Write the equation for a cosine function with amplitude \( 3 \pi \), period 1 , and phase shift \( =8 \).

1. The general cosine function is given by \[ y = A \cos\left( B \left( x - C \right) \right) \] where: - \( A \) is the amplitude. - \( B \) is related to the period by the formula \(\text{period} = \frac{2\pi}{|B|}\). - \( C \) is the phase shift. 2. Given that the amplitude is \( 3\pi \), we have \[ A = 3\pi. \] 3. The period is given as \( 1 \). Using the relationship between \( B \) and the period, we set \[ \frac{2\pi}{|B|} = 1. \] Solving for \( B \), we get \[ |B| = 2\pi. \] We choose \( B = 2\pi \) (the sign here does not affect the amplitude or period). 4. The phase shift is given as \( 8 \), so we have \[ C = 8. \] 5. Substituting all values into the general formula, we obtain the equation: \[ y = 3\pi \cos\left( 2\pi(x - 8) \right). \]