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: Find the x - axis intersection points

Set \(y = 0\), then \(2\sin(x)=0\), \(\sin(x)=0\). For \(0^{\circ}\leq x\leq360^{\circ}\), \(x = 0^{\circ},180^{\circ},360^{\circ}\). When \(x = 0^{\circ}\), \(y=2\sin(0^{\circ}) = 0\); when \(x = 180^{\circ}\), \(y=2\sin(180^{\circ})=0\); when \(x = 360^{\circ}\), \(y=2\sin(360^{\circ}) = 0\).

Step2: Find the maximum and minimum points

The general form of the sine function is \(y = A\sin(x)+k\), here \(A = 2,k = 0\). The maximum value of \(\sin(x)\) is \(1\) (when \(x = 90^{\circ}\)), so \(y=2\times1=2\). The minimum value of \(\sin(x)\) is \(-1\) (when \(x = 270^{\circ}\)), so \(y=2\times(-1)=-2\).

Answer:

The key points are \((0^{\circ},0)\), \((90^{\circ},2)\), \((180^{\circ},0)\), \((270^{\circ}, - 2)\), \((360^{\circ},0)\).