1. **State the problem:** We need to sketch a sinusoidal function with period 8, amplitude 5, axis at $y=-1$, and 2 cycles. 2. **Recall the sinusoidal function formula:** A general sine function is given by: $$y = A \sin\left(\frac{2\pi}{T}x\right) + D$$ where $A$ is amplitude, $T$ is period, and $D$ is the vertical shift (axis). 3. **Identify given values:** - Amplitude $A = 5$ - Period $T = 8$ - Axis $D = -1$ - Number of cycles = 2 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. **Check maximum and minimum values:** - Maximum: $D + A = -1 + 5 = 4$ - Minimum: $D - A = -1 - 5 = -6$ 6. **Check the user's max and min:** User states max = 4 + 5 = 4 (incorrect addition, should be $-1 + 5 = 4$) User states min = -1 + -5 = -6 (correct) 7. **Horizontal axis scale:** - Period is 8, so 2 cycles span $2 \times 8 = 16$ units. - User's horizontal axis goes from 0 to 30, which is enough to show 2 full cycles. 8. **Vertical axis scale:** - From -6 to 5 covers the range from minimum to slightly above maximum. 9. **Conclusion:** - The function and graph description are consistent except the user's max calculation typo. - The graph is correct with the function: $$y = 5 \sin\left(\frac{\pi}{4}x\right) - 1$$ 10. **Label axes:** - Vertical axis labeled "height" from -6 to 5. - Horizontal axis labeled "Time" from 0 to 30. 11. **Summary:** The sinusoidal wave oscillates between 4 and -6, has period 8, amplitude 5, axis at -1, and 2 cycles within the time frame. **Final answer:** $$y = 5 \sin\left(\frac{\pi}{4}x\right) - 1$$