1. **Problem Statement:** We need to understand why the period of a sinusoidal function with given characteristics is 8 and not 4. 2. **Given Information:** - Period: 8 - Amplitude: 5 - Equation of the axis (midline): $y = -1$ - Number of cycles: 2 3. **Formula for Period of Sinusoidal Function:** The general form of a sinusoidal function is: $$y = A \sin(Bx + C) + D$$ where: - $A$ is the amplitude - $B$ affects the period - $D$ is the vertical shift (axis) The period $T$ is given by: $$T = \frac{2\pi}{|B|}$$ 4. **Explanation:** - The period is the length of one full cycle of the wave. - If the period were 4, then the function would complete one cycle every 4 units along the x-axis. - Since the problem states the function completes 2 full cycles over the x-range from 0 to 16 (because 2 cycles × period = total length), the period must be: $$T = \frac{16}{2} = 8$$ 5. **Why not 4?** - If the period were 4, then 2 cycles would only cover 8 units on the x-axis. - But the graph shows 2 cycles over 16 units, so the period cannot be 4. 6. **Summary:** The period is 8 because the function completes 2 full cycles over an x-range of 16 units, and period is the length of one cycle, so: $$\text{Period} = \frac{\text{Total x-range}}{\text{Number of cycles}} = \frac{16}{2} = 8$$ **Final answer:** The period is 8, not 4, because the function completes 2 cycles over 16 units on the x-axis, making each cycle 8 units long.