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 \( 90^\circ \) (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 \( 360^\circ \).