Question

question: consider the function ( y = 2sin(x) ) for ( 0^{circ}leq xleq360^{circ} ). 1. 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 its maximum, or minimum within the given interval.

Explanation:

Step1: Determine the general form of the sine function

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

Step2: Find the x - intercepts

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

Step3: Find the maximum and minimum points

The amplitude is \(A = 2\). The maximum value of \(y = 2\sin(x)\) occurs when \(\sin(x)=1\). Solving \(\sin(x)=1\) for \(0^{\circ}\leq x\leq360^{\circ}\), we get \(x = 90^{\circ}\), and \(y=2\). The coordinate of the maximum point is \((90^{\circ},2)\).
The minimum value of \(y = 2\sin(x)\) occurs when \(\sin(x)= - 1\). Solving \(\sin(x)=-1\) for \(0^{\circ}\leq x\leq360^{\circ}\), we get \(x = 270^{\circ}\), and \(y=-2\). The coordinate of the minimum point is \((270^{\circ},-2)\).

Step4: Plot the key 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". Then, connect these points with a smooth curve that has a period of \(360^{\circ}\) (since it is a sine function with \(B = 1\)).

Answer:

The x - intercepts are \((0^{\circ},0)\), \((180^{\circ},0)\), \((360^{\circ},0)\); the maximum point is \((90^{\circ},2)\); the minimum point is \((270^{\circ},-2)\). Plot these points and connect them with a smooth sine - wave curve over the interval \(0^{\circ}\leq x\leq360^{\circ}\).