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