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 amplitude \( 1 \), period \( 360^\circ \), x - intercepts at \( 0^\circ, 180^\circ, 360^\circ \), maximum at \( (90^\circ, 1) \), minimum at \( (270^\circ, - 1) \).

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

The function \( y = 2\sin(x) \) is a vertical stretch of the parent sine function by a factor of \( 2 \). The amplitude becomes \( 2 \), the period remains \( 360^\circ \) (since there is no horizontal scaling), the x - intercepts remain at \( x = 0^\circ, 180^\circ, 360^\circ \) (because multiplying the function by \( 2 \) does not change where \( y = 0 \)), the maximum value of \( y=\sin(x) \) is \( 1 \), so for \( y = 2\sin(x) \), the maximum value is \( 2\times1 = 2 \) and it occurs at \( x = 90^\circ \), so the point is \( (90^\circ, 2) \). The minimum value of \( y=\sin(x) \) is \( - 1 \), so for \( y = 2\sin(x) \), the minimum value is \( 2\times(-1)=-2 \) and it occurs at \( x = 270^\circ \), so the point is \( (270^\circ, - 2) \).

Step3: Plot the key points

• X - intercepts: \( (0^\circ, 0) \), \( (180^\circ, 0) \), \( (360^\circ, 0) \)
• Maximum point: \( (90^\circ, 2) \)
• Minimum point: \( (270^\circ, - 2) \)
• Then, we can sketch the curve of \( y = 2\sin(x) \) by connecting these key points smoothly, following the shape of the sine wave, starting from \( (0^\circ, 0) \), rising to \( (90^\circ, 2) \), falling to \( (180^\circ, 0) \), falling further to \( (270^\circ, - 2) \), and then rising back to \( (360^\circ, 0) \) over the interval \( 0^\circ\leq x\leq360^\circ \). Label the x - axis as "Angle (degrees)" and the y - axis as "y - value".

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 curve is a sine - wave - shaped curve passing through these points with amplitude \( 2 \) and period \( 360^\circ \), with the x - axis labeled "Angle (degrees)" and the y - axis labeled "y - value".