find the amplitude, phase shift, and period of the function. y = - 1 - 4 cos(3x + π/2) give the exact…

Explanation:

Step1: Recall amplitude formula

For $y = A\cos(Bx - C)+D$, amplitude is $|A|$. In $y=-1 - 4\cos(3x+\frac{\pi}{2})$, $A=-4$. So amplitude $= |-4| = 4$.

Step2: Recall phase - shift formula

Phase - shift is given by $\frac{C}{B}$. Rewrite $y=-1 - 4\cos(3x+\frac{\pi}{2})$ as $y=-1 - 4\cos(3(x+\frac{\pi}{6}))$. Here $B = 3$ and $C=-\frac{\pi}{2}$, so phase - shift $=-\frac{\pi}{6}$.

Step3: Recall period formula

The period of $y = A\cos(Bx - C)+D$ is $\frac{2\pi}{|B|}$. Since $B = 3$, period $=\frac{2\pi}{3}$.

Answer:

Amplitude: $4$ Phase shift: $-\frac{\pi}{6}$ Period: $\frac{2\pi}{3}$