sketch a graph of the function $f(x)=-2sin(x)$. -π/4 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π -1 -2 -3 clear all…

Explanation:

Step1: Recall properties of sine function

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

Step2: Find key points

For the sine function $y = \sin(x)$, key points in one period $[0,2\pi]$ are $(0,0)$, $(\frac{\pi}{2},1)$, $(\pi,0)$, $(\frac{3\pi}{2}, - 1)$, $(2\pi,0)$. For the function $y=-2\sin(x)$, we multiply the $y$ - values of the key - points of $y = \sin(x)$ by $-2$. So the key points are:

• When $x = 0$, $y=-2\sin(0)=0$.
• When $x=\frac{\pi}{2}$, $y=-2\sin(\frac{\pi}{2})=-2$.
• When $x=\pi$, $y=-2\sin(\pi)=0$.
• When $x = \frac{3\pi}{2}$, $y=-2\sin(\frac{3\pi}{2})=2$.
• When $x = 2\pi$, $y=-2\sin(2\pi)=0$.

Step3: Sketch the graph

Plot the key points $(0,0)$, $(\frac{\pi}{2},-2)$, $(\pi,0)$, $(\frac{3\pi}{2},2)$, $(2\pi,0)$ on the given coordinate grid. Then, connect these points with a smooth curve. The graph of $y = - 2\sin(x)$ is a sine - wave with an amplitude of 2, reflected about the $x$ - axis compared to the graph of $y=\sin(x)$.

Answer:

Sketch the points $(0,0)$, $(\frac{\pi}{2},-2)$, $(\pi,0)$, $(\frac{3\pi}{2},2)$, $(2\pi,0)$ on the provided grid and connect them with a smooth curve to form a sine - wave with amplitude 2 and reflected about the $x$ - axis.