Question

consider the function ( y = 2 sin(x) ) for ( 0^circ leq x leq 360^circ ).

1. graph the function:

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.

Explanation:

Step1: Recall the parent sine function

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

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

For the function \( y = A\sin(x) \), the amplitude is \( |A| \). Here \( A = 2 \), so the amplitude is \( 2 \). The period remains \( 360^\circ \) (since there is no horizontal stretch/compression, \( B = 1 \) in \( y = A\sin(Bx) \)).

Step3: Find key points

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

Step4: 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 key points: \( (0^\circ, 0) \), \( (90^\circ, 2) \), \( (180^\circ, 0) \), \( (270^\circ, - 2) \), \( (360^\circ, 0) \).
• Connect the points with a smooth curve, following the shape of the sine wave, with amplitude \( 2 \) and period \( 360^\circ \).

Answer:

The key points for the function \( y = 2\sin(x) \) in the interval \( 0^\circ\leq x\leq360^\circ \) are \( (0^\circ, 0) \), \( (90^\circ, 2) \), \( (180^\circ, 0) \), \( (270^\circ, - 2) \), \( (360^\circ, 0) \). The graph is a sine - wave with amplitude \( 2 \), period \( 360^\circ \), passing through these key points, with the x - axis labeled "Angle (degrees)" and the y - axis labeled "y - value".