Find the amplitude, period, and phase shift of the function. \( y=5 \cos \left(6 x+\frac{\pi}{2}\right) \)

To analyze the function \( y = 5 \cos \left( 6x + \frac{\pi}{2} \right) \), we can identify the amplitude, period, and phase shift from its standard form. 1. **Amplitude**: The amplitude of a cosine function \( y = A \cos(Bx + C) \) is given by the absolute value of \( A \). In this case, \( A = 5 \), so the amplitude is: \[ \text{Amplitude} = |5| = 5 \] 2. **Period**: The period of a cosine function is calculated using the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \). Here, \( B = 6 \), so the period is: \[ \text{Period} = \frac{2\pi}{|6|} = \frac{2\pi}{6} = \frac{\pi}{3} \] 3. **Phase Shift**: The phase shift is determined from the expression \( Bx + C \). The phase shift can be calculated using the formula \( -\frac{C}{B} \). In this case, \( C = \frac{\pi}{2} \) and \( B = 6 \), so the phase shift is: \[ \text{Phase Shift} = -\frac{\frac{\pi}{2}}{6} = -\frac{\pi}{12} \] This indicates a shift to the left by \( \frac{\pi}{12} \). In summary, for the function \( y = 5 \cos \left( 6x + \frac{\pi}{2} \right) \): - Amplitude: \( 5 \) - Period: \( \frac{\pi}{3} \) - Phase Shift: \( -\frac{\pi}{12} \) (to the left)