Questions: Find the amplitude, period, and phase shift of the function. y=-4 sin(2 x + (π/2)) What is the amplitude? What is the period? What is the phase shift?

Solution Steps

To find the amplitude, period, and phase shift of the function \( y = -4 \sin \left(2 x + \frac{\pi}{2}\right) \):

1. Amplitude: The amplitude of a sine function \( y = a \sin(bx + c) \) is given by the absolute value of the coefficient \( a \).
2. Period: The period of a sine function \( y = a \sin(bx + c) \) is given by \( \frac{2\pi}{|b|} \).
3. Phase Shift: The phase shift of a sine function \( y = a \sin(bx + c) \) is given by \( -\frac{c}{b} \).

The amplitude of the function \( y = -4 \sin \left(2 x + \frac{\pi}{2}\right) \) is given by the absolute value of the coefficient \( a \). Thus, we have: \[ \text{Amplitude} = |a| = |-4| = 4 \]

The period of the function is calculated using the formula \( \frac{2\pi}{|b|} \). For our function, where \( b = 2 \): \[ \text{Period} = \frac{2\pi}{|2|} = \frac{2\pi}{2} = \pi \]

The phase shift is determined by the formula \( -\frac{c}{b} \). Given \( c = \frac{\pi}{2} \) and \( b = 2 \): \[ \text{Phase Shift} = -\frac{\frac{\pi}{2}}{2} = -\frac{\pi}{4} \]

Final Answer