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