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 standard sine function

The standard sine function is \( y = \sin(x) \), with amplitude \( 1 \), period \( 360^\circ \), x - intercepts at \( 0^\circ, 180^\circ, 360^\circ \), maximum at \( 90^\circ \) with \( y = 1 \), and minimum at \( 270^\circ \) with \( y=- 1 \). For the function \( y = 2\sin(x) \), the amplitude is \( 2 \) (since the amplitude of \( A\sin(x) \) is \( |A| \)), and the period remains \( 360^\circ \) (the period of \( \sin(Bx) \) is \( \frac{360^\circ}{|B|} \), here \( B = 1 \)).

Step2: 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 when \( x = 90^\circ \). The coordinate is \( (90^\circ,2) \).
• Minimum point: The minimum value of \( \sin(x) \) is \( - 1 \), so \( y=2\times(- 1)=-2 \). This occurs when \( x = 270^\circ \). The coordinate is \( (270^\circ,- 2) \).

Step3: Plot the graph

• Draw the coordinate plane. Label the x - axis as "Angle (degrees)" and the y - axis as "y - value".
• Mark 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, starting from \( (0^\circ,0) \), rising to \( (90^\circ,2) \), falling to \( (180^\circ,0) \), then falling to \( (270^\circ,-2) \), and rising back to \( (360^\circ,0) \).

(Note: Since this is a graphing problem, the final answer is the description of how to plot the graph with the key points as above. If we were to represent the key points in a table for clarity:

x (degrees)	y - value
\( 90^\circ \)	\( 2 \)
\( 180^\circ \)	\( 0 \)
\( 270^\circ \)	\( - 2 \)
\( 360^\circ \)	\( 0 \)

And the graph is a sine - wave with amplitude 2, period \( 360^\circ \), passing through these points.)

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. Mark Key Points:

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

1. Draw the Graph: Connect the points with a smooth, wave - like curve (sine curve) passing through these points, with amplitude 2 and period \( 360^{\circ} \).