In order to sketch the graph of the function [tex]\( y = 3\sin(2x) \)[/tex], we need to determine its amplitude and period, and then plot a full cycle.
Step 1: Determine Amplitude
The amplitude of a sine function in the form [tex]\( y = A\sin(Bx) \)[/tex] is the coefficient [tex]\( |A| \)[/tex]. For [tex]\( y = 3\sin(2x) \)[/tex], [tex]\( A = 3 \)[/tex], so the amplitude is [tex]\( |3| \)[/tex], which is simply [tex]\( 3 \)[/tex]. This means the graph will have peaks at [tex]\( y = 3 \)[/tex] and troughs at [tex]\( y = -3 \)[/tex].
Step 2: Determine Period
The period of a sine function in the form [tex]\( y = A\sin(Bx) \)[/tex] is [tex]\( \frac{2\pi}{|B|} \)[/tex]. For [tex]\( y = 3\sin(2x) \)[/tex], [tex]\( B = 2 \)[/tex], so the period is [tex]\( \frac{2\pi}{2} \)[/tex], which simplifies to [tex]\( \pi \)[/tex]. This means that the function will repeat every [tex]\( \pi \)[/tex] units along the x-axis.
Step 3: Plot a Full Cycle
Starting from [tex]\( x = 0 \)[/tex] and going to [tex]\( x = \pi \)[/tex], we will see one complete cycle of the sine function.
1. When [tex]\( x = 0 \)[/tex], [tex]\( y = 3\sin(0) = 0 \)[/tex].
2. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] rises to a maximum at [tex]\( y = 3 \)[/tex] when [tex]\( 2x = \frac{\pi}{2} \)[/tex] (or [tex]\( x = \frac{\pi}{4} \)[/tex]).
3. Then [tex]\( y \)[/tex] decreases back to [tex]\( 0 \)[/tex] when [tex]\( 2x = \pi \)[/tex] (or [tex]\( x = \frac{\pi}{2} \)[/tex]).
4. As [tex]\( x \)[/tex] continues to increase, [tex]\( y \)[/tex] goes to a minimum at [tex]\( y = -3 \)[/tex] when [tex]\( 2x = \frac{3\pi}{2} \)[/tex] (or [tex]\( x = \frac{3\pi}{4} \)[/tex]).
5. Finally, [tex]\( y \)[/tex] returns to [tex]\( 0 \)[/tex] when [tex]\( 2x = 2\pi \)[/tex] (or [tex]\( x = \pi \)[/tex]).
The points mentioned above are the key points of the sine function within one cycle. Joining these points with a smooth, wave-like curve will give us the graph of [tex]\( y = 3\sin(2x) \)[/tex] over one period.
The graph begins at the origin [tex]\( (0,0) \)[/tex], rises to the maximum at [tex]\( (\frac{\pi}{4}, 3) \)[/tex], crosses through [tex]\( (\frac{\pi}{2}, 0) \)[/tex], goes to the minimum at [tex]\( (\frac{3\pi}{4}, -3) \)[/tex], and then finally returns to [tex]\( (\pi, 0) \)[/tex].
Amplitude: [tex]\( 3 \)[/tex]
Period: [tex]\( \pi \)[/tex]
These properties are the key features of the function [tex]\( y = 3\sin(2x) \)[/tex] and should be clearly marked on the graph for full understanding.
