Question

consider the function $y = 2sin(x)$ for $0^{circ}leq xleq360^{circ}$. graph the function: plot the function $y = 2sin(x)$ on a coordinate plane. label the x - axis as \angle (degrees)\ and the y - axis as \y - value\. mark the coordinates of all key points where the graph intersects the x - axis, reaches a maximum, or minimum within the given interval.

Explanation:

Step1: Recall properties of sine - function

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

Step3: Find maximum and minimum points

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

Answer:

The x - intercepts are $(0,0)$, $(180,0)$, $(360,0)$. The maximum point is $(90,2)$ and the minimum point is $(270,-2)$. To graph, plot these key - points and draw a smooth sine - wave connecting them with the x - axis labeled "Angle (degrees)" and the y - axis labeled "y - value".