Write the equation for a cosine function with amplitude \( 4 \pi \), period 2 , and phase shift -3 .

The standard form for a cosine function is \[ y = A \cos\left(B(x - C)\right), \] where: - \( A \) is the amplitude. - The period \( T \) is given by \( T = \frac{2\pi}{B} \). - \( C \) is the phase shift. Given: - Amplitude \( A = 4\pi \). - Period \( T = 2 \). - Phase shift \( C = -3 \) (which means the function is shifted to the left by 3 units). First, find \( B \) using the period formula: \[ T = \frac{2\pi}{B} \quad \Longrightarrow \quad 2 = \frac{2\pi}{B} \quad \Longrightarrow \quad B = \frac{2\pi}{2} = \pi. \] Now substitute \( A \), \( B \), and \( C \) into the standard form. Remember that a phase shift of \(-3\) means \( C = -3 \), so the expression inside the cosine becomes \[ \pi\left(x - (-3)\right) = \pi(x + 3). \] Thus, the equation is \[ y = 4\pi \cos\left(\pi(x + 3)\right). \]