Transformation of the function y = sin(x -3pi/4)
Phase shift to the right of 3(pi)/4, so y=0 at this value of x instead of at x=0.
Phase shift to the right of (pi)/6, so y=1 (max) at x=(pi)/6 instead of at zero.
Vertical displacement of whole graph downwards by 2 units so that at x=0, y=-2 instead of zero, and at x=(pi)/4 y=-1 instead of 1.
Phase shift to the right of 3(pi)/4 and a vertical displacement upwards by 2 units. The curve is also compressed by a factor of 2 (1/2) (amplitude is halved, although csc function extends to infinity).
Vertical displacement upwards so that the sine wave sits on the x axis.
Vertical displacement upwards of 2+(pi) (5.14 units approx), but no phase shift. The amplitude is 4 times that of the cosine curve so the maximum and minimum value is 4 and -4 added to the vertical displacement: 6+(pi) (9.14) and (pi)-2 (1.14). The curve sits over but not on the x axis.
The sine curve is inverted and displaced vertically upwards by 1 unit. When x=0, y=1; x=(pi)/2, y=0; x=-(pi)/2, y=2, so the sine wave undulates between y=0 and 2 sitting on the x axis as in (5) but starting below rather than above the line y=1, resembling a phase shift of (pi)/2 or cos(x).
[cos(x) and sin(x) are phase-shifted versions of one another because sin(x+(pi)/2)=cos(x).] Read More: ...