GIVEN AMPLITUDE, PERIOD, and PHASE SHIFT, WRITE AN EQUATION

The process is illustrated with an example:

Write an equation of a sine curve with amplitude $\,5\,,$ period $\,3\,,$ and phase shift $\,2\,.$

• Start with the basic model (sine or cosine):
  We want a sine curve, so the ‘basic model’ is:   $y = \sin x$
• Apply a vertical stretch/shrink to get the desired amplitude:
  new equation:   $y = 5\sin x$
• Apply a horizontal stretch/shrink to get the desired period:
  • For $\,k > 0\,,$ the curve $\,y = \sin kx\,$ has period $\displaystyle\,\frac{2\pi}{k}\,.$
    
    The basic sine curve, $\,y = \sin t\,,$ goes through one complete cycle as its inputs ($\,t\,$) go from $\,\color{purple}{0}\,$ to $\,\color{green}{2\pi}\,.$
    So, how long does it take for $\,y = \sin kx\,$ to go through one complete cycle?
    
    The inputs for $\,y = \sin \color{blue}{kx}\,$ are $\,\color{blue}{kx}\,.$
    

    When are the inputs $\,\color{purple}{0}\,$?	$\color{blue}{kx} = \color{purple}{0}$	when	$\color{red}{x = 0}$
    When are the inputs $\,\color{green}{2\pi}\,$?	$\color{blue}{kx} = \color{green}{2\pi}$	when	$\displaystyle\color{red}{x = \frac{2\pi}k}$

    Thus, $\,y =\sin kx\,$ goes through one complete cycle as $\,x\,$ goes from $\,\color{red}{0}\,$ to $\displaystyle\,\color{red}{\frac{2\pi}k}\,.$
    Thus, the period of $\,y =\sin kx\,$ is: $\,\displaystyle\color{red}{\frac{2\pi}k} - \color{red}{0} = \frac{2\pi}k\,$
  • We want the period to be $\,3\,$:   $\displaystyle \frac{2\pi}{k} = 3\ \ \implies\ \ k = \frac{2\pi}{3}$
  • new equation: $\displaystyle\,y = 5\sin \frac{2\pi}3x\,$
• Apply the desired phase shift:
  To shift right $\,2\,,$ replace every $\,x\,$ by $\,x - 2\,.$
  Remember—transformations involving $\,x\,$ are counter-intuitive!
  new equation:   $\displaystyle y = 5\sin \frac{2\pi}{3}(x - 2)$
• Thus, the curve   $\displaystyle y = 5\sin \frac{2\pi}{3}(x - 2)$ has amplitude $\,5\,,$ period $\,3\,,$ and phase shift $\,2\,.$

Important: Change the period BEFORE the phase shift!

If you apply the phase shift first, then the subsequent horizontal stretch/shrink to adjust the period will mess up this phase shift.
So, be sure to adjust the period before applying the phase shift!
The amplitude adjustment (the vertical stretch/shrink transformation) can be applied at any time.

Here is a second example:

Write an equation of a cosine curve with amplitude $\,4\,,$ period $\displaystyle\,\frac{\pi}{3}\,,$ and phase shift $\displaystyle\,-\frac 12\,.$

• basic model:   $y = \cos x$
• period: $\displaystyle\frac{2\pi}{k} = \frac{\pi}{3}\ \ \implies\ \ k = 6$
  new equation:   $\displaystyle\,y = \cos 6x$
• phase shift:   $\displaystyle\,y = \cos 6(x+\frac 12)$
• amplitude:   $\displaystyle\,y = 4\cos 6(x+\frac 12)$
• Thus, the curve   $\displaystyle\,y = 4\cos 6(x+\frac 12)$ has amplitude $\,4\,,$ period $\displaystyle\,\frac{\pi}{3}\,,$ and phase shift $\displaystyle\,-\frac 12\,.$