Determine the amplitude, period, and phase shift of the function. Graph the function. \( y=-5 \cos (3 x+\pi) \) The amplitude is \( \square \). (Simplify your answer.) The period is \( \square \). (Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.) The phase shift is \( \square \). (Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.)

1. **Amplitude** The amplitude is the absolute value of the coefficient in front of the cosine function. Here the coefficient is \(-5\). Thus, the amplitude is \[ |{-5}| = 5. \] 2. **Period** The period of a cosine function in the form \[ y = a \cos(bx+c) \] is given by \[ \text{Period} = \frac{2\pi}{|b|}. \] For the function \( y=-5 \cos (3x+\pi) \), the value of \( b \) is \( 3 \). Hence, the period is \[ \frac{2\pi}{3}. \] 3. **Phase Shift** To find the phase shift, factor the argument of the cosine function: \[ 3x+\pi = 3\left(x+\frac{\pi}{3}\right). \] The function can be written as \[ y=-5 \cos\left(3\left(x+\frac{\pi}{3}\right)\right). \] The phase shift is given by the value that is added to \( x \) inside the cosine function. Here it is \[ -\frac{\pi}{3}. \] This indicates a shift of \(\frac{\pi}{3}\) units to the left. 4. **Graph of the Function** To graph \( y=-5 \cos (3x+\pi) \): - **Amplitude:** 5 (vertical stretch). - **Period:** \(\frac{2\pi}{3}\) (the distance required for one complete cycle). - **Phase Shift:** \(-\frac{\pi}{3}\) (shift to the left by \(\frac{\pi}{3}\) units). - The negative sign in front of the 5 reflects the cosine graph across the horizontal axis. Plot the key points of the cosine function over one period and then reflect and shift as needed: - Start at the phase shift: \( x = -\frac{\pi}{3} \) with maximum \( y = -5 \cos(0) = -5 \). - Identify subsequent points using the period \(\frac{2\pi}{3}\) divided into quarters. The resulting graph will show the standard cosine curve, modified by these transformations. **Answers:** - The amplitude is \(\boxed{5}\). - The period is \(\boxed{\frac{2\pi}{3}}\). - The phase shift is \(\boxed{-\frac{\pi}{3}}\).