graph each function using radians. 11) $y = 4\\sin\\theta$

Explanation:

Step1: Identify amplitude

The general form of a sine - function is $y = A\sin(B\theta - C)+D$. For the function $y = 4\sin\theta$, $A = 4$, $B = 1$, $C = 0$, $D = 0$. The amplitude is $|A|$, so the amplitude is $|4|=4$.

Step2: Find key - points

The period of the sine function $y=\sin\theta$ is $T = 2\pi$. For $y = 4\sin\theta$, the period is also $T = 2\pi$ since $B = 1$. We find the values of $y$ for some key - values of $\theta$: When $\theta=0$, $y = 4\sin(0)=0$. When $\theta=\frac{\pi}{2}$, $y = 4\sin(\frac{\pi}{2})=4$. When $\theta=\pi$, $y = 4\sin(\pi)=0$. When $\theta=\frac{3\pi}{2}$, $y = 4\sin(\frac{3\pi}{2})=-4$. When $\theta = 2\pi$, $y = 4\sin(2\pi)=0$.

Step3: Plot the points

Plot the points $(0,0)$, $(\frac{\pi}{2},4)$, $(\pi,0)$, $(\frac{3\pi}{2}, - 4)$, $(2\pi,0)$ on the given coordinate grid and connect them with a smooth curve. The curve oscillates between $y=-4$ and $y = 4$ with a period of $2\pi$.

Answer:

Plot the points $(0,0)$, $(\frac{\pi}{2},4)$, $(\pi,0)$, $(\frac{3\pi}{2}, - 4)$, $(2\pi,0)$ and draw a smooth sine - curve.