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 2π|b|. In this case, b = 1, so the period is 2π.
Phase shift: The phase shift is the horizontal displacement. It can be found by solving the equation x+c=0.
x−π2=0
x=π2
Since π2 is positive, the horizontal displacement is π2 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 π2 we start at (π2,1). 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: π24=π8. The point will be (5π8,−3), since the amplitude is 4 and the line of rotation at y = 1. The next point will be at (3π4,5). The final point in our cycle is (2π2,1), since we have finished our period.
Hopefully this helps!
