# How do you write the equation of a cos function: amplitude=2/3 period=pi/3 phase shift= -pi/3 vert. shift= 5?

Jan 4, 2017

$y = \left(A\right) \cos \left(B x + C\right) + D$
Please see the explanation for how you change it.

#### Explanation:

A is the amplitude:

$y = \frac{2}{3} \cos \left(B x + C\right) + D$

The period is $T = \frac{2 \pi}{B}$

Substitute $\frac{\pi}{3}$ for T and then solve for B:

$\frac{\pi}{3} = 2 \frac{\pi}{B}$

$B = 6$

$y = \frac{2}{3} \cos \left(6 x + C\right) + D$

The phase shift is $\phi = - \frac{C}{B}$

Substitute, $- \frac{\pi}{3} \text{ for } \phi$, 6 for B, and then solve for C:

$- \frac{\pi}{3} = - \frac{C}{6}$

$C = 2 \pi$

$y = \frac{2}{3} \cos \left(6 x + 2 \pi\right) + D$

The vertical shift is $D = 5$:

$y = \frac{2}{3} \cos \left(6 x + 2 \pi\right) + 5$ Answer

Here is a graph of the above equation:

graph{y = 2/3cos(6x + 2pi) + 5 [-11.25, 11.25, -5.625, 5.625]}

Jan 4, 2017

$y = 5 + \frac{2}{3} \sin \left(6 t + 2 \pi\right)$

#### Explanation:

let's write out something really general first

$y - {y}_{o} = A \sin \left(\omega t + \psi\right)$

Easy ones first:

Amplitude: $A = \frac{2}{3}$

Vertical Shift upward: ${y}_{o} = 5$

Ergo: $y - 5 = \frac{2}{3} \sin \left(\omega t + \psi\right)$ !!

For the period (am assuming you mean temporal, not spatial, period, though the approach is the same), note that $\omega = \frac{2 \pi}{T} = \frac{2 \pi}{\frac{\pi}{3}} = 6$

$\implies y - 5 = \frac{2}{3} \sin \left(6 t + \psi\right)$

For phase shift $- \frac{\pi}{3}$, you need to be careful. If we re-write this as $y - 5 = \frac{2}{3} \sin \left(6 \left(t + \overline{\psi}\right)\right)$, hopefully you will see that $\overline{\psi}$ represents the actual shift along the t axis.

We want that shift to be $\frac{\pi}{3}$ to the left so we can say that

$y - 5 = \frac{2}{3} \sin \left(6 \left(t + \frac{\pi}{3}\right)\right)$

$= \frac{2}{3} \sin \left(6 t + 2 \pi\right)$

ie we can say that $\psi = 2 \pi$

That may seem odd but the period of the function is $T = \frac{\pi}{3}$ so it is a full phase shift