Given Amplitude, Period, and Phase Shift, Write an Equation

In the prior section, you learned how to find the amplitude, period, and phase shift of a given (generalized) sine or cosine curve.

In this section, you will write an equation of a curve with a specified amplitude, period, and phase shift.

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

Specifying a sine (or cosine) curve with a given amplitude, period, and phase shift defines a unique set of points in the plane. However, there are infinitely many equations that can describe that set of points!

For example, the set of points given by the sine curve with amplitude $\,5\,,$ period $\,3\,,$ and phase shift $\,2\,$ can be described by any of these equations (and many more):

In this section, we find a simple (natural) equation that works, by using graphical transformations to change a basic model into a curve with the desired attributes.

The process is illustrated with an example:

Example

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

Solution

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 $\,k \gt 0\,,$ the curve $\,y = \sin kx\,$ has period $\displaystyle\,\frac{2\pi}{k}\,.$

We want the period to be $\,3\,$:

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 $\,y = 5\sin \frac{2\pi}{3}(x - 2)\,$ 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:

Example

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

Solution

• Basic model:  $y = \cos x$

• Period:  $\displaystyle\frac{2\pi}{k} = \frac{\pi}{3}\ \implies\ k = 6$

  New equation:    $y = \cos 6x$

• 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 $\displaystyle\,-\frac 12\,.$