Determine the amplitude, period, and phase shift of the following trigonometric equation, \[ y=\frac{-1}{2} \cos (x-7) \]

To analyze the trigonometric equation \[ y = \frac{-1}{2} \cos (x - 7), \] we can identify the amplitude, period, and phase shift by comparing it to the standard form of a cosine function: \[ y = A \cos(B(x - C)) + D, \] where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. ### Step 1: Determine the Amplitude The amplitude is given by the absolute value of \( A \): \[ \text{Amplitude} = |A| = \left| \frac{-1}{2} \right| = \frac{1}{2}. \] ### Step 2: Determine the Period The period of the cosine function is calculated using the formula: \[ \text{Period} = \frac{2\pi}{|B|}. \] In our equation, \( B = 1 \) (since there is no coefficient in front of \( x \)). Thus, the period is: \[ \text{Period} = \frac{2\pi}{|1|} = 2\pi. \] ### Step 3: Determine the Phase Shift The phase shift is calculated using the formula: \[ \text{Phase Shift} = C, \] where \( C \) is the value that shifts the function horizontally. In our equation, we have \( (x - 7) \), which means: \[ C = 7. \] Since the cosine function shifts to the right, the phase shift is: \[ \text{Phase Shift} = 7 \text{ units to the right}. \] ### Summary Putting it all together, we have: - Amplitude: \( \frac{1}{2} \) - Period: \( 2\pi \) - Phase Shift: \( 7 \) units to the right Thus, the final results are: - **Amplitude**: \( \frac{1}{2} \) - **Period**: \( 2\pi \) - **Phase Shift**: \( 7 \) units to the right.