Determine the amplitude, period, and phase shift (if any) of the given function. Graph the function. \( y=-2 \sin x \) The amplitude is \( \begin{array}{l}\text { (Simplify your answer. Type an } \\ \text { exact answer, using } \pi \text { as needed.) }\end{array} \)

To analyze the function \( y = -2 \sin x \), we can determine the amplitude, period, and phase shift as follows: 1. **Amplitude**: The amplitude of a sine function \( y = A \sin(Bx + C) + D \) is given by the absolute value of \( A \). In this case, \( A = -2 \), so the amplitude is: \[ \text{Amplitude} = |A| = |-2| = 2 \] 2. **Period**: The period of a sine function is given by the formula \( \frac{2\pi}{|B|} \). In this case, \( B = 1 \) (since there is no coefficient in front of \( x \)), so the period is: \[ \text{Period} = \frac{2\pi}{|1|} = 2\pi \] 3. **Phase Shift**: The phase shift is determined by the term \( C \) in the function \( y = A \sin(Bx + C) + D \). In this case, there is no \( C \) term (it is effectively 0), so the phase shift is: \[ \text{Phase Shift} = -\frac{C}{B} = -\frac{0}{1} = 0 \] ### Summary: - Amplitude: \( 2 \) - Period: \( 2\pi \) - Phase Shift: \( 0 \) ### Graphing the Function: To graph \( y = -2 \sin x \): - The sine function normally oscillates between -1 and 1. However, since we have a coefficient of -2, the function will oscillate between -2 and 2, but inverted (i.e., it will reach its maximum at -0 and minimum at -2). - The function will complete one full cycle over the interval \( [0, 2\pi] \). **Key Points to Plot**: - At \( x = 0 \), \( y = -2 \sin(0) = 0 \) - At \( x = \frac{\pi}{2} \), \( y = -2 \sin\left(\frac{\pi}{2}\right) = -2 \) - At \( x = \pi \), \( y = -2 \sin(\pi) = 0 \) - At \( x = \frac{3\pi}{2} \), \( y = -2 \sin\left(\frac{3\pi}{2}\right) = 2 \) - At \( x = 2\pi \), \( y = -2 \sin(2\pi) = 0 \) ### Graph: The graph will start at \( (0, 0) \), go down to \( ( \frac{\pi}{2}, -2) \), return to \( ( \pi, 0) \), rise to \( ( \frac{3\pi}{2}, 2) \), and return to \( (2\pi, 0) \). You can sketch the graph based on these points, ensuring to reflect the sine wave's typical shape but inverted and scaled vertically.