\( 16 \quad \begin{array}{l}\text { Determine the amplitude, period, and phase shift of the } \\ \text { function. Graph the function. } \\ y=4 \cos (3 x+\pi) \\ \text { The amplitude is } 4 \text {. } \\ \text { (Simplity your answer.) } \\ \text { The period is } \\ \text { (Type an exact answer, using } \pi \text { as needed. Use integers or } \\ \text { fractions for any numbers in the expression.) }\end{array} \)

**Step 1: Identify the amplitude** The given function is \[ y = 4 \cos(3x+\pi). \] The amplitude is the absolute value of the coefficient in front of the cosine, so \[ \text{Amplitude} = |4| = 4. \] **Step 2: Determine the period** For a function of the form \[ y = \cos(Bx + C), \] the period is given by \[ \text{Period} = \frac{2\pi}{|B|}. \] Here, \(B = 3\). Thus, the period is \[ \text{Period} = \frac{2\pi}{3}. \] **Step 3: Find the phase shift** To find the phase shift, factor the argument inside the cosine: \[ 3x+\pi = 3\left(x+\frac{\pi}{3}\right). \] The phase shift for a cosine function \[ y = \cos(B(x - \text{shift})) \] is given by the value that makes the inside of the cosine zero. Comparing \[ 3\left(x+\frac{\pi}{3}\right) \] with \[ 3\left(x - \left(-\frac{\pi}{3}\right)\right), \] we see that the phase shift is \[ \text{Phase shift} = -\frac{\pi}{3}. \] This indicates that the graph is shifted \(\frac{\pi}{3}\) units to the left. **Summary of the results:** - Amplitude: \(4\) - Period: \(\frac{2\pi}{3}\) - Phase shift: \(-\frac{\pi}{3}\) **Graphing the function:** To graph \(y = 4 \cos(3x+\pi)\): 1. Start with the standard cosine function \(y = \cos x\). 2. Apply the amplitude by stretching the graph vertically by a factor of 4. 3. Compress the graph horizontally according to the period \(\frac{2\pi}{3}\) (since the standard period \(2\pi\) is divided by 3). 4. Shift the graph \(\frac{\pi}{3}\) units to the left. Plot key points over one period by calculating values at appropriate \(x\)-coordinates, then replicate the pattern for other periods.