0)$, and the peak is 12 which is not an option, the closest is 0, which matches the dips at $x=4$ and between 12 and 16. 8. Therefore, $g(81) = g(1) = 8$ (D), and $g(-32) = g(0) = 0$ (A). --- **Final answers:** - $g(81) = 8$ (D) - $g(-32) = 0$ (A)