consider the function $y = 2sin(x)$ for $0^{circ}leq xleq360^{circ}$. graph the function: plot the function…

Explanation:

Step1: Recall sine - function properties

The general form of a sine function is $y = A\sin(Bx - C)+D$. For $y = 2\sin(x)$, $A = 2$, $B = 1$, $C = 0$, $D = 0$. The amplitude is $|A|=2$, the period is $T=\frac{2\pi}{B}=360^{\circ}$ (since $B = 1$ and we are working in degrees).

Step2: Find x - intercepts

Set $y = 0$. Then $2\sin(x)=0$. So $\sin(x)=0$. In the interval $0^{\circ}\leq x\leq360^{\circ}$, $x = 0^{\circ},180^{\circ},360^{\circ}$. The coordinates of the x - intercepts are $(0^{\circ},0),(180^{\circ},0),(360^{\circ},0)$.

Step3: Find maximum and minimum points

The amplitude is 2. The maximum value of $y = 2\sin(x)$ occurs when $\sin(x)=1$. So $y = 2$ when $x = 90^{\circ}$ (coordinate: $(90^{\circ},2)$). The minimum value of $y = 2\sin(x)$ occurs when $\sin(x)= - 1$. So $y=-2$ when $x = 270^{\circ}$ (coordinate: $(270^{\circ},-2)$).

Step4: Plot the points and draw the graph

Plot the points $(0^{\circ},0),(90^{\circ},2),(180^{\circ},0),(270^{\circ},-2),(360^{\circ},0)$ on the coordinate plane with the x - axis labeled "Angle (degrees)" and the y - axis labeled "y - value" and connect them with a smooth curve to get the graph of $y = 2\sin(x)$ for $0^{\circ}\leq x\leq360^{\circ}$.

Answer:

The key - points are $(0^{\circ},0),(90^{\circ},2),(180^{\circ},0),(270^{\circ},-2),(360^{\circ},0)$. Plot these points and draw a smooth sine - curve within the interval $0^{\circ}\leq x\leq360^{\circ}$ with the x - axis labeled "Angle (degrees)" and the y - axis labeled "y - value".