graph $g(x)=0.5sinleft(\frac{pi}{3}x+\frac{4pi}{3}\right)-2$ in the interactive widget. note that one…

Explanation:

Step1: Identify the general form of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$, where for the function $g(x)=0.5\sin(\frac{\pi}{3}x+\frac{4\pi}{3})-2$, we have $A = 0.5$, $B=\frac{\pi}{3}$, $C =-\frac{4\pi}{3}$, and $D=-2$.

Step2: Calculate the amplitude

The amplitude is given by $|A|$. Since $A = 0.5$, the amplitude $|A|=0.5$.

Step3: Calculate the period

The period of a sine - function $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Here, $B=\frac{\pi}{3}$, so $T=\frac{2\pi}{\frac{\pi}{3}}=6$.

Step4: Calculate the phase shift

The phase shift is $\frac{C}{B}$. Substituting $C =-\frac{4\pi}{3}$ and $B=\frac{\pi}{3}$, we get $\frac{-\frac{4\pi}{3}}{\frac{\pi}{3}}=- 4$.

Step5: Identify the mid - line

The mid - line of the function is $y = D$. Since $D=-2$, the mid - line is $y=-2$.

Step6: Find the maximum and minimum values

The maximum value of the function occurs when $\sin(\frac{\pi}{3}x+\frac{4\pi}{3}) = 1$. Then $y_{max}=0.5\times1 - 2=-1.5$. The minimum value occurs when $\sin(\frac{\pi}{3}x+\frac{4\pi}{3})=-1$. Then $y_{min}=0.5\times(-1)-2=-2.5$.

To graph the function:

1. Mark the mid - line $y = - 2$.
2. Plot points based on the period, phase shift, amplitude, maximum and minimum values. For example, when $x=-4$, $g(-4)=0.5\sin(0)-2=-2$. When $x=-1$, $g(-1)=0.5\sin(\pi)-2=-2$. When $x = 2$, $g(2)=0.5\sin(2\pi)-2=-2$. The maximum points occur at $x=-4 + 3+6k$ ($k\in\mathbb{Z}$) with $y=-1.5$ and minimum points occur at $x=-4 + 6k$ ($k\in\mathbb{Z}$) with $y=-2.5$.

Answer:

The graph of $g(x)=0.5\sin(\frac{\pi}{3}x+\frac{4\pi}{3})-2$ has an amplitude of $0.5$, a period of $6$, a phase shift of $-4$, a mid - line of $y = - 2$, a maximum value of $-1.5$ and a minimum value of $-2.5$.