Find the period and phase shift (horizontal shift) of the function. Answer exactly. \( f(x)=3 \csc \left(\frac{\pi}{2} x+\pi\right) \) Period: Phase shift:

To determine the **period** and **phase shift** of the function \( f(x) = 3 \csc \left(\frac{\pi}{2} x + \pi\right) \), we can compare it to the general form of a trigonometric function: \[ y = A \csc(Bx + C) + D \] Here: - \( A \) is the amplitude (not relevant for period and phase shift). - \( B \) affects the period. - \( C \) affects the phase shift. - \( D \) is the vertical shift (not relevant for period and phase shift). ### 1. **Period** The period of a cosecant function is given by: \[ \text{Period} = \frac{2\pi}{|B|} \] In the given function: - \( B = \frac{\pi}{2} \) So, \[ \text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 4 \] ### 2. **Phase Shift** The phase shift is determined by: \[ \text{Phase Shift} = -\frac{C}{B} \] In the given function: - \( C = \pi \) - \( B = \frac{\pi}{2} \) So, \[ \text{Phase Shift} = -\frac{\pi}{\frac{\pi}{2}} = -2 \] This means the graph is shifted **2 units to the left**. ### **Final Answer** - **Period:** \( 4 \) - **Phase shift:** \( -2 \)