Question

find the phase shift, period, and amplitude of the function. y = - 3+3 cos(2x + π/2) give the exact values, not decimal approximations. amplitude: period: phase shift:

Explanation:

Step1: Recall cosine - function form

The general form of a cosine function is $y = A + B\cos(Cx - D)$. For the function $y=-3 + 3\cos(2x+\frac{\pi}{2})$, we can rewrite it as $y=-3 + 3\cos(2x-(-\frac{\pi}{2}))$.

Step2: Find the amplitude

The amplitude of a cosine function $y = A + B\cos(Cx - D)$ is given by $|B|$. Here, $B = 3$, so the amplitude is $|3|=3$.

Step3: Find the period

The period of a cosine function $y = A + B\cos(Cx - D)$ is given by $T=\frac{2\pi}{|C|}$. Here, $C = 2$, so $T=\frac{2\pi}{2}=\pi$.

Step4: Find the phase - shift

The phase - shift of a cosine function $y = A + B\cos(Cx - D)$ is given by $\frac{D}{C}$. Here, $C = 2$ and $D=-\frac{\pi}{2}$, so the phase - shift is $\frac{-\frac{\pi}{2}}{2}=-\frac{\pi}{4}$.

Answer:

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