y = a * sin(b * (x-c)) + d
a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift
phase shift is the same as horizontal shift.
you have:
amplitude = 5 = a
vertical shift = -4 = d
phase shift / horizontal shift = (3/4)*pi = c
period = 8*pi
frequency is equal to 2*pi / period
period = 8*pi.
frequency = 2*pi/(8*pi) = 4 = b
formula becomes:
y = 5 * sin(4*(x-(3/4)*pi)) - 4
the amplitude and the vertical shift and the frequency / period are fairly easy to show.
the phase shift / horizontal shift is a little harder.
to show the horizontal shift, i created 2 graphs.
the first graph is without the horizontal shift and the second graph is with the horizontal shift so you can the effects of the horizontal shift.
the vertical shift moves the center line of the graph the number of units indicated.
that's why the graphs shows the center line at y = -4 rather than at y = 0.
the amplitude shows the maximum / minimum height of the graph in relationship to the center line.
since the amplitude of the sine wave is normally 1, and the amplitude was shown as 5, then the amplitude of the sine wave is 5 * 1 = 5 instead of 1.
in the graphs, you will see that the maximum value of the sine wave is y = 1 and the minimum value of the sine wave is - 9.
y = 1 is 5 units above the center line at y = -4 and y = -9 is 5 units below the center line at y = -4.
the frequency is equal to 2 * pi divided by the period which becomes 2*pi / (8*pi) which then becomes 1/4.
the frequency is 1/4.
the period is equal to 2 * pi divided by the frequency which becomes 2 * pi / (1/4) which then becomes 8 * pi.
in the graph without the horizontal shift, you can see that one full cycle of the sine wave goes from 0 to 8 * pi.
8*pi - 0*pi results in a period of 8*pi.
in the graph with the horizontal shift, you can see that one full cycle of the sine wave goes from 3/4 * pi to 35/4 * pi.
35/4 * pi - 3/4 * pi is equal to 32/4 * pi which is equal to 8 * pi.
the period in both graphs is the same.
the starting point of the sine wave in the second graph has been shifted to the right by 3/4 * pi radians.
that difference is your phase shift.
you can see that the second full cycle of the sine wave has been shifted to the right by 3/4 * pi radians.
here's the graphs.
