(4-1)  ①   (b): the only difference here is that the amplitude is increased. The amplitude – and therefore every value of y - is 8x bigger than in the parent graph (a). Therefore the equation is:

           y  =  8sin x

(c):  the additional change this time is that the x-axis has been scaled, so that the wavelength, λ = 5, rather than 2π. Therefore any value of x in this curve has been scaled by a factor of 5/2π, relative to curve (b). This means that if we multiply any x value by 2π/5, we will get the correct number whose sine (multiplied by 8) is the y-value. So the equation is:

           y = 8sin(2πx/5)

(d):  now the whole curve has been shifted to the right by a quarter of a wavelength. Compared to curve (c), each peak is shifted by 1.25 units. Therefore if we subtract 1.25 from an x value, we will get the correct number to compute y by the same function as in (c).  This gives us the equation:

           y = 8sin(2π(x-1.25)/5)

If you’re not convinced by any of these transformations, try it with some numbers and see how it works out.

②  If we define the displacement of the curve to the right as D,  the general equation is:

                          y = Asin(2π(x-D)/λ)