1. **State the problem:** We need to sketch a sinusoidal function with period 8, amplitude 5, axis at $y=-1$, and 2 full cycles. 2. **Formula and explanation:** A sinusoidal function can be written as $$y = A \sin\left(\frac{2\pi}{T}x\right) + D$$ where $A$ is amplitude, $T$ is period, and $D$ is the vertical shift (axis). 3. **Apply given values:** Amplitude $A=5$, period $T=8$, axis $D=-1$. 4. **Write the function:** $$y = 5 \sin\left(\frac{2\pi}{8}x\right) - 1 = 5 \sin\left(\frac{\pi}{4}x\right) - 1$$ 5. **Calculate max and min values:** Maximum: $D + A = -1 + 5 = 4$ Minimum: $D - A = -1 - 5 = -6$ 6. **Number of cycles:** Two full cycles means the function completes 2 periods over the x-range. Since one period is 8, two periods span $2 \times 8 = 16$ units. 7. **Graph description:** - The sinusoidal wave oscillates between 4 and -6. - The midline is at $y=-1$. - The wave completes 2 cycles over the x-axis. - The x-axis is labeled "Time" with increments every 2 units. - The y-axis is labeled "height" with values from -6 to 5. This matches the problem's description and confirms the function and graph characteristics. **Final answer:** $$y = 5 \sin\left(\frac{\pi}{4}x\right) - 1$$