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

To find the **period** and **phase shift** of the function \( f(x) = 2 \csc\left(\frac{\pi}{3}x + \pi\right) \), we'll analyze the argument of the cosecant function, which is similar to the sine function. ### Standard Form The general form for a sine or cosecant function is: \[ f(x) = A \csc(Bx + C) + D \] where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) affects the phase shift, - \( D \) is the vertical shift. ### Given Function For the given function: \[ f(x) = 2 \csc\left(\frac{\pi}{3}x + \pi\right) \] we identify: - \( B = \frac{\pi}{3} \) - \( C = \pi \) ### Period The period \( P \) of the function is calculated using the formula: \[ P = \frac{2\pi}{|B|} \] Substituting \( B = \frac{\pi}{3} \): \[ P = \frac{2\pi}{\frac{\pi}{3}} = 6 \] ### Phase Shift The phase shift \( \phi \) is given by: \[ \phi = -\frac{C}{B} \] Substituting \( C = \pi \) and \( B = \frac{\pi}{3} \): \[ \phi = -\frac{\pi}{\frac{\pi}{3}} = -3 \] This means the graph is shifted 3 units to the **left**. ### Final Answer - **Period:** \( 6 \) - **Phase shift:** \( -3 \) **Answer:** - **Period:** 6 - **Phase shift:** –3