Determine the amplitude and period of the following function without graphing. \[ y=8 \cos (\pi x) \]

To determine the amplitude and period of the function \( y = 8 \cos (\pi x) \), we can analyze the general form of a cosine function, which is given by: \[ y = A \cos(Bx) \] where: - \( A \) is the amplitude, - \( B \) affects the period of the function. 1. **Amplitude**: The amplitude of a cosine function is the absolute value of \( A \). In this case, \( A = 8 \). Therefore, the amplitude is: \[ \text{Amplitude} = |A| = |8| = 8 \] 2. **Period**: The period of a cosine function is calculated using the formula: \[ \text{Period} = \frac{2\pi}{B} \] In our function, we have \( B = \pi \). Thus, the period is: \[ \text{Period} = \frac{2\pi}{\pi} = 2 \] In summary, for the function \( y = 8 \cos (\pi x) \): - The amplitude is \( 8 \). - The period is \( 2 \).