Question

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

The standard sine function \( y=\sin(x) \) has a period of \( 360^\circ \), amplitude \( 1 \), x - intercepts at \( x = 0^\circ,180^\circ,360^\circ \), maximum value \( 1 \) at \( x = 90^\circ \), and minimum value \( - 1 \) at \( x=270^\circ \) in the interval \( 0^\circ\leq x\leq360^\circ \).

Step2: Analyze the transformation for \( y = 2\sin(x) \)

For the function \( y = A\sin(x) \), the amplitude is \( |A| \). Here \( A = 2 \), so the amplitude of \( y = 2\sin(x) \) is \( 2 \). The period remains the same as the standard sine function, which is \( 360^\circ \) since there is no horizontal scaling (the coefficient of \( x \) is \( 1 \)).

Step3: Find key points

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

Step4: Sketch the graph

• Draw the coordinate plane with the x - axis labeled "Angle (degrees)" and the y - axis labeled "y - value".
• 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 the amplitude of \( 2 \) (the distance from the mid - line (x - axis) to the peak or trough is \( 2 \)) and period \( 360^\circ \).

Answer:

To graph \( y = 2\sin(x) \) for \( 0^\circ\leq x\leq360^\circ \):

1. Label Axes: Label the x - axis as "Angle (degrees)" and the y - axis as "y - value".
2. Plot Key Points:

• X - intercepts: \( (0^\circ,0) \), \( (180^\circ,0) \), \( (360^\circ,0) \)
• Maximum: \( (90^\circ,2) \)
• Minimum: \( (270^\circ,-2) \)

1. Draw the Curve: Connect the plotted points with a smooth, periodic (sine - shaped) curve that has an amplitude of \( 2 \) and a period of \( 360^\circ \).