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):

$\displaystyle y = 5\sin \frac{2\pi}{3}(x - 2)$                     $\displaystyle y = -5\sin \frac{2\pi}{3}(x - 3.5)$                     $\displaystyle y = 5\sin \frac{2\pi}{3}(x + 1)$

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:

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:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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