identify the amplitude, period, phase shift, vertical shift, and the maximum and minimum values of the…

For the function $y = \frac{2}{3}\sin(x+\frac{\pi}{3})+3$:

Explanation:

Step1: Find the amplitude

The amplitude $A$ of a sine function $y = A\sin(Bx - C)+D$ is given by the absolute - value of the coefficient of the sine term. Here, $A=\left|\frac{2}{3}\right|=\frac{2}{3}$.

Step2: Find the period

The period $T$ of a sine function $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Since $B = 1$, $T=\frac{2\pi}{|1|}=2\pi$.

Step3: Find the phase - shift

The phase - shift of a sine function $y = A\sin(Bx - C)+D$ is $\frac{C}{B}$. For $y=\frac{2}{3}\sin(x+\frac{\pi}{3})+3$, which can be written as $y=\frac{2}{3}\sin(1x-(-\frac{\pi}{3})) + 3$, the phase - shift is $-\frac{\pi}{3}$ (a shift to the left).

Step4: Find the vertical shift

The vertical shift $D$ of the function $y = A\sin(Bx - C)+D$ is the constant term. Here, $D = 3$.

Step5: Find the maximum and minimum values

The maximum value of $y = A\sin(Bx - C)+D$ is $A + D$ and the minimum value is $-A+D$. Maximum value: $\frac{2}{3}+3=\frac{2 + 9}{3}=\frac{11}{3}$ Minimum value: $-\frac{2}{3}+3=\frac{-2 + 9}{3}=\frac{7}{3}$

For the function $y=\frac{1}{2}\cos[2(x-\frac{\pi}{4})]-1$:

Step1: Find the amplitude

The amplitude $A$ of a cosine function $y = A\cos(Bx - C)+D$ is the absolute - value of the coefficient of the cosine term. So, $A=\left|\frac{1}{2}\right|=\frac{1}{2}$.

Step2: Find the period

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

Step3: Find the phase - shift

The phase - shift of a cosine function $y = A\cos(Bx - C)+D$ is $\frac{C}{B}$. For $y=\frac{1}{2}\cos[2(x-\frac{\pi}{4})]-1=\frac{1}{2}\cos(2x-\frac{\pi}{2})-1$, the phase - shift is $\frac{\frac{\pi}{2}}{2}=\frac{\pi}{4}$ (a shift to the right).

Step4: Find the vertical shift

The vertical shift $D$ of the function $y = A\cos(Bx - C)+D$ is the constant term. Here, $D=-1$.

Step5: Find the maximum and minimum values

The maximum value of $y = A\cos(Bx - C)+D$ is $A + D$ and the minimum value is $-A+D$. Maximum value: $\frac{1}{2}-1=-\frac{1}{2}$ Minimum value: $-\frac{1}{2}-1=-\frac{3}{2}$

Answer:

For $y = \frac{2}{3}\sin(x+\frac{\pi}{3})+3$: Amplitude: $\frac{2}{3}$, Period: $2\pi$, Phase - shift: $-\frac{\pi}{3}$, Vertical shift: $3$, Maximum value: $\frac{11}{3}$, Minimum value: $\frac{7}{3}$ For $y=\frac{1}{2}\cos[2(x-\frac{\pi}{4})]-1$: Amplitude: $\frac{1}{2}$, Period: $\pi$, Phase - shift: $\frac{\pi}{4}$, Vertical shift: $-1$, Maximum value: $-\frac{1}{2}$, Minimum value: $-\frac{3}{2}$