1. **Problem 1:** Determine an equation for the sinusoidal function given the data points: x: 0°, 45°, 90°, 135°, 180°, 225°, 270° y: 9, 7, 5, 7, 9, 7, 5 2. **Step 1: Identify the amplitude, midline, and period.** - Amplitude $A = \frac{\text{max} - \text{min}}{2} = \frac{9 - 5}{2} = 2$ - Midline $D = \frac{\text{max} + \text{min}}{2} = \frac{9 + 5}{2} = 7$ - Period $P = 360^\circ$ (since the pattern repeats every 360°) 3. **Step 2: Choose the sinusoidal function type and write the general form.** - Since the data starts at a maximum (9 at 0°), the cosine function fits well. - General form: $$y = A \cos\left(\frac{2\pi}{P}x\right) + D$$ 4. **Step 3: Substitute values.** $$y = 2 \cos\left(\frac{2\pi}{360}x\right) + 7 = 2 \cos\left(\frac{\pi}{180}x\right) + 7$$ 5. **Step 4: Verify with points.** - At $x=0^\circ$, $y=2\cos(0)+7=2(1)+7=9$ matches data. - At $x=90^\circ$, $y=2\cos(\frac{\pi}{2})+7=2(0)+7=7$ matches data. **Final equation:** $$y = 2 \cos\left(\frac{\pi}{180}x\right) + 7$$ --- **Problem 2:** Determine the function that models the data in the table without horizontal translation. This is the same as Problem 1, so the answer is the same: $$y = 2 \cos\left(\frac{\pi}{180}x\right) + 7$$ --- **Problem 3:** A sinusoidal function has amplitude 4, period 120°, and maximum at $(0,9)$. Find the equation. 1. Amplitude $A=4$ 2. Period $P=120^\circ$ 3. Maximum at $x=0$ means cosine function with no phase shift. 4. Midline $D = \text{max} - A = 9 - 4 = 5$ 5. General form: $$y = A \cos\left(\frac{2\pi}{P}x\right) + D = 4 \cos\left(\frac{2\pi}{120}x\right) + 5 = 4 \cos\left(\frac{\pi}{60}x\right) + 5$$ **Final equation:** $$y = 4 \cos\left(\frac{\pi}{60}x\right) + 5$$