In the function #y = sin(b(x + c)) + d#, the two parameters that will influence the range are #a# and #d#, which are the amplitude and vertical transformation, respectively.
The function #y = sinx# has an amplitude of #1# and a vertical transformation of #0#, for example. Since the sine function rotates around the centre line which will be at #y = 0# (due to the vertical transformation), the range will be #-1 ≤ y≤ 1#.
With our function, the amplitude is #|-2| = 2# and the vertical transformation brings the centre line down to #y = -3#. Hence, the minimum values will be #y = -3 - 2 =- 5# and our maximum values will be #y = -3+ 2 = -1#.
The range is therefore #-5 ≤ y≤ -1#.
Hopefully this helps!
