Amplitude: in a function of the for #y = asinb(x + c) + d#, the amplitude is at #|a|#. So, the amplitude is at 4. The amplitude is the distance between the maximum and minimum points and the horizontal line of rotation. This line of rotation would be 0 if d = 0, but since there is a vertical displacement of +1, the line is y = 1.
Period: the period is the distance before the function's movement repeats itself. It can be found by #(2pi)/|b|#. In this case, b = 1, so the period is #2pi#.
Phase shift: The phase shift is the horizontal displacement. It can be found by solving the equation #x + c = 0#.
#x - pi/2 = 0#
#x = pi/2#
Since #pi/2# is positive, the horizontal displacement is #pi/2# units right.
Now, we have enough information to graph. If there was no phase shift, you would start on the line of rotation at (0, 1). However, since there is a phase shift of #pi/2# we start at #(pi/2, 1)#. Usually, you will only be asked to do one complete cycle, so that's what I'll do. In a Sine function, there are 4 parts on the x value in one cycle, so the first point can be found by dividing the period by 4: #(pi/2)/4 = pi/8#. The point will be #(5pi/8, -3)#, since the amplitude is 4 and the line of rotation at y = 1. The next point will be at #(3pi/4, 5)#. The final point in our cycle is #(2pi/2, 1)#, since we have finished our period.
Hopefully this helps!
