Question

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:

The function \( y = 2\sin(x) \) is a vertical stretch of the basic sine function \( y = \sin(x) \) by a factor of 2. Key points in \( 0^\circ \leq x \leq 360^\circ \):

• X-intercepts: Where \( \sin(x) = 0 \), so \( x = 0^\circ, 180^\circ, 360^\circ \) (coordinates: \( (0^\circ, 0) \), \( (180^\circ, 0) \), \( (360^\circ, 0) \)).
• Maximum: \( \sin(x) = 1 \) at \( x = 90^\circ \), so \( y = 2 \times 1 = 2 \) (coordinate: \( (90^\circ, 2) \)).
• Minimum: \( \sin(x) = -1 \) at \( x = 270^\circ \), so \( y = 2 \times (-1) = -2 \) (coordinate: \( (270^\circ, -2) \)).

To graph: Plot these key points, connect them with a smooth sinusoidal curve, label axes as "Angle (degrees)" (x-axis) and "y-value" (y-axis).

Answer:

Key points: \( (0^\circ, 0) \), \( (90^\circ, 2) \), \( (180^\circ, 0) \), \( (270^\circ, -2) \), \( (360^\circ, 0) \). The graph is a sine wave with amplitude 2, oscillating between -2 and 2 over \( 0^\circ \) to \( 360^\circ \). (Graph plotting requires a coordinate plane with labeled axes and connected key points.)