1. **Problem Statement:** We analyze the height function of a Ferris wheel over time, which is sinusoidal and periodic. 2. **Periodicity:** A function is periodic if it repeats its values in regular intervals or cycles. Here, the height repeats every 8 seconds, so the function is periodic. 3. **Range and Domain:** The range is the set of possible heights, given as $R = \{ y \in \mathbb{R} \mid 1 \leq y \leq 7 \}$. The domain is the time interval $D = \{ x \in \mathbb{R} \mid 0 \leq x \leq 24 \}$. 4. **Period:** The period $T$ is the length of one full cycle, which is 8 seconds. 5. **Axis of the Wave:** The midline or axis is the average of the maximum and minimum heights: $$\frac{7 + 1}{2} = \frac{8}{2} = 4$$ So, the axis is $y = 4$. 6. **Amplitude:** The amplitude is half the distance between the maximum and minimum heights: $$\frac{7 - 1}{2} = 3$$ 7. **Summary:** The function is periodic with period 8 seconds, amplitude 3, midline $y=4$, domain $0 \leq x \leq 24$, and range $1 \leq y \leq 7$. One cycle spans from $t=0$ to $t=8$ seconds. Final answers: - Periodic: Yes - Range: $1 \leq y \leq 7$ - Period: 8 seconds - Axis: $y=4$ - Amplitude: 3