How do you graph #y=4sin(3x+pi/4)-1#?

graph{4 sin(3x + (pi/4)) - 1 [-10, 10, -5, 5]}

To graph the function ( y = 4\sin(3x + \frac{\pi}{4}) - 1 ):

1. Start by identifying the key components:

   • The amplitude is 4.
   • The period of the function is ( \frac{2\pi}{3} ) (since ( \omega = 3 ) in ( y = \sin(\omega x) )).
   • The phase shift is ( -\frac{\pi}{12} ) to the left (since ( \frac{\pi}{4} ) corresponds to ( \frac{1}{12} ) of the period).
   • The vertical shift is down 1 unit.

2. Plot the initial point:

   • The initial point for sine function is at the origin (0,0) since there's no phase shift in the amplitude.

3. Plot additional points using the period, amplitude, and phase shift:

   • Since the period is ( \frac{2\pi}{3} ), plot points every ( \frac{\pi}{3} ) units.
   • Apply the amplitude to determine the maximum and minimum points.
   • Apply the phase shift to shift the graph to the left.

4. Connect the points smoothly to form the graph of the function.

5. Label the axes and any significant points if necessary.

6. Optionally, you can extend the graph beyond one period if you need to illustrate more cycles.

Following these steps will allow you to accurately graph the function ( y = 4\sin(3x + \frac{\pi}{4}) - 1 ).