Find the period and phase shift (horizontal shift) of the function. Answer exactly. \( f(x)=2 \sec \left(\frac{\pi}{4}(x+3)\right) \) Period: Phase shift:

To determine the **period** and **phase shift** of the function \( f(x) = 2 \sec \left(\frac{\pi}{4}(x + 3)\right) \), we'll analyze its structure based on the general form of a transformed secant function: \[ f(x) = A \sec\left(B(x - C)\right) + D \] Where: - \( A \) affects the vertical stretch/compression. - \( B \) affects the period of the function. - \( C \) represents the phase shift (horizontal shift). - \( D \) represents the vertical shift. ### 1. **Period** The period of the basic secant function \( \sec(x) \) is \( 2\pi \). When the function is transformed to \( \sec(Bx) \), the period changes according to the formula: \[ \text{Period} = \frac{2\pi}{|B|} \] In the given function: \[ B = \frac{\pi}{4} \] Thus, the period is: \[ \text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 8 \] ### 2. **Phase Shift** The phase shift is determined by the horizontal translation of the function, represented by \( C \) in the general form. To identify \( C \), we rearrange the argument of the secant function to match the form \( B(x - C) \): \[ \frac{\pi}{4}(x + 3) = \frac{\pi}{4}\left(x - (-3)\right) \] Here, \( C = -3 \), which means the graph is shifted **3 units to the left**. ### **Final Answer:** - **Period:** \( 8 \) - **Phase shift:** \( -3 \)