graph the function f(x) = 4 sin (x).

Explanation:

Step1: Recall properties of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $f(x)=4\sin(x)$, we have $A = 4$, $B = 1$, $C = 0$, and $D = 0$. The amplitude is given by $|A|$. Here, the amplitude $|A|=4$, which means the graph of $y = 4\sin(x)$ oscillates between $y=-4$ and $y = 4$.

Step2: Find key points

The period of the function $y=\sin(x)$ is $T = 2\pi$. For $y = 4\sin(x)$, the period remains $T = 2\pi$ since $B = 1$. We find key - points in one period $[0,2\pi]$. When $x = 0$, $y=4\sin(0)=0$; when $x=\frac{\pi}{2}$, $y = 4\sin(\frac{\pi}{2})=4$; when $x=\pi$, $y=4\sin(\pi)=0$; when $x=\frac{3\pi}{2}$, $y = 4\sin(\frac{3\pi}{2})=-4$; when $x = 2\pi$, $y=4\sin(2\pi)=0$.

Step3: Plot the points and draw the graph

Plot the points $(0,0),(\frac{\pi}{2},4),(\pi,0),(\frac{3\pi}{2},-4),(2\pi,0)$ on the given coordinate grid. Then, connect these points with a smooth curve. Since the sine - function is periodic with period $2\pi$, we can repeat the pattern to the left and right of the interval $[0,2\pi]$.

Answer:

Plot the points $(0,0),(\frac{\pi}{2},4),(\pi,0),(\frac{3\pi}{2},-4),(2\pi,0)$ and connect them with a smooth curve, repeating the pattern every $2\pi$ units along the $x$ - axis.