1. **State the problem:** We need to graph the function $$y = 4 \sin\left(4\left(\theta - \frac{\pi}{2}\right)\right) + 2$$ using its five critical points and characteristics. 2. **Identify key characteristics:** - Vertical shift: 2 (the midline is $$y=2$$) - Amplitude: 4 (the sine wave oscillates 4 units above and below the midline) - Phase shift: $$\frac{\pi}{2}$$ to the right - Period: $$\frac{\pi}{2}$$ (since period $$= \frac{2\pi}{b}$$ and here $$b=4$$) - Range: from $$2 - 4 = -2$$ to $$2 + 4 = 6$$ 3. **Formula for sine function with transformations:** $$y = A \sin\left(B(\theta - C)\right) + D$$ where: - $$A$$ is amplitude - $$B$$ affects period $$= \frac{2\pi}{B}$$ - $$C$$ is phase shift - $$D$$ is vertical shift (midline) 4. **Calculate critical points:** Critical points for sine occur at multiples of quarter periods starting from phase shift: - Period $$= \frac{\pi}{2}$$, so quarter period $$= \frac{\pi}{8}$$ - Starting at phase shift $$\frac{\pi}{2}$$, critical points are: $$\theta = \frac{\pi}{2}, \frac{\pi}{2} + \frac{\pi}{8} = \frac{5\pi}{8}, \frac{\pi}{2} + \frac{2\pi}{8} = \frac{3\pi}{4}, \frac{\pi}{2} + \frac{3\pi}{8} = \frac{7\pi}{8}, \frac{\pi}{2} + \frac{4\pi}{8} = \pi$$ 5. **Evaluate function at critical points:** Using $$y = 4 \sin(4(\theta - \frac{\pi}{2})) + 2$$: - At $$\theta = \frac{\pi}{2}$$: $$y = 4 \sin(4(\frac{\pi}{2} - \frac{\pi}{2})) + 2 = 4 \sin(0) + 2 = 2$$ - At $$\theta = \frac{5\pi}{8}$$: $$y = 4 \sin(4(\frac{5\pi}{8} - \frac{\pi}{2})) + 2 = 4 \sin(4(\frac{\pi}{8})) + 2 = 4 \sin(\frac{\pi}{2}) + 2 = 4(1) + 2 = 6$$ - At $$\theta = \frac{3\pi}{4}$$: $$y = 4 \sin(4(\frac{3\pi}{4} - \frac{\pi}{2})) + 2 = 4 \sin(4(\frac{\pi}{4})) + 2 = 4 \sin(\pi) + 2 = 4(0) + 2 = 2$$ - At $$\theta = \frac{7\pi}{8}$$: $$y = 4 \sin(4(\frac{7\pi}{8} - \frac{\pi}{2})) + 2 = 4 \sin(4(\frac{3\pi}{8})) + 2 = 4 \sin(\frac{3\pi}{2}) + 2 = 4(-1) + 2 = -2$$ - At $$\theta = \pi$$: $$y = 4 \sin(4(\pi - \frac{\pi}{2})) + 2 = 4 \sin(4(\frac{\pi}{2})) + 2 = 4 \sin(2\pi) + 2 = 4(0) + 2 = 2$$ 6. **Interpretation:** - The sine wave starts at the midline value 2 at $$\theta=\frac{\pi}{2}$$. - It reaches a maximum of 6 at $$\theta=\frac{5\pi}{8}$$. - Returns to midline 2 at $$\theta=\frac{3\pi}{4}$$. - Hits minimum -2 at $$\theta=\frac{7\pi}{8}$$. - Returns to midline 2 at $$\theta=\pi$$. 7. **Summary:** The function oscillates between -2 and 6 with period $$\frac{\pi}{2}$$, shifted right by $$\frac{\pi}{2}$$, and vertical shift 2. The critical points correspond to midline, max, midline, min, midline values respectively. Final answer: The graph of $$y = 4 \sin\left(4\left(\theta - \frac{\pi}{2}\right)\right) + 2$$ has amplitude 4, midline $$y=2$$, period $$\frac{\pi}{2}$$, phase shift $$\frac{\pi}{2}$$ right, and critical points at $$\theta = \frac{\pi}{2}, \frac{5\pi}{8}, \frac{3\pi}{4}, \frac{7\pi}{8}, \pi$$ with corresponding $$y$$ values $$2, 6, 2, -2, 2$$ respectively.