1. **State the problem:** We are given a sinusoidal function with period 8, amplitude 5, axis at $y=-1$, and 2 cycles shown. We want to write the equation of this sinusoidal function and understand its properties. 2. **Recall the general form of a sinusoidal function:** $$y = A \sin\left(B(x - C)\right) + D$$ where: - $A$ is the amplitude (height from the axis to peak), - $B = \frac{2\pi}{\text{period}}$ controls the period, - $C$ is the horizontal shift (phase shift), - $D$ is the vertical shift (axis of the wave). 3. **Identify given values:** - Amplitude $A = 5$ - Period $T = 8$, so $B = \frac{2\pi}{8} = \frac{\pi}{4}$ - Axis $D = -1$ - Number of cycles = 2 (over 16 units, consistent with period 8) 4. **Determine the phase shift $C$:** Since the maximum occurs at $x=3$ (given max = 3 + 5 = 8), the peak is at $x=3$. For a sine function, the peak occurs at $x = C + \frac{\pi}{2B}$. Solve for $C$: $$3 = C + \frac{\pi}{2B} = C + \frac{\pi}{2 \cdot \frac{\pi}{4}} = C + 2$$ So, $$C = 3 - 2 = 1$$ 5. **Write the equation:** $$y = 5 \sin\left(\frac{\pi}{4}(x - 1)\right) - 1$$ 6. **Check max and min values:** - Max: $D + A = -1 + 5 = 4$ - Min: $D - A = -1 - 5 = -6$ These match the given max and min values. **Final answer:** $$y = 5 \sin\left(\frac{\pi}{4}(x - 1)\right) - 1$$