consider the function $y = 2\\sin(x)$ for $0^\\circ \\leq x \\leq 360^\\circ$.\n1. graph the function:\n…

consider the function $y = 2\\sin(x)$ for $0^\\circ \\leq x \\leq 360^\\circ$.\n1. graph the function:\n plot the function $y = 2\\sin(x)$ 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.

consider the function $y = 2\\sin(x)$ for $0^\\circ \\leq x \\leq 360^\\circ$.\n1. graph the function:\n plot the function $y = 2\\sin(x)$ 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.

Answer

Explanation:

Step1: Recall the parent sine function

The parent function is ( y = \sin(x) ), with amplitude 1, period ( 360^\circ ), x - intercepts at ( 0^\circ, 180^\circ, 360^\circ ), maximum at ( 90^\circ ) (value 1) and minimum at ( 270^\circ ) (value - 1).

Step2: Analyze the transformed function ( y = 2\sin(x) )

The function ( y = 2\sin(x) ) is a vertical stretch of the parent function ( y=\sin(x) ) by a factor of 2. The period remains ( 360^\circ ) (since there is no horizontal scaling), the amplitude is 2.

Key points calculation:

  • x - intercepts: Set ( y = 0 ), then ( 2\sin(x)=0\Rightarrow\sin(x) = 0 ). For ( 0^\circ\leq x\leq360^\circ ), ( x = 0^\circ, 180^\circ, 360^\circ ). The coordinates are ( (0^\circ, 0) ), ( (180^\circ, 0) ), ( (360^\circ, 0) ).
  • Maximum point: The maximum value of ( \sin(x) ) is 1, so the maximum value of ( y = 2\sin(x) ) is ( 2\times1 = 2 ). This occurs when ( \sin(x)=1\Rightarrow x = 90^\circ ). The coordinate is ( (90^\circ, 2) ).
  • Minimum point: The minimum value of ( \sin(x) ) is - 1, so the minimum value of ( y = 2\sin(x) ) is ( 2\times(- 1)=-2 ). This occurs when ( \sin(x)=-1\Rightarrow x = 270^\circ ). The coordinate is ( (270^\circ, - 2) ).

Step3: Plot the graph

  • Draw the x - axis (labeled "Angle (degrees)") with values from ( 0^\circ ) to ( 360^\circ ) and the y - axis (labeled "y - value") with values from - 2 to 2.
  • Plot the points ( (0^\circ, 0) ), ( (90^\circ, 2) ), ( (180^\circ, 0) ), ( (270^\circ, - 2) ), ( (360^\circ, 0) ).
  • Connect the points with a smooth curve, starting at ( (0^\circ, 0) ), rising to ( (90^\circ, 2) ), falling to ( (180^\circ, 0) ), falling further to ( (270^\circ, - 2) ), and then rising back to ( (360^\circ, 0) ).

Answer:

The graph of ( y = 2\sin(x) ) for ( 0^\circ\leq x\leq360^\circ ) has x - intercepts at ( (0^\circ, 0) ), ( (180^\circ, 0) ), ( (360^\circ, 0) ), a maximum at ( (90^\circ, 2) ) and a minimum at ( (270^\circ, - 2) ). The graph is a sine - wave - shaped curve passing through these points with amplitude 2 and period ( 360^\circ ).