1. **Problem Statement:** We need to graph the function $y = \cos(3x + \frac{\pi}{2})$. 2. **Starting Graph:** The base graph is $y = \cos(x)$, which has a period of $2\pi$ and amplitude 1. 3. **Transformation - Frequency Change:** The function inside the cosine is $3x$, which means the frequency is multiplied by 3. - The period of $y = \cos(bx)$ is given by $\frac{2\pi}{|b|}$. - Here, $b = 3$, so the period is $$\frac{2\pi}{3}$$. 4. **Transformation - Phase Shift:** The function has a phase shift of $\frac{\pi}{2}$ inside the cosine. - The phase shift is calculated by solving $3x + \frac{\pi}{2} = 0$ for $x$. - So, $$x = -\frac{\pi}{6}$$. - This means the graph shifts to the left by $\frac{\pi}{6}$. 5. **Summary of Transformations:** - Start with $y = \cos(x)$. - Compress horizontally by a factor of $\frac{1}{3}$ to get $y = \cos(3x)$. - Shift left by $\frac{\pi}{6}$ to get $y = \cos(3x + \frac{\pi}{2})$. 6. **Final Graph Description:** The graph oscillates three times faster than $y = \cos(x)$, with period $\frac{2\pi}{3}$, and is shifted left by $\frac{\pi}{6}$. This matches the third graph from the left in the bottom row, which shows a high-frequency cosine wave.