graph the trigonometric function. y = 3/2 sin x plot all points corresponding to x - intercepts, minima, and…

Explanation:

Step1: Recall properties of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $y=\frac{3}{2}\sin x$, we have $A = \frac{3}{2}$, $B = 1$, $C = 0$, and $D = 0$. The period of $y=\sin x$ is $T=\frac{2\pi}{B}$, so the period of $y=\frac{3}{2}\sin x$ is $T = 2\pi$.

Step2: Find x - intercepts

Set $y = 0$. Then $\frac{3}{2}\sin x=0$, which implies $\sin x = 0$. In the interval $[0,2\pi]$, $x = 0,\pi,2\pi$. The points are $(0,0),(\pi,0),(2\pi,0)$.

Step3: Find maxima

The maximum value of $\sin x$ is 1. When $\sin x = 1$, $x=\frac{\pi}{2}$, and $y=\frac{3}{2}\times1=\frac{3}{2}$. The point is $(\frac{\pi}{2},\frac{3}{2})$.

Step4: Find minima

The minimum value of $\sin x$ is - 1. When $\sin x=-1$, $x = \frac{3\pi}{2}$, and $y=\frac{3}{2}\times(-1)=-\frac{3}{2}$. The point is $(\frac{3\pi}{2},-\frac{3}{2})$.

Answer:

Plot the points $(0,0),(\frac{\pi}{2},\frac{3}{2}),(\pi,0),(\frac{3\pi}{2},-\frac{3}{2}),(2\pi,0)$ on the graph.