How do you graph #y =3sin2x+4#?

graph{3sin(x)+4 [-22.58, 23, -11.3, 11.54]}

To graph the function ( y = 3\sin(2x) + 4 ):

1. Identify the properties of the parent function ( y = \sin(x) ). The sine function has a period of ( 2\pi ), an amplitude of 1, and its range is between -1 and 1.
2. The coefficient before ( x ) inside the sine function, ( 2 ), indicates a horizontal compression/stretch. Since it's ( \sin(2x) ), the period is ( \frac{2\pi}{2} = \pi ), which means the graph completes one full cycle in the interval ( [0, \pi] ).
3. The coefficient ( 3 ) outside the sine function indicates a vertical stretch by a factor of 3, and the ( +4 ) shifts the graph vertically upwards by 4 units.
4. Plot key points for one period of the function. These include the maximum, minimum, and zero crossings. Since ( \sin(2x) ) completes one full cycle in ( [0, \pi] ), you can choose points within this interval.
5. Connect the points smoothly to graph the function.

Key points for one period (( [0, \pi] )) of ( y = 3\sin(2x) + 4 ):

• When ( x = 0 ), ( y = 3\sin(0) + 4 = 4 ) (maximum)
• When ( x = \frac{\pi}{4} ), ( y = 3\sin\left(2 \times \frac{\pi}{4}\right) + 4 = 3\sin\left(\frac{\pi}{2}\right) + 4 = 3 + 4 = 7 )
• When ( x = \frac{\pi}{2} ), ( y = 3\sin\left(2 \times \frac{\pi}{2}\right) + 4 = 3\sin(\pi) + 4 = 0 + 4 = 4 ) (minimum)
• When ( x = \pi ), ( y = 3\sin(2\pi) + 4 = 0 + 4 = 4 ) (maximum again)

Graph the points and connect them to form a smooth curve. The graph should oscillate between the maximum and minimum values within each period, and it's shifted up by 4 units.

Remember, this process gives you one period of the function. To graph the entire function, repeat the pattern to the left and right of the graphed period.