The commonly form of sine function used is the following:
[tex]y=A\sin (\frac{2\pi}{B}(x+C))+D[/tex]
Where:
- A is the amplitute
- B is the period
- C is the phase shift
- D is the vertical shift, or vertical translation.
For the cosine, we only change the sin to cos.
a) We have a sine function with amplitute 4 and period 3π. It is implied that there is no vertical trnslation and no phase shift, so they are 0.
Thus:
A = 4
B = 3π
C = 0
D = 0
[tex]\begin{gathered} y=4\sin (\frac{2\pi}{3\pi}(x+0))+0 \\ y=4\sin (\frac{2}{3}x) \end{gathered}[/tex]
b) We have a sine function, amplitude 3, period 2π, vertical translation -4 and phase shift π.
Thus:
A = 3
B = 2π
C = -4
D = π
[tex]\begin{gathered} y=3\sin (\frac{2\pi}{2\pi}(x+\pi))-4 \\ y=3\sin (x+\pi)-4 \end{gathered}[/tex]
c) We have a cosine function, amplitude 5, period π, vertical translation 1 and phase shift -π/2.
Thus:
A = 5
B = π
C = 1
D = -π/2
[tex]\begin{gathered} y=5\cos (\frac{2\pi}{\pi}(x-\frac{\pi}{2}))+1 \\ y=5\cos (2x-\pi)+1 \end{gathered}[/tex]
