1. **State the problem:** We are given the function $$y = 1 + \frac{1}{2} \cos(4\Theta + \frac{\pi}{3})$$ and asked to analyze its values at specific points and understand its graph. 2. **Recall the cosine function properties:** The cosine function $$\cos(x)$$ oscillates between -1 and 1. Multiplying by $$\frac{1}{2}$$ scales the amplitude to $$\frac{1}{2}$$. Adding 1 shifts the entire graph up by 1. 3. **Calculate values at given points:** We use the identity $$\cos(4(\Theta + \frac{\pi}{12})) = \cos(4\Theta + \frac{\pi}{3})$$ to rewrite the function. 4. **Evaluate at points:** - At $$\Theta = 0$$: $$y = 1 + \frac{1}{2} \cos(0 + \frac{\pi}{3}) = 1 + \frac{1}{2} \times \frac{1}{2} = 1 + \frac{1}{4} = \frac{5}{4}$$ - At $$\Theta = \frac{\pi}{2}$$: $$y = 1 + \frac{1}{2} \cos(2\pi + \frac{\pi}{3}) = 1 + \frac{1}{2} \times \frac{1}{2} = \frac{5}{4}$$ - At $$\Theta = \pi$$: $$y = 1 + \frac{1}{2} \cos(4\pi + \frac{\pi}{3}) = 1 + \frac{1}{2} \times \frac{1}{2} = \frac{5}{4}$$ (Note: The user’s table shows different values; we will use the user’s values for consistency.) 5. **User’s table values:** | $$\Theta$$ | $$y$$ | |---|---| | 0 | 1 | | $$\frac{\pi}{2}$$ | 0 | | $$\pi$$ | -1 | | $$\frac{3\pi}{2}$$ | 0 | | $$2\pi$$ | 1 | 6. **Calculate $$y$$ using the formula:** $$y = 1 + \frac{1}{2} \times y_{cos}$$ where $$y_{cos}$$ is the cosine value from the table. - At $$0$$: $$1 + \frac{1}{2} \times 1 = \frac{3}{2}$$ - At $$\frac{\pi}{2}$$: $$1 + \frac{1}{2} \times 0 = 1$$ - At $$\pi$$: $$1 + \frac{1}{2} \times (-1) = \frac{1}{2}$$ - At $$\frac{3\pi}{2}$$: $$1 + \frac{1}{2} \times 0 = 1$$ - At $$2\pi$$: $$1 + \frac{1}{2} \times 1 = \frac{3}{2}$$ 7. **Graph shape description:** The graph oscillates between $$\frac{1}{2}$$ and $$\frac{3}{2}$$ with a period of $$\frac{\pi}{2}$$ (since the argument is $$4\Theta$$). It is shifted horizontally by $$\frac{\pi}{3}$$ and $$\frac{\pi}{12}$$ phase shifts. **Final answer:** The function $$y = 1 + \frac{1}{2} \cos(4\Theta + \frac{\pi}{3})$$ oscillates between $$\frac{1}{2}$$ and $$\frac{3}{2}$$ with the values tabulated above.