1) \( y=3 \sin \left(\frac{\theta}{3}-\frac{\pi}{2}\right)+2 \)

Function by following steps: - step0: Determine the period: \(y=-3\cos\left(\frac{\theta }{3}\right)+2\) - step1: Calculate: \(y=-3\cos\left(\frac{1}{3}\theta \right)+2\) - step2: The period of the function is \(\frac{2\pi }{\left|\frac{1}{3}\right|}:\) \(\frac{2\pi }{\left|\frac{1}{3}\right|}\) - step3: Calculate: \(\frac{2\pi }{\frac{1}{3}}\) - step4: Divide the terms: \(6\pi \) It seems there was an issue retrieving the results for the analysis of the function \( y=3 \sin \left(\frac{\theta}{3}-\frac{\pi}{2}\right)+2 \) for domain, range, critical points, inflection points, and intervals. However, I was able to determine the periodicity of the function. ### Periodicity The periodicity of the function is \( 6\pi \). This means that the function repeats its values every \( 6\pi \) units along the \(\theta\) axis. ### Further Analysis Let's analyze the function step by step to find the domain, range, critical points, and inflection points manually. 1. **Domain**: - The sine function is defined for all real numbers, so the domain of \( y \) is all real numbers: \[ \text{Domain: } (-\infty, \infty) \] 2. **Range**: - The sine function oscillates between -1 and 1. Therefore, multiplying by 3 gives us a range of -3 to 3. Adding 2 shifts this range up by 2: \[ \text{Range: } [-1, 5] \] 3. **Critical Points**: - To find critical points, we need to take the derivative of \( y \) and set it to zero. The derivative is: \[ y' = 3 \cdot \cos\left(\frac{\theta}{3} - \frac{\pi}{2}\right) \cdot \frac{1}{3} = \cos\left(\frac{\theta}{3} - \frac{\pi}{2}\right) \] Setting \( y' = 0 \): \[ \cos\left(\frac{\theta}{3} - \frac{\pi}{2}\right) = 0 \] This occurs when: \[ \frac{\theta}{3} - \frac{\pi}{2} = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \] Solving for \(\theta\): \[ \theta = 3\left(n\pi + \pi\right) = 3(n + 1)\pi \] Thus, the critical points occur at \( \theta = 3(n + 1)\pi \). 4. **Inflection Points**: - Inflection points occur where the second derivative changes sign. The second derivative is: \[ y'' = -\frac{1}{3} \sin\left(\frac{\theta}{3} - \frac{\pi}{2}\right) \] Setting \( y'' = 0 \): \[ -\frac{1}{3} \sin\left(\frac{\theta}{3} - \frac{\pi}{2}\right) = 0 \] This occurs when: \[ \sin\left(\frac{\theta}{3} - \frac{\pi}{2}\right) = 0 \] This occurs at: \[ \frac{\theta}{3} - \frac{\pi}{2} = n\pi \quad (n \in \mathbb{Z}) \] Solving for \(\theta\): \[ \theta = 3n\pi + \frac{3\pi}{2} \] ### Summary - **Domain**: \( (-\infty, \infty) \) - **Range**: \( [-1, 5] \) - **Critical Points**: \( \theta = 3(n + 1)\pi \) - **Inflection Points**: \( \theta = 3n\pi + \frac{3\pi}{2} \) - **Periodicity**: \( 6\pi \) If you need further analysis or specific calculations, feel free to ask!